![Page 1: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/1.jpg)
Large Deviations for Multi-valued Stochastic DifferentialEquations
Joint work with Siyan Xu, Xicheng Zhang
![Page 2: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/2.jpg)
Large Deviations for Multi-valued Stochastic DifferentialEquations
Joint work with Siyan Xu, Xicheng Zhang
![Page 3: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/3.jpg)
We shall talk about the Freidlin-Wentzell Large Deviation Principlefor the following multivalued stochastic differential equation
{dXε(t) ∈ b(Xε(t)) dt +
√εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
I.e., we seek a result of the following type:
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), where
F o = the interior of F
F = the closure of F
![Page 4: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/4.jpg)
We shall talk about the Freidlin-Wentzell Large Deviation Principlefor the following multivalued stochastic differential equation{
dXε(t) ∈ b(Xε(t)) dt +√
εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
I.e., we seek a result of the following type:
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), where
F o = the interior of F
F = the closure of F
![Page 5: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/5.jpg)
We shall talk about the Freidlin-Wentzell Large Deviation Principlefor the following multivalued stochastic differential equation{
dXε(t) ∈ b(Xε(t)) dt +√
εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
I.e., we seek a result of the following type:
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), where
F o = the interior of F
F = the closure of F
![Page 6: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/6.jpg)
We shall talk about the Freidlin-Wentzell Large Deviation Principlefor the following multivalued stochastic differential equation{
dXε(t) ∈ b(Xε(t)) dt +√
εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
I.e., we seek a result of the following type:
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), where
F o = the interior of F
F = the closure of F
![Page 7: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/7.jpg)
We shall talk about the Freidlin-Wentzell Large Deviation Principlefor the following multivalued stochastic differential equation{
dXε(t) ∈ b(Xε(t)) dt +√
εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
I.e., we seek a result of the following type:
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)),
where
F o = the interior of F
F = the closure of F
![Page 8: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/8.jpg)
We shall talk about the Freidlin-Wentzell Large Deviation Principlefor the following multivalued stochastic differential equation{
dXε(t) ∈ b(Xε(t)) dt +√
εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
I.e., we seek a result of the following type:
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), where
F o = the interior of F
F = the closure of F
![Page 9: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/9.jpg)
We shall talk about the Freidlin-Wentzell Large Deviation Principlefor the following multivalued stochastic differential equation{
dXε(t) ∈ b(Xε(t)) dt +√
εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
I.e., we seek a result of the following type:
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), where
F o = the interior of F
F = the closure of F
![Page 10: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/10.jpg)
We shall talk about the Freidlin-Wentzell Large Deviation Principlefor the following multivalued stochastic differential equation{
dXε(t) ∈ b(Xε(t)) dt +√
εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
I.e., we seek a result of the following type:
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), where
F o = the interior of F
F = the closure of F
![Page 11: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/11.jpg)
• I: a rate function on Cx([0, 1], D(A));
(I ∈ [0,∞], {I 6 c} is compact ∀c > 0).
and
I(A) := infx∈A
I(x)
Let us begin by explaining the notion of
![Page 12: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/12.jpg)
• I: a rate function on Cx([0, 1], D(A));
(I ∈ [0,∞], {I 6 c} is compact ∀c > 0).
and
I(A) := infx∈A
I(x)
Let us begin by explaining the notion of
![Page 13: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/13.jpg)
• I: a rate function on Cx([0, 1], D(A));
(I ∈ [0,∞], {I 6 c} is compact ∀c > 0).
and
I(A) := infx∈A
I(x)
Let us begin by explaining the notion of
![Page 14: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/14.jpg)
• I: a rate function on Cx([0, 1], D(A));
(I ∈ [0,∞], {I 6 c} is compact ∀c > 0).
and
I(A) := infx∈A
I(x)
Let us begin by explaining the notion of
![Page 15: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/15.jpg)
• I: a rate function on Cx([0, 1], D(A));
(I ∈ [0,∞], {I 6 c} is compact ∀c > 0).
and
I(A) := infx∈A
I(x)
Let us begin by explaining the notion of
![Page 16: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/16.jpg)
• I: a rate function on Cx([0, 1], D(A));
(I ∈ [0,∞], {I 6 c} is compact ∀c > 0).
and
I(A) := infx∈A
I(x)
Let us begin by explaining the notion of
![Page 17: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/17.jpg)
MSDE: MULTIVALUED STOCHASTIC DIFFERENTIALEQUATIONS
dX(t) ∈ b(X(t))dt + σ(X(t))dw(t)−A(X(t))dt
Difference: An additional AWhat is it?A: a maximal monotone operator.main features:• A is multivalued: ∀x, A(x) is a set, not necessarily a single point• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)
〈x− y, u− v〉 > 0
• A is maximal:
(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).
![Page 18: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/18.jpg)
MSDE: MULTIVALUED STOCHASTIC DIFFERENTIALEQUATIONS
dX(t) ∈ b(X(t))dt + σ(X(t))dw(t)−A(X(t))dt
Difference: An additional AWhat is it?A: a maximal monotone operator.main features:• A is multivalued: ∀x, A(x) is a set, not necessarily a single point• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)
〈x− y, u− v〉 > 0
• A is maximal:
(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).
![Page 19: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/19.jpg)
MSDE: MULTIVALUED STOCHASTIC DIFFERENTIALEQUATIONS
dX(t) ∈ b(X(t))dt + σ(X(t))dw(t)−A(X(t))dt
Difference: An additional A
What is it?A: a maximal monotone operator.main features:• A is multivalued: ∀x, A(x) is a set, not necessarily a single point• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)
〈x− y, u− v〉 > 0
• A is maximal:
(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).
![Page 20: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/20.jpg)
MSDE: MULTIVALUED STOCHASTIC DIFFERENTIALEQUATIONS
dX(t) ∈ b(X(t))dt + σ(X(t))dw(t)−A(X(t))dt
Difference: An additional AWhat is it?
A: a maximal monotone operator.main features:• A is multivalued: ∀x, A(x) is a set, not necessarily a single point• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)
〈x− y, u− v〉 > 0
• A is maximal:
(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).
![Page 21: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/21.jpg)
MSDE: MULTIVALUED STOCHASTIC DIFFERENTIALEQUATIONS
dX(t) ∈ b(X(t))dt + σ(X(t))dw(t)−A(X(t))dt
Difference: An additional AWhat is it?A: a maximal monotone operator.
main features:• A is multivalued: ∀x, A(x) is a set, not necessarily a single point• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)
〈x− y, u− v〉 > 0
• A is maximal:
(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).
![Page 22: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/22.jpg)
MSDE: MULTIVALUED STOCHASTIC DIFFERENTIALEQUATIONS
dX(t) ∈ b(X(t))dt + σ(X(t))dw(t)−A(X(t))dt
Difference: An additional AWhat is it?A: a maximal monotone operator.main features:
• A is multivalued: ∀x, A(x) is a set, not necessarily a single point• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)
〈x− y, u− v〉 > 0
• A is maximal:
(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).
![Page 23: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/23.jpg)
MSDE: MULTIVALUED STOCHASTIC DIFFERENTIALEQUATIONS
dX(t) ∈ b(X(t))dt + σ(X(t))dw(t)−A(X(t))dt
Difference: An additional AWhat is it?A: a maximal monotone operator.main features:• A is multivalued: ∀x, A(x) is a set, not necessarily a single point
• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)
〈x− y, u− v〉 > 0
• A is maximal:
(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).
![Page 24: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/24.jpg)
MSDE: MULTIVALUED STOCHASTIC DIFFERENTIALEQUATIONS
dX(t) ∈ b(X(t))dt + σ(X(t))dw(t)−A(X(t))dt
Difference: An additional AWhat is it?A: a maximal monotone operator.main features:• A is multivalued: ∀x, A(x) is a set, not necessarily a single point• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)
〈x− y, u− v〉 > 0
• A is maximal:
(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).
![Page 25: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/25.jpg)
MSDE: MULTIVALUED STOCHASTIC DIFFERENTIALEQUATIONS
dX(t) ∈ b(X(t))dt + σ(X(t))dw(t)−A(X(t))dt
Difference: An additional AWhat is it?A: a maximal monotone operator.main features:• A is multivalued: ∀x, A(x) is a set, not necessarily a single point• A is monotone: ∀x, y, all u ∈ A(x), v ∈ A(y)
〈x− y, u− v〉 > 0
• A is maximal:
(x1, y1) ∈ Gr(A) ⇔ 〈y1 − y2, x1 − x2〉 > 0, ∀(x2, y2) ∈ Gr(A).
![Page 26: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/26.jpg)
Definition of Solution
Definition: A pair of continuous and Ft−adapted processes(X, K) is called a strong solution of{
dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t) dt,
X(0) = x ∈ D(A).
if
![Page 27: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/27.jpg)
Definition of Solution
Definition: A pair of continuous and Ft−adapted processes(X, K) is called a strong solution of
{dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t) dt,
X(0) = x ∈ D(A).
if
![Page 28: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/28.jpg)
Definition of Solution
Definition: A pair of continuous and Ft−adapted processes(X, K) is called a strong solution of{
dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t) dt,
X(0) = x ∈ D(A).
if
![Page 29: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/29.jpg)
Definition of Solution
Definition: A pair of continuous and Ft−adapted processes(X, K) is called a strong solution of{
dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t) dt,
X(0) = x ∈ D(A).
if
![Page 30: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/30.jpg)
(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;
(ii) K = {K(t),Ft; t ∈ R+} is of finite variation andK(0) = 0 a.s.;(iii) dX(t) = b(X(t))dt + σ(X(t))dw(t)− dK(t), t ∈ R+, a.s.;(iv) for any continuous processes (α, β) satisfying
(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,
the measure
〈X(t)− α(t), dK(t)− β(t)dt〉 > 0
(Formally, this amounts to saying
〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0
or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )
![Page 31: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/31.jpg)
(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;(ii) K = {K(t),Ft; t ∈ R+} is of finite variation andK(0) = 0 a.s.;
(iii) dX(t) = b(X(t))dt + σ(X(t))dw(t)− dK(t), t ∈ R+, a.s.;(iv) for any continuous processes (α, β) satisfying
(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,
the measure
〈X(t)− α(t), dK(t)− β(t)dt〉 > 0
(Formally, this amounts to saying
〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0
or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )
![Page 32: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/32.jpg)
(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;(ii) K = {K(t),Ft; t ∈ R+} is of finite variation andK(0) = 0 a.s.;(iii) dX(t) = b(X(t))dt + σ(X(t))dw(t)− dK(t), t ∈ R+, a.s.;
(iv) for any continuous processes (α, β) satisfying
(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,
the measure
〈X(t)− α(t), dK(t)− β(t)dt〉 > 0
(Formally, this amounts to saying
〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0
or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )
![Page 33: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/33.jpg)
(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;(ii) K = {K(t),Ft; t ∈ R+} is of finite variation andK(0) = 0 a.s.;(iii) dX(t) = b(X(t))dt + σ(X(t))dw(t)− dK(t), t ∈ R+, a.s.;(iv) for any continuous processes (α, β) satisfying
(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,
the measure
〈X(t)− α(t), dK(t)− β(t)dt〉 > 0
(Formally, this amounts to saying
〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0
or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )
![Page 34: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/34.jpg)
(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;(ii) K = {K(t),Ft; t ∈ R+} is of finite variation andK(0) = 0 a.s.;(iii) dX(t) = b(X(t))dt + σ(X(t))dw(t)− dK(t), t ∈ R+, a.s.;(iv) for any continuous processes (α, β) satisfying
(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,
the measure
〈X(t)− α(t), dK(t)− β(t)dt〉
> 0
(Formally, this amounts to saying
〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0
or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )
![Page 35: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/35.jpg)
(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;(ii) K = {K(t),Ft; t ∈ R+} is of finite variation andK(0) = 0 a.s.;(iii) dX(t) = b(X(t))dt + σ(X(t))dw(t)− dK(t), t ∈ R+, a.s.;(iv) for any continuous processes (α, β) satisfying
(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,
the measure
〈X(t)− α(t), dK(t)− β(t)dt〉 > 0
(Formally, this amounts to saying
〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0
or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )
![Page 36: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/36.jpg)
(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;(ii) K = {K(t),Ft; t ∈ R+} is of finite variation andK(0) = 0 a.s.;(iii) dX(t) = b(X(t))dt + σ(X(t))dw(t)− dK(t), t ∈ R+, a.s.;(iv) for any continuous processes (α, β) satisfying
(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,
the measure
〈X(t)− α(t), dK(t)− β(t)dt〉 > 0
(Formally, this amounts to saying
〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0
or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )
![Page 37: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/37.jpg)
(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;(ii) K = {K(t),Ft; t ∈ R+} is of finite variation andK(0) = 0 a.s.;(iii) dX(t) = b(X(t))dt + σ(X(t))dw(t)− dK(t), t ∈ R+, a.s.;(iv) for any continuous processes (α, β) satisfying
(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,
the measure
〈X(t)− α(t), dK(t)− β(t)dt〉 > 0
(Formally, this amounts to saying
〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0
or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )
![Page 38: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/38.jpg)
(i) X0 = x0 and X(t) ∈ D(A) a.s., ∀t;(ii) K = {K(t),Ft; t ∈ R+} is of finite variation andK(0) = 0 a.s.;(iii) dX(t) = b(X(t))dt + σ(X(t))dw(t)− dK(t), t ∈ R+, a.s.;(iv) for any continuous processes (α, β) satisfying
(α(t), β(t)) ∈ Gr(A), ∀t ∈ R+,
the measure
〈X(t)− α(t), dK(t)− β(t)dt〉 > 0
(Formally, this amounts to saying
〈X(t)− α(t),K ′(t)dt− β(t)dt〉 > 0
or〈X(t)− α(t),K ′(t)− β(t)〉 > 0 )
![Page 39: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/39.jpg)
Existence and Uniqueness
Theorem (Cepa, 1995) If σ and b are Lipschitz, then{dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t)−A(X(t)) dt,
X(0) = x ∈ D(A), ε ∈ (0, 1].
admits a unique solution, (X, K).
The unique solution of{dXε(t) ∈ b(Xε(t)) dt +
√εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
will be denoted by(Xε,Kε)
![Page 40: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/40.jpg)
Existence and Uniqueness
Theorem (Cepa, 1995) If σ and b are Lipschitz, then{dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t)−A(X(t)) dt,
X(0) = x ∈ D(A), ε ∈ (0, 1].
admits a unique solution, (X, K).
The unique solution of{dXε(t) ∈ b(Xε(t)) dt +
√εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
will be denoted by(Xε,Kε)
![Page 41: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/41.jpg)
Existence and Uniqueness
Theorem (Cepa, 1995) If σ and b are Lipschitz, then{dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t)−A(X(t)) dt,
X(0) = x ∈ D(A), ε ∈ (0, 1].
admits a unique solution, (X, K).
The unique solution of{dXε(t) ∈ b(Xε(t)) dt +
√εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
will be denoted by(Xε,Kε)
![Page 42: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/42.jpg)
Existence and Uniqueness
Theorem (Cepa, 1995) If σ and b are Lipschitz, then{dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t)−A(X(t)) dt,
X(0) = x ∈ D(A), ε ∈ (0, 1].
admits a unique solution, (X, K).
The unique solution of
{dXε(t) ∈ b(Xε(t)) dt +
√εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
will be denoted by(Xε,Kε)
![Page 43: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/43.jpg)
Existence and Uniqueness
Theorem (Cepa, 1995) If σ and b are Lipschitz, then{dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t)−A(X(t)) dt,
X(0) = x ∈ D(A), ε ∈ (0, 1].
admits a unique solution, (X, K).
The unique solution of{dXε(t) ∈ b(Xε(t)) dt +
√εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
will be denoted by(Xε,Kε)
![Page 44: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/44.jpg)
Existence and Uniqueness
Theorem (Cepa, 1995) If σ and b are Lipschitz, then{dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t)−A(X(t)) dt,
X(0) = x ∈ D(A), ε ∈ (0, 1].
admits a unique solution, (X, K).
The unique solution of{dXε(t) ∈ b(Xε(t)) dt +
√εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
will be denoted by
(Xε,Kε)
![Page 45: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/45.jpg)
Existence and Uniqueness
Theorem (Cepa, 1995) If σ and b are Lipschitz, then{dX(t) ∈ b(X(t)) dt + σ(X(t)) dW (t)−A(X(t)) dt,
X(0) = x ∈ D(A), ε ∈ (0, 1].
admits a unique solution, (X, K).
The unique solution of{dXε(t) ∈ b(Xε(t)) dt +
√εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
will be denoted by(Xε,Kε)
![Page 46: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/46.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 47: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/47.jpg)
normally reflected SDE as a special case of MSDE
Suppose
O: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 48: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/48.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm,
IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 49: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/49.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,
i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 50: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/50.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 51: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/51.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,
+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 52: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/52.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 53: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/53.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 54: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/54.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 55: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/55.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 56: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/56.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,
{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 57: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/57.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),
Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 58: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/58.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 59: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/59.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
where
Int(O) is the interior of OΛx is the exterior normal cone at x.
![Page 60: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/60.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of O
Λx is the exterior normal cone at x.
![Page 61: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/61.jpg)
normally reflected SDE as a special case of MSDE
SupposeO: a closed convex subset of Rm, IO : the indicator function of O,i.e,
IO(x) ={
0, if x ∈ O,+∞, if x /∈ O.
The subdifferential of IO is given by
∂IO(x) = {y ∈ Rm : 〈y, x− z〉Rm > 0,∀z ∈ O}
=
∅, if x /∈ O,{0}, if x ∈ Int(O),Λx, if x ∈ ∂O,
whereInt(O) is the interior of OΛx is the exterior normal cone at x.
![Page 62: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/62.jpg)
Then ∂IO is a multivalued maximal monotone operator.
In this case, an MSDE is an SDE normally reflected at theboundary of O.
More generally, the subdifferential of any convex and lowersemi-continuous function is a maximal monotone operator.
∂ϕ(x) := {y ∈ Rm : 〈y, z − x〉Rm > ϕ(z)− ϕ(x),∀z ∈ Rm}
![Page 63: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/63.jpg)
Then ∂IO is a multivalued maximal monotone operator.
In this case, an MSDE is an SDE normally reflected at theboundary of O.
More generally, the subdifferential of any convex and lowersemi-continuous function is a maximal monotone operator.
∂ϕ(x) := {y ∈ Rm : 〈y, z − x〉Rm > ϕ(z)− ϕ(x),∀z ∈ Rm}
![Page 64: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/64.jpg)
Then ∂IO is a multivalued maximal monotone operator.
In this case, an MSDE is an SDE normally reflected at theboundary of O.
More generally,
the subdifferential of any convex and lowersemi-continuous function is a maximal monotone operator.
∂ϕ(x) := {y ∈ Rm : 〈y, z − x〉Rm > ϕ(z)− ϕ(x),∀z ∈ Rm}
![Page 65: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/65.jpg)
Then ∂IO is a multivalued maximal monotone operator.
In this case, an MSDE is an SDE normally reflected at theboundary of O.
More generally, the subdifferential of any convex and lowersemi-continuous function is a maximal monotone operator.
∂ϕ(x) := {y ∈ Rm : 〈y, z − x〉Rm > ϕ(z)− ϕ(x),∀z ∈ Rm}
![Page 66: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/66.jpg)
Then ∂IO is a multivalued maximal monotone operator.
In this case, an MSDE is an SDE normally reflected at theboundary of O.
More generally, the subdifferential of any convex and lowersemi-continuous function is a maximal monotone operator.
∂ϕ(x) := {y ∈ Rm : 〈y, z − x〉Rm > ϕ(z)− ϕ(x),∀z ∈ Rm}
![Page 67: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/67.jpg)
Then ∂IO is a multivalued maximal monotone operator.
In this case, an MSDE is an SDE normally reflected at theboundary of O.
More generally, the subdifferential of any convex and lowersemi-continuous function is a maximal monotone operator.
∂ϕ(x) := {y ∈ Rm : 〈y, z − x〉Rm > ϕ(z)− ϕ(x),∀z ∈ Rm}
![Page 68: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/68.jpg)
Towards a Freidlin-Wentzell theory
We now look at the problem of small perturbation:{dXε(t) ∈ b(Xε(t)) dt +
√εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
![Page 69: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/69.jpg)
Towards a Freidlin-Wentzell theory
We now look at the problem of small perturbation:
{dXε(t) ∈ b(Xε(t)) dt +
√εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
![Page 70: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/70.jpg)
Towards a Freidlin-Wentzell theory
We now look at the problem of small perturbation:{dXε(t) ∈ b(Xε(t)) dt +
√εσ(Xε(t)) dW (t)−A(Xε(t)) dt,
Xε(0) = x ∈ D(A), ε ∈ (0, 1].
![Page 71: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/71.jpg)
Classical theory
Recall the classical case: A ≡ 0{dXε(t) = b(Xε(t)) dt +
√εσ(Xε(t)) dW (t),
Xε(0) = x, ε ∈ (0, 1].
The theory begins in the seminal paper
Wentzell-Freidlin:On small random perturbation of dynamical system, 1969
There are two standard and well known approaches to this classicalproblem.
![Page 72: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/72.jpg)
Classical theory
Recall the classical case:
A ≡ 0{dXε(t) = b(Xε(t)) dt +
√εσ(Xε(t)) dW (t),
Xε(0) = x, ε ∈ (0, 1].
The theory begins in the seminal paper
Wentzell-Freidlin:On small random perturbation of dynamical system, 1969
There are two standard and well known approaches to this classicalproblem.
![Page 73: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/73.jpg)
Classical theory
Recall the classical case: A ≡ 0
{dXε(t) = b(Xε(t)) dt +
√εσ(Xε(t)) dW (t),
Xε(0) = x, ε ∈ (0, 1].
The theory begins in the seminal paper
Wentzell-Freidlin:On small random perturbation of dynamical system, 1969
There are two standard and well known approaches to this classicalproblem.
![Page 74: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/74.jpg)
Classical theory
Recall the classical case: A ≡ 0{dXε(t) = b(Xε(t)) dt +
√εσ(Xε(t)) dW (t),
Xε(0) = x, ε ∈ (0, 1].
The theory begins in the seminal paper
Wentzell-Freidlin:On small random perturbation of dynamical system, 1969
There are two standard and well known approaches to this classicalproblem.
![Page 75: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/75.jpg)
Classical theory
Recall the classical case: A ≡ 0{dXε(t) = b(Xε(t)) dt +
√εσ(Xε(t)) dW (t),
Xε(0) = x, ε ∈ (0, 1].
The theory begins in the seminal paper
Wentzell-Freidlin:On small random perturbation of dynamical system, 1969
There are two standard and well known approaches to this classicalproblem.
![Page 76: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/76.jpg)
Classical theory
Recall the classical case: A ≡ 0{dXε(t) = b(Xε(t)) dt +
√εσ(Xε(t)) dW (t),
Xε(0) = x, ε ∈ (0, 1].
The theory begins in the seminal paper
Wentzell-Freidlin:On small random perturbation of dynamical system, 1969
There are two standard and well known approaches to this classicalproblem.
![Page 77: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/77.jpg)
Classical theory
Recall the classical case: A ≡ 0{dXε(t) = b(Xε(t)) dt +
√εσ(Xε(t)) dW (t),
Xε(0) = x, ε ∈ (0, 1].
The theory begins in the seminal paper
Wentzell-Freidlin:On small random perturbation of dynamical system, 1969
There are two standard and well known approaches to this classicalproblem.
![Page 78: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/78.jpg)
1. Use of Euler Scheme
D.W.Stroock: An introduction to the theory of large deviation,1984
You look at the Euler Scheme
Xεn(t) = x +
∫ t
0b(Xε
n(n−1[sn])) ds +√
ε
∫ t
0σ(Xε
n(n−1[sn])) dW (s),
Then you have to estimate the quantity
P ( max06t61
|Xε(t, x)−Xεn| > δ)
If you have for any δ > 0
limn→∞
lim supε↓0
ε log P ( max06t61
|Xε(t, x)−Xεn| > δ) = −∞,
then you are done.
![Page 79: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/79.jpg)
1. Use of Euler Scheme
D.W.Stroock: An introduction to the theory of large deviation,1984
You look at the Euler Scheme
Xεn(t) = x +
∫ t
0b(Xε
n(n−1[sn])) ds +√
ε
∫ t
0σ(Xε
n(n−1[sn])) dW (s),
Then you have to estimate the quantity
P ( max06t61
|Xε(t, x)−Xεn| > δ)
If you have for any δ > 0
limn→∞
lim supε↓0
ε log P ( max06t61
|Xε(t, x)−Xεn| > δ) = −∞,
then you are done.
![Page 80: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/80.jpg)
1. Use of Euler Scheme
D.W.Stroock: An introduction to the theory of large deviation,1984
You look at the Euler Scheme
Xεn(t) = x +
∫ t
0b(Xε
n(n−1[sn])) ds +√
ε
∫ t
0σ(Xε
n(n−1[sn])) dW (s),
Then you have to estimate the quantity
P ( max06t61
|Xε(t, x)−Xεn| > δ)
If you have for any δ > 0
limn→∞
lim supε↓0
ε log P ( max06t61
|Xε(t, x)−Xεn| > δ) = −∞,
then you are done.
![Page 81: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/81.jpg)
1. Use of Euler Scheme
D.W.Stroock: An introduction to the theory of large deviation,1984
You look at the Euler Scheme
Xεn(t) = x +
∫ t
0b(Xε
n(n−1[sn])) ds +√
ε
∫ t
0σ(Xε
n(n−1[sn])) dW (s),
Then you have to estimate the quantity
P ( max06t61
|Xε(t, x)−Xεn| > δ)
If you have for any δ > 0
limn→∞
lim supε↓0
ε log P ( max06t61
|Xε(t, x)−Xεn| > δ) = −∞,
then you are done.
![Page 82: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/82.jpg)
1. Use of Euler Scheme
D.W.Stroock: An introduction to the theory of large deviation,1984
You look at the Euler Scheme
Xεn(t) = x +
∫ t
0b(Xε
n(n−1[sn])) ds +√
ε
∫ t
0σ(Xε
n(n−1[sn])) dW (s),
Then you have to estimate the quantity
P ( max06t61
|Xε(t, x)−Xεn| > δ)
If you have for any δ > 0
limn→∞
lim supε↓0
ε log P ( max06t61
|Xε(t, x)−Xεn| > δ) = −∞,
then you are done.
![Page 83: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/83.jpg)
1. Use of Euler Scheme
D.W.Stroock: An introduction to the theory of large deviation,1984
You look at the Euler Scheme
Xεn(t) = x +
∫ t
0b(Xε
n(n−1[sn])) ds +√
ε
∫ t
0σ(Xε
n(n−1[sn])) dW (s),
Then you have to estimate the quantity
P ( max06t61
|Xε(t, x)−Xεn| > δ)
If you have for any δ > 0
limn→∞
lim supε↓0
ε log P ( max06t61
|Xε(t, x)−Xεn| > δ) = −∞,
then you are done.
![Page 84: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/84.jpg)
1. Use of Euler Scheme
D.W.Stroock: An introduction to the theory of large deviation,1984
You look at the Euler Scheme
Xεn(t) = x +
∫ t
0b(Xε
n(n−1[sn])) ds +√
ε
∫ t
0σ(Xε
n(n−1[sn])) dW (s),
Then you have to estimate the quantity
P ( max06t61
|Xε(t, x)−Xεn| > δ)
If you have for any δ > 0
limn→∞
lim supε↓0
ε log P ( max06t61
|Xε(t, x)−Xεn| > δ) = −∞,
then you are done.
![Page 85: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/85.jpg)
1. Use of Euler Scheme
D.W.Stroock: An introduction to the theory of large deviation,1984
You look at the Euler Scheme
Xεn(t) = x +
∫ t
0b(Xε
n(n−1[sn])) ds +√
ε
∫ t
0σ(Xε
n(n−1[sn])) dW (s),
Then you have to estimate the quantity
P ( max06t61
|Xε(t, x)−Xεn| > δ)
If you have for any δ > 0
limn→∞
lim supε↓0
ε log P ( max06t61
|Xε(t, x)−Xεn| > δ) = −∞,
then you are done.
![Page 86: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/86.jpg)
1. Use of Euler Scheme
D.W.Stroock: An introduction to the theory of large deviation,1984
You look at the Euler Scheme
Xεn(t) = x +
∫ t
0b(Xε
n(n−1[sn])) ds +√
ε
∫ t
0σ(Xε
n(n−1[sn])) dW (s),
Then you have to estimate the quantity
P ( max06t61
|Xε(t, x)−Xεn| > δ)
If you have for any δ > 0
limn→∞
lim supε↓0
ε log P ( max06t61
|Xε(t, x)−Xεn| > δ) = −∞,
then you are done.
![Page 87: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/87.jpg)
Unfortunately, this approach does NOT apply to MSDE case
simply because the Euler scheme cannot be defined for MSDEsince either A(Xn(t)) is not defined or may be a set.
![Page 88: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/88.jpg)
Unfortunately, this approach does NOT apply to MSDE casesimply because the Euler scheme cannot be defined for MSDE
since either A(Xn(t)) is not defined or may be a set.
![Page 89: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/89.jpg)
Unfortunately, this approach does NOT apply to MSDE casesimply because the Euler scheme cannot be defined for MSDEsince either A(Xn(t)) is not defined or may be a set.
![Page 90: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/90.jpg)
2. Use of Freidlin-Wentzell estimate
(R. Azencott: Grandes deviations et applications, 1980
P. Priouret: Remarques sur les petite perturbations de systemesdynamiques, 1982)
Consider an ODE{dg(t) = b(g(t)) dt + σ(g(t))f ′(t) dt,g(0) = x.
where f satisfies ∫ 1
0f ′(t)2dt < ∞.
![Page 91: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/91.jpg)
2. Use of Freidlin-Wentzell estimate
(R. Azencott: Grandes deviations et applications, 1980
P. Priouret: Remarques sur les petite perturbations de systemesdynamiques, 1982)
Consider an ODE{dg(t) = b(g(t)) dt + σ(g(t))f ′(t) dt,g(0) = x.
where f satisfies ∫ 1
0f ′(t)2dt < ∞.
![Page 92: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/92.jpg)
2. Use of Freidlin-Wentzell estimate
(R. Azencott: Grandes deviations et applications, 1980
P. Priouret: Remarques sur les petite perturbations de systemesdynamiques, 1982)
Consider an ODE{dg(t) = b(g(t)) dt + σ(g(t))f ′(t) dt,g(0) = x.
where f satisfies ∫ 1
0f ′(t)2dt < ∞.
![Page 93: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/93.jpg)
2. Use of Freidlin-Wentzell estimate
(R. Azencott: Grandes deviations et applications, 1980
P. Priouret: Remarques sur les petite perturbations de systemesdynamiques, 1982)
Consider an ODE
{dg(t) = b(g(t)) dt + σ(g(t))f ′(t) dt,g(0) = x.
where f satisfies ∫ 1
0f ′(t)2dt < ∞.
![Page 94: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/94.jpg)
2. Use of Freidlin-Wentzell estimate
(R. Azencott: Grandes deviations et applications, 1980
P. Priouret: Remarques sur les petite perturbations de systemesdynamiques, 1982)
Consider an ODE{dg(t) = b(g(t)) dt + σ(g(t))f ′(t) dt,g(0) = x.
where f satisfies ∫ 1
0f ′(t)2dt < ∞.
![Page 95: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/95.jpg)
2. Use of Freidlin-Wentzell estimate
(R. Azencott: Grandes deviations et applications, 1980
P. Priouret: Remarques sur les petite perturbations de systemesdynamiques, 1982)
Consider an ODE{dg(t) = b(g(t)) dt + σ(g(t))f ′(t) dt,g(0) = x.
where f satisfies
∫ 1
0f ′(t)2dt < ∞.
![Page 96: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/96.jpg)
2. Use of Freidlin-Wentzell estimate
(R. Azencott: Grandes deviations et applications, 1980
P. Priouret: Remarques sur les petite perturbations de systemesdynamiques, 1982)
Consider an ODE{dg(t) = b(g(t)) dt + σ(g(t))f ′(t) dt,g(0) = x.
where f satisfies ∫ 1
0f ′(t)2dt < ∞.
![Page 97: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/97.jpg)
If you can prove
∀R > 0, ρ > 0, ∃ε0 > 0, α > 0, r > 0 such that
ε log P (‖Xε − g‖ > ρ, ‖ε12 W − f‖ < α) 6 −R, ∀ε 6 ε0
then you have the large deviation principle.
![Page 98: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/98.jpg)
If you can prove ∀R > 0,
ρ > 0, ∃ε0 > 0, α > 0, r > 0 such that
ε log P (‖Xε − g‖ > ρ, ‖ε12 W − f‖ < α) 6 −R, ∀ε 6 ε0
then you have the large deviation principle.
![Page 99: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/99.jpg)
If you can prove ∀R > 0, ρ > 0,
∃ε0 > 0, α > 0, r > 0 such that
ε log P (‖Xε − g‖ > ρ, ‖ε12 W − f‖ < α) 6 −R, ∀ε 6 ε0
then you have the large deviation principle.
![Page 100: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/100.jpg)
If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0,
α > 0, r > 0 such that
ε log P (‖Xε − g‖ > ρ, ‖ε12 W − f‖ < α) 6 −R, ∀ε 6 ε0
then you have the large deviation principle.
![Page 101: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/101.jpg)
If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0, α > 0,
r > 0 such that
ε log P (‖Xε − g‖ > ρ, ‖ε12 W − f‖ < α) 6 −R, ∀ε 6 ε0
then you have the large deviation principle.
![Page 102: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/102.jpg)
If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0, α > 0, r > 0
such that
ε log P (‖Xε − g‖ > ρ, ‖ε12 W − f‖ < α) 6 −R, ∀ε 6 ε0
then you have the large deviation principle.
![Page 103: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/103.jpg)
If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0, α > 0, r > 0 such that
ε log P (‖Xε − g‖ > ρ, ‖ε12 W − f‖ < α) 6 −R, ∀ε 6 ε0
then you have the large deviation principle.
![Page 104: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/104.jpg)
If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0, α > 0, r > 0 such that
ε log P (‖Xε − g‖ > ρ, ‖ε12 W − f‖ < α) 6 −R, ∀ε 6 ε0
then you have the large deviation principle.
![Page 105: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/105.jpg)
If you can prove ∀R > 0, ρ > 0, ∃ε0 > 0, α > 0, r > 0 such that
ε log P (‖Xε − g‖ > ρ, ‖ε12 W − f‖ < α) 6 −R, ∀ε 6 ε0
then you have the large deviation principle.
![Page 106: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/106.jpg)
But this approach does NOT apply neither
since to obtain the Freidlin-Wentzell estimate directlyone has to suppose b and σ are Lipschitz(Priouret had to.).
![Page 107: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/107.jpg)
But this approach does NOT apply neithersince to obtain the Freidlin-Wentzell estimate directly
one has to suppose b and σ are Lipschitz(Priouret had to.).
![Page 108: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/108.jpg)
But this approach does NOT apply neithersince to obtain the Freidlin-Wentzell estimate directlyone has to suppose b and σ are Lipschitz
(Priouret had to.).
![Page 109: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/109.jpg)
But this approach does NOT apply neithersince to obtain the Freidlin-Wentzell estimate directlyone has to suppose b and σ are Lipschitz(Priouret had to.).
![Page 110: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/110.jpg)
3. Use of weak convergence
This approach was developed a few years ago by
P. Dupuis and R.S. Ellis: A weak convergence approach to thetheory of large deviatins, 1997
and was tested in several recent work to treat Wentzell-Freidlinlarge deviation principle for SDEs with irregular coefficients. It hasproved to be highly efficient.
The hard core of this approach is the following observation
![Page 111: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/111.jpg)
3. Use of weak convergence
This approach was developed a few years ago by
P. Dupuis and R.S. Ellis: A weak convergence approach to thetheory of large deviatins, 1997
and was tested in several recent work to treat Wentzell-Freidlinlarge deviation principle for SDEs with irregular coefficients. It hasproved to be highly efficient.
The hard core of this approach is the following observation
![Page 112: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/112.jpg)
3. Use of weak convergence
This approach was developed a few years ago by
P. Dupuis and R.S. Ellis: A weak convergence approach to thetheory of large deviatins, 1997
and was tested in several recent work to treat Wentzell-Freidlinlarge deviation principle for SDEs with irregular coefficients. It hasproved to be highly efficient.
The hard core of this approach is the following observation
![Page 113: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/113.jpg)
3. Use of weak convergence
This approach was developed a few years ago by
P. Dupuis and R.S. Ellis: A weak convergence approach to thetheory of large deviatins, 1997
and was tested in several recent work to treat Wentzell-Freidlinlarge deviation principle for SDEs with irregular coefficients. It hasproved to be highly efficient.
The hard core of this approach is the following observation
![Page 114: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/114.jpg)
3. Use of weak convergence
This approach was developed a few years ago by
P. Dupuis and R.S. Ellis: A weak convergence approach to thetheory of large deviatins, 1997
and was tested in several recent work to treat Wentzell-Freidlinlarge deviation principle for SDEs with irregular coefficients. It hasproved to be highly efficient.
The hard core of this approach is the following observation
![Page 115: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/115.jpg)
• Xn: a sequence of r.v., valued in a complete metric space X ;
• I: a rate function on X ;{Xn} is said to satisfy the LDP with rate function I if
lim supn→∞
1n
log P (Xn ∈ F ) 6 −I(F ), F closed
lim infn→∞
1n
log P (Xn ∈ G) > −I(G), F open
{Xn} is said to satisfy the Laplace Principle with rate function I if
limn→∞
1n
log E{exp[−nh(Xn)]} = − infx∈X
{h(x) + I(x)}.
![Page 116: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/116.jpg)
• Xn: a sequence of r.v., valued in a complete metric space X ;• I: a rate function on X ;
{Xn} is said to satisfy the LDP with rate function I if
lim supn→∞
1n
log P (Xn ∈ F ) 6 −I(F ), F closed
lim infn→∞
1n
log P (Xn ∈ G) > −I(G), F open
{Xn} is said to satisfy the Laplace Principle with rate function I if
limn→∞
1n
log E{exp[−nh(Xn)]} = − infx∈X
{h(x) + I(x)}.
![Page 117: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/117.jpg)
• Xn: a sequence of r.v., valued in a complete metric space X ;• I: a rate function on X ;{Xn} is said to satisfy the LDP with rate function I if
lim supn→∞
1n
log P (Xn ∈ F ) 6 −I(F ), F closed
lim infn→∞
1n
log P (Xn ∈ G) > −I(G), F open
{Xn} is said to satisfy the Laplace Principle with rate function I if
limn→∞
1n
log E{exp[−nh(Xn)]} = − infx∈X
{h(x) + I(x)}.
![Page 118: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/118.jpg)
• Xn: a sequence of r.v., valued in a complete metric space X ;• I: a rate function on X ;{Xn} is said to satisfy the LDP with rate function I if
lim supn→∞
1n
log P (Xn ∈ F ) 6 −I(F ), F closed
lim infn→∞
1n
log P (Xn ∈ G) > −I(G), F open
{Xn} is said to satisfy the Laplace Principle with rate function I if
limn→∞
1n
log E{exp[−nh(Xn)]} = − infx∈X
{h(x) + I(x)}.
![Page 119: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/119.jpg)
• Xn: a sequence of r.v., valued in a complete metric space X ;• I: a rate function on X ;{Xn} is said to satisfy the LDP with rate function I if
lim supn→∞
1n
log P (Xn ∈ F ) 6 −I(F ), F closed
lim infn→∞
1n
log P (Xn ∈ G) > −I(G), F open
{Xn} is said to satisfy the Laplace Principle with rate function I if
limn→∞
1n
log E{exp[−nh(Xn)]} = − infx∈X
{h(x) + I(x)}.
![Page 120: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/120.jpg)
• Xn: a sequence of r.v., valued in a complete metric space X ;• I: a rate function on X ;{Xn} is said to satisfy the LDP with rate function I if
lim supn→∞
1n
log P (Xn ∈ F ) 6 −I(F ), F closed
lim infn→∞
1n
log P (Xn ∈ G) > −I(G), F open
{Xn} is said to satisfy the Laplace Principle with rate function I if
limn→∞
1n
log E{exp[−nh(Xn)]} = − infx∈X
{h(x) + I(x)}.
![Page 121: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/121.jpg)
Theorem: LP⇐⇒ LDP
(⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)Now let us return to our MSDE. Suppose D ⊂ Rd is an open set.Consider the following reflected SDE
dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x
where a increases only when Xε is on the boundary of D, v(x) isin the outward normal cone of D at x ∈ ∂D.For the result of the following type
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), the story is
![Page 122: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/122.jpg)
Theorem: LP⇐⇒ LDP (⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)
Now let us return to our MSDE. Suppose D ⊂ Rd is an open set.Consider the following reflected SDE
dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x
where a increases only when Xε is on the boundary of D, v(x) isin the outward normal cone of D at x ∈ ∂D.For the result of the following type
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), the story is
![Page 123: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/123.jpg)
Theorem: LP⇐⇒ LDP (⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)Now let us return to our MSDE.
Suppose D ⊂ Rd is an open set.Consider the following reflected SDE
dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x
where a increases only when Xε is on the boundary of D, v(x) isin the outward normal cone of D at x ∈ ∂D.For the result of the following type
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), the story is
![Page 124: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/124.jpg)
Theorem: LP⇐⇒ LDP (⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)Now let us return to our MSDE. Suppose D ⊂ Rd is an open set.Consider the following reflected SDE
dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x
where a increases only when Xε is on the boundary of D, v(x) isin the outward normal cone of D at x ∈ ∂D.For the result of the following type
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), the story is
![Page 125: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/125.jpg)
Theorem: LP⇐⇒ LDP (⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)Now let us return to our MSDE. Suppose D ⊂ Rd is an open set.Consider the following reflected SDE
dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x
where a increases only when Xε is on the boundary of D, v(x) isin the outward normal cone of D at x ∈ ∂D.For the result of the following type
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), the story is
![Page 126: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/126.jpg)
Theorem: LP⇐⇒ LDP (⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)Now let us return to our MSDE. Suppose D ⊂ Rd is an open set.Consider the following reflected SDE
dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x
where a increases only when Xε is on the boundary of D, v(x) isin the outward normal cone of D at x ∈ ∂D.
For the result of the following type
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), the story is
![Page 127: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/127.jpg)
Theorem: LP⇐⇒ LDP (⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)Now let us return to our MSDE. Suppose D ⊂ Rd is an open set.Consider the following reflected SDE
dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x
where a increases only when Xε is on the boundary of D, v(x) isin the outward normal cone of D at x ∈ ∂D.For the result of the following type
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), the story is
![Page 128: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/128.jpg)
Theorem: LP⇐⇒ LDP (⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)Now let us return to our MSDE. Suppose D ⊂ Rd is an open set.Consider the following reflected SDE
dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x
where a increases only when Xε is on the boundary of D, v(x) isin the outward normal cone of D at x ∈ ∂D.For the result of the following type
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), the story is
![Page 129: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/129.jpg)
Theorem: LP⇐⇒ LDP (⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)Now let us return to our MSDE. Suppose D ⊂ Rd is an open set.Consider the following reflected SDE
dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x
where a increases only when Xε is on the boundary of D, v(x) isin the outward normal cone of D at x ∈ ∂D.For the result of the following type
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)),
the story is
![Page 130: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/130.jpg)
Theorem: LP⇐⇒ LDP (⇐=: Varadhan, 1966; =⇒: Dupuis-Ellis,1997)Now let us return to our MSDE. Suppose D ⊂ Rd is an open set.Consider the following reflected SDE
dXε(t) ∈ b(Xε(t))dt+σ(Xε(t))◦dw(t)−v(Xε(t))da(t), X(0) = x
where a increases only when Xε is on the boundary of D, v(x) isin the outward normal cone of D at x ∈ ∂D.For the result of the following type
−I(F o) 6 lim infε↓0
ε log P (Xε ∈ F )
6 lim supε↓0
ε log P (Xε ∈ F )
6 −I(F )
for F ⊂ Cx([0, 1], D(A)), the story is
![Page 131: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/131.jpg)
(i) by Anderson and Orey in 1976 when D is (in addition toconvex) smooth and σσ∗ > 0;
(ii) by Doss and Priouret when σ may be degenerated;
(iii) by Dupuis for general convex sets D
![Page 132: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/132.jpg)
(i) by Anderson and Orey in 1976 when D is (in addition toconvex) smooth and σσ∗ > 0;
(ii) by Doss and Priouret when σ may be degenerated;
(iii) by Dupuis for general convex sets D
![Page 133: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/133.jpg)
(i) by Anderson and Orey in 1976 when D is (in addition toconvex) smooth and σσ∗ > 0;
(ii) by Doss and Priouret when σ may be degenerated;
(iii) by Dupuis for general convex sets D
![Page 134: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/134.jpg)
(i) by Anderson and Orey in 1976 when D is (in addition toconvex) smooth and σσ∗ > 0;
(ii) by Doss and Priouret when σ may be degenerated;
(iii) by Dupuis for general convex sets D
![Page 135: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/135.jpg)
(i) by Anderson and Orey in 1976 when D is (in addition toconvex) smooth and σσ∗ > 0;
(ii) by Doss and Priouret when σ may be degenerated;
(iii) by Dupuis for general convex sets D
![Page 136: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/136.jpg)
Return to our original problem.
If we manage to prove a Laplace principle, then we are done.
But how to obtain the Laplace principle?
Budhiraja and Dupuis pointed out the way for us.
![Page 137: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/137.jpg)
Return to our original problem.
If we manage to prove a Laplace principle, then we are done.
But how to obtain the Laplace principle?
Budhiraja and Dupuis pointed out the way for us.
![Page 138: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/138.jpg)
Return to our original problem.
If we manage to prove a Laplace principle, then we are done.
But how to obtain the Laplace principle?
Budhiraja and Dupuis pointed out the way for us.
![Page 139: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/139.jpg)
Return to our original problem.
If we manage to prove a Laplace principle, then we are done.
But how to obtain the Laplace principle?
Budhiraja and Dupuis pointed out the way for us.
![Page 140: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/140.jpg)
Things to do
SetB := C([0, 1], Rd)
H := {h ∈ B such that h is a. c. and ‖h‖2H :=
∫ 1
0|h(s)|2ds < ∞}
µ the standard Wiener measure on B
Then (B,H, µ) is a classical Wiener space.
∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}
Then DN with weak topology is a metrizable compact Polish space.
AN := {h : h is an adapted process , h(·) ∈ DN a.s.}
![Page 141: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/141.jpg)
Things to do
SetB := C([0, 1], Rd)
H := {h ∈ B such that h is a. c. and ‖h‖2H :=
∫ 1
0|h(s)|2ds < ∞}
µ the standard Wiener measure on B
Then (B,H, µ) is a classical Wiener space.
∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}
Then DN with weak topology is a metrizable compact Polish space.
AN := {h : h is an adapted process , h(·) ∈ DN a.s.}
![Page 142: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/142.jpg)
Things to do
SetB := C([0, 1], Rd)
H := {h ∈ B such that h is a. c. and ‖h‖2H :=
∫ 1
0|h(s)|2ds < ∞}
µ the standard Wiener measure on B
Then (B,H, µ) is a classical Wiener space.
∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}
Then DN with weak topology is a metrizable compact Polish space.
AN := {h : h is an adapted process , h(·) ∈ DN a.s.}
![Page 143: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/143.jpg)
Things to do
SetB := C([0, 1], Rd)
H := {h ∈ B such that h is a. c. and ‖h‖2H :=
∫ 1
0|h(s)|2ds < ∞}
µ the standard Wiener measure on B
Then (B,H, µ) is a classical Wiener space.
∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}
Then DN with weak topology is a metrizable compact Polish space.
AN := {h : h is an adapted process , h(·) ∈ DN a.s.}
![Page 144: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/144.jpg)
Things to do
SetB := C([0, 1], Rd)
H := {h ∈ B such that h is a. c. and ‖h‖2H :=
∫ 1
0|h(s)|2ds < ∞}
µ the standard Wiener measure on B
Then (B,H, µ) is a classical Wiener space.
∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}
Then DN with weak topology is a metrizable compact Polish space.
AN := {h : h is an adapted process , h(·) ∈ DN a.s.}
![Page 145: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/145.jpg)
Things to do
SetB := C([0, 1], Rd)
H := {h ∈ B such that h is a. c. and ‖h‖2H :=
∫ 1
0|h(s)|2ds < ∞}
µ the standard Wiener measure on B
Then (B,H, µ) is a classical Wiener space.
∀N > 0, let
DN := {h ∈ H : ‖h‖H 6 N}
Then DN with weak topology is a metrizable compact Polish space.
AN := {h : h is an adapted process , h(·) ∈ DN a.s.}
![Page 146: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/146.jpg)
Things to do
SetB := C([0, 1], Rd)
H := {h ∈ B such that h is a. c. and ‖h‖2H :=
∫ 1
0|h(s)|2ds < ∞}
µ the standard Wiener measure on B
Then (B,H, µ) is a classical Wiener space.
∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}
Then DN with weak topology is a metrizable compact Polish space.
AN := {h : h is an adapted process , h(·) ∈ DN a.s.}
![Page 147: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/147.jpg)
Things to do
SetB := C([0, 1], Rd)
H := {h ∈ B such that h is a. c. and ‖h‖2H :=
∫ 1
0|h(s)|2ds < ∞}
µ the standard Wiener measure on B
Then (B,H, µ) is a classical Wiener space.
∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}
Then DN with weak topology is a metrizable compact Polish space.
AN := {h : h is an adapted process , h(·) ∈ DN a.s.}
![Page 148: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/148.jpg)
Things to do
SetB := C([0, 1], Rd)
H := {h ∈ B such that h is a. c. and ‖h‖2H :=
∫ 1
0|h(s)|2ds < ∞}
µ the standard Wiener measure on B
Then (B,H, µ) is a classical Wiener space.
∀N > 0, letDN := {h ∈ H : ‖h‖H 6 N}
Then DN with weak topology is a metrizable compact Polish space.
AN := {h : h is an adapted process , h(·) ∈ DN a.s.}
![Page 149: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/149.jpg)
The way pointed out by Budhiraja and Duuis
Let Yn : B 7→ X , n = 1, 2, · · · . If there exists Y0 : H 7→ X suchthat
A For any N > 0, if a family {hn} ⊂ AN (as random variables inDN ) converges in distribution to h ∈ AN , then for some
subsequence nk, Ynk
(w + hnk
(w)√n−1
k
)converges in distribution to
Y0(h) in X .
B For any N > 0, if {hn, n ∈ N} ⊂ AN converges weakly toh ∈ H, then for some subsequence hnk
, Y0(hnk) converges
weakly to Y0(h) in X .
![Page 150: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/150.jpg)
The way pointed out by Budhiraja and Duuis
Let Yn : B 7→ X , n = 1, 2, · · · .
If there exists Y0 : H 7→ X suchthat
A For any N > 0, if a family {hn} ⊂ AN (as random variables inDN ) converges in distribution to h ∈ AN , then for some
subsequence nk, Ynk
(w + hnk
(w)√n−1
k
)converges in distribution to
Y0(h) in X .
B For any N > 0, if {hn, n ∈ N} ⊂ AN converges weakly toh ∈ H, then for some subsequence hnk
, Y0(hnk) converges
weakly to Y0(h) in X .
![Page 151: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/151.jpg)
The way pointed out by Budhiraja and Duuis
Let Yn : B 7→ X , n = 1, 2, · · · . If there exists Y0 : H 7→ X suchthat
A For any N > 0, if a family {hn} ⊂ AN (as random variables inDN ) converges in distribution to h ∈ AN , then for some
subsequence nk, Ynk
(w + hnk
(w)√n−1
k
)converges in distribution to
Y0(h) in X .
B For any N > 0, if {hn, n ∈ N} ⊂ AN converges weakly toh ∈ H, then for some subsequence hnk
, Y0(hnk) converges
weakly to Y0(h) in X .
![Page 152: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/152.jpg)
The way pointed out by Budhiraja and Duuis
Let Yn : B 7→ X , n = 1, 2, · · · . If there exists Y0 : H 7→ X suchthat
A For any N > 0, if a family {hn} ⊂ AN (as random variables inDN ) converges in distribution to h ∈ AN , then for some
subsequence nk, Ynk
(w + hnk
(w)√n−1
k
)converges in distribution to
Y0(h) in X .
B For any N > 0, if {hn, n ∈ N} ⊂ AN converges weakly toh ∈ H, then for some subsequence hnk
, Y0(hnk) converges
weakly to Y0(h) in X .
![Page 153: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/153.jpg)
The way pointed out by Budhiraja and Duuis
Let Yn : B 7→ X , n = 1, 2, · · · . If there exists Y0 : H 7→ X suchthat
A For any N > 0, if a family {hn} ⊂ AN (as random variables inDN ) converges in distribution to h ∈ AN , then for some
subsequence nk, Ynk
(w + hnk
(w)√n−1
k
)converges in distribution to
Y0(h) in X .
B For any N > 0, if {hn, n ∈ N} ⊂ AN converges weakly toh ∈ H, then for some subsequence hnk
, Y0(hnk) converges
weakly to Y0(h) in X .
![Page 154: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/154.jpg)
Theorem: Under (A) and (B), the LP holds:
limn→∞
log E{exp[−nf(Yn)]} = − infx∈X
{g(x) + I(x)}
![Page 155: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/155.jpg)
Verification of (A) and (B)
We replace n by ε−1 and we take
Y0 = X
Yε := Xε,hε .
Verification of (A).Now Yε satisfies:
Yε(t) = x +∫ t
0b(Yε(s)) ds +
∫ t
0σ(Yε(s))hε(s) ds
+√
ε
∫ t
0σ(Yε(s)) dW (s)−Kε(t)
Y0(t) = x +∫ t
0b(Y0(s)) ds +
∫ t
0σ(Y0(s))h(s) ds−K0(t).
![Page 156: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/156.jpg)
Verification of (A) and (B)
We replace n by ε−1 and we take
Y0 = X
Yε := Xε,hε .
Verification of (A).Now Yε satisfies:
Yε(t) = x +∫ t
0b(Yε(s)) ds +
∫ t
0σ(Yε(s))hε(s) ds
+√
ε
∫ t
0σ(Yε(s)) dW (s)−Kε(t)
Y0(t) = x +∫ t
0b(Y0(s)) ds +
∫ t
0σ(Y0(s))h(s) ds−K0(t).
![Page 157: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/157.jpg)
Verification of (A) and (B)
We replace n by ε−1 and we take
Y0 = X
Yε := Xε,hε .
Verification of (A).Now Yε satisfies:
Yε(t) = x +∫ t
0b(Yε(s)) ds +
∫ t
0σ(Yε(s))hε(s) ds
+√
ε
∫ t
0σ(Yε(s)) dW (s)−Kε(t)
Y0(t) = x +∫ t
0b(Y0(s)) ds +
∫ t
0σ(Y0(s))h(s) ds−K0(t).
![Page 158: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/158.jpg)
Verification of (A) and (B)
We replace n by ε−1 and we take
Y0 = X
Yε := Xε,hε .
Verification of (A).Now Yε satisfies:
Yε(t) = x +∫ t
0b(Yε(s)) ds +
∫ t
0σ(Yε(s))hε(s) ds
+√
ε
∫ t
0σ(Yε(s)) dW (s)−Kε(t)
Y0(t) = x +∫ t
0b(Y0(s)) ds +
∫ t
0σ(Y0(s))h(s) ds−K0(t).
![Page 159: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/159.jpg)
Verification of (A) and (B)
We replace n by ε−1 and we take
Y0 = X
Yε := Xε,hε .
Verification of (A).
Now Yε satisfies:
Yε(t) = x +∫ t
0b(Yε(s)) ds +
∫ t
0σ(Yε(s))hε(s) ds
+√
ε
∫ t
0σ(Yε(s)) dW (s)−Kε(t)
Y0(t) = x +∫ t
0b(Y0(s)) ds +
∫ t
0σ(Y0(s))h(s) ds−K0(t).
![Page 160: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/160.jpg)
Verification of (A) and (B)
We replace n by ε−1 and we take
Y0 = X
Yε := Xε,hε .
Verification of (A).Now Yε satisfies:
Yε(t) = x +∫ t
0b(Yε(s)) ds +
∫ t
0σ(Yε(s))hε(s) ds
+√
ε
∫ t
0σ(Yε(s)) dW (s)−Kε(t)
Y0(t) = x +∫ t
0b(Y0(s)) ds +
∫ t
0σ(Y0(s))h(s) ds−K0(t).
![Page 161: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/161.jpg)
Verification of (A) and (B)
We replace n by ε−1 and we take
Y0 = X
Yε := Xε,hε .
Verification of (A).Now Yε satisfies:
Yε(t) = x +∫ t
0b(Yε(s)) ds +
∫ t
0σ(Yε(s))hε(s) ds
+√
ε
∫ t
0σ(Yε(s)) dW (s)−Kε(t)
Y0(t) = x +∫ t
0b(Y0(s)) ds +
∫ t
0σ(Y0(s))h(s) ds−K0(t).
![Page 162: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/162.jpg)
Verification of (A) and (B)
We replace n by ε−1 and we take
Y0 = X
Yε := Xε,hε .
Verification of (A).Now Yε satisfies:
Yε(t) = x +∫ t
0b(Yε(s)) ds +
∫ t
0σ(Yε(s))hε(s) ds
+√
ε
∫ t
0σ(Yε(s)) dW (s)−Kε(t)
Y0(t) = x +∫ t
0b(Y0(s)) ds +
∫ t
0σ(Y0(s))h(s) ds−K0(t).
![Page 163: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/163.jpg)
Note (A) is equivalent to the apparently weaker condition
A’ For any N > 0, if a family {hn} ⊂ AN (as random variables inDN ) converges almost surely to h ∈ AN , in H, then for some
subsequence nk, Ynk
(w + hnk
(w)√n−1
k
)converges in distribution to
Y0(h) in X .
This equivalence can be easily achieved by using Skorohodrepresentation.
![Page 164: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/164.jpg)
Note (A) is equivalent to the apparently weaker condition
A’ For any N > 0, if a family {hn} ⊂ AN (as random variables inDN ) converges almost surely to h ∈ AN , in H, then for some
subsequence nk, Ynk
(w + hnk
(w)√n−1
k
)converges in distribution to
Y0(h) in X .
This equivalence can be easily achieved by using Skorohodrepresentation.
![Page 165: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/165.jpg)
Note (A) is equivalent to the apparently weaker condition
A’ For any N > 0, if a family {hn} ⊂ AN (as random variables inDN ) converges almost surely to h ∈ AN , in H, then for some
subsequence nk, Ynk
(w + hnk
(w)√n−1
k
)converges in distribution to
Y0(h) in X .
This equivalence can be easily achieved by using Skorohodrepresentation.
![Page 166: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/166.jpg)
Hence, to prove (A), it suffices to prove that
Yε −→ Y0, weakly
if hε −→ h0 almost surely in H. This can be done by Ito calculus.Similarly, B can be verified.
![Page 167: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/167.jpg)
Hence, to prove (A), it suffices to prove that
Yε −→ Y0, weakly
if hε −→ h0 almost surely in H.
This can be done by Ito calculus.Similarly, B can be verified.
![Page 168: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/168.jpg)
Hence, to prove (A), it suffices to prove that
Yε −→ Y0, weakly
if hε −→ h0 almost surely in H. This can be done by Ito calculus.
Similarly, B can be verified.
![Page 169: Large Deviations for Multi-valued Stochastic Differential ...jiagang.pdf · Large Deviations for Multi-valued Stochastic Differential Equations Joint work with Siyan Xu, Xicheng](https://reader035.vdocuments.mx/reader035/viewer/2022071006/5fc398c72bb21d53085e8f62/html5/thumbnails/169.jpg)
Hence, to prove (A), it suffices to prove that
Yε −→ Y0, weakly
if hε −→ h0 almost surely in H. This can be done by Ito calculus.Similarly, B can be verified.