stochastic functional differential equations · 2011-08-31 · stochastic functional di erential...

Stochastic functional differential equations Explicit Solutions and the method of steps Ito formulas and Fokker-Planck equations Oscillatory behaviour Stability issues Open Problems

Stochastic Functional Differential Equations

Evelyn Buckwar

Heriot-Watt University, until 31st of August 2011, then off to JKU Linz!

MFO Oberwolfach, 26th August 2011


Stochastic functional differential equations

X (t) = X (0) +∫ t

0 F (s,Xs)ds +∫ t

0 G (s,Xs)dW (s)

• memory functional or segment processXt(u) = X (t + u) : u ∈ [−τ, 0] for Xt ∈ C ([−τ, 0]; Rn);

• initial data: X (t) = ψ(t) for t ∈ [−τ, 0];

• coefficients: (globally Lipschitz)F : [0,T ]× C ([−τ, 0]; Rn)→ Rn,G = (G1, . . . ,Gm) : [0,T ]× C ([−τ, 0]; Rn)→ Rn×m;

• Wiener process: W = W (t, ω), t ∈ [0,T ], ω ∈ Ω is anm-dim. Wiener process on (Ω,F , Ftt∈[0,T ],P).


The memory functional often takes special forms, e.g., by settingF and/or G to

H1(s,X (s)), (no delay)

H2(s,X (s),X (s − τ)), (constant/discrete delay)

H3(s,X (s),X (s − τ(s))), (variable delay)

H4(s,X (s),

0∫−τ

K (s, u,X (s + u))du), (distributed delay).

We assume that there exists a path-wise unique strong solutionX (·) of the above equation. (Analysis: Ito, Nisio (1964); Mao;Mohammed)


Some references to SDDEs in Biosciences

• Human pupil light reflex (Longtin et al, 90)

• Human postural sway (Eurich et al, 96)

• Neurological diseases (Beuter et al, 93, Tass et al 05)

• Infectious diseases (Beretta et al, 98)

• Chemical kinetics (Burrage et al, 07)

• Population dynamics (Carletti, Beretta, Tapaswi &Mukhopadhyay);

• Neural field models (Hutt et al)

• Cowan-Wilson networks with delayed feedback (Jirsa et al)

• FitzHugh-Nagumo networks with delayed feedback (Scholl etal)

• Hematopoietic Stem Cell Regulation System (Mackey et al,07)

• .... and many more


Explicit Solutions and the method of steps


Solutions of stochastic differential equations

• dX (t)=aX (t)dt+bdW (t), BX (t)=eat(1 + b∫ t

0 e−asdW (s))

• dX (t)=aX (t)dt+bX (t)dW (t),BX (t)=exp((a− 1

2b2)t + bW (t))


Solutions of SDDEs via “Method of steps”B dX (t)=X (t − 1)dt+βdW (t), t ≥ 0 andX (t)=Φ1(t)=1 + t, t ∈ [−1, 0]

→ t ∈ [0, 1] dX (t) = Φ1(t − 1)dt + βdW (t) = t dt + βdW (t)

⇒ X (t) = 1 +t2

2+ βW (t) =: Φ2(t)

→ t ∈ [1, 2] dX (t) = Φ2(t − 1)dt + βdW (t)

⇒ X (t) =1

6t3 − 1

2t2 +

3

2t +

1

3+ β(

∫ t

1

W (s − 1)ds + W (t))


Ito formulas and Fokker-Planck equations


Stochastic ordinary differential equations

Consider

X (t) = X (0) +

∫ t0 F (X (s)) ds +

∫ t0 G (X (s)) dW (s)

Ito formula for φ(x) function, suff. differentiable, everything scalar:

φ(X (t)) = φ(X (0))+

∫ t

0φx (X (s))F (X (s))ds+φx (X (s))G (X (s))dW (s)

+1

2

∫ t

0φxx (X (s)) G 2(X (s)) ds


Ito-formula, distributed delay

X (t) = X (0) +∫ t

0F (s,X (s),Y (s)) ds +

∫ t

0G (s,X (s),Y (s)) dW (s)

Y (t) =t∫

t−τ

K (t, s − t,X (s)) ds

(A. Friedman (1975), M. Arriojas, PhD thesis (1997)):

φ[t]− φ[t ′] =

∫ t

t′D1φ[s] + D2φ[s]F [s] +

1

2D2

2φ[s](G [s])2

+ D3φ[s]a2(s) ds +

∫ t

t′D2φ[s] G [s] dW (s)

a2(t) := ddt

t∫t−τ

K (t, s − t,X (s)) ds =

t∫t−τ

D1K (t, s − t,X (s)− D2K (t, s − t,X (s)) ds

+K (t, 0,X (t))− K (t,−τ,X (t − τ))

f [t] := f (t,X (t),Y (t)), Di derivative w.r.t. the i ’th argument


Ito-formula, discrete delay

Ito-formula from Hu, Mohammed & Yan (AoP 32(1A), 2004)for SDDE with m = 1, n = 1, r = 2, s = 2, τ1 = σ1 = 0dX (t)=F (X (t),X (t − τ2))dt + G (X (t),X (t − σ2))dW (t), t > 0X (t) = ψ(t), −τ < t < 0, τ := τ2 ∨ σ2,and function φ(X (t),X (t − δ)) δ > 0


dφ(X (t),X (t − δ)

)=

∂φ

∂x2(X (t),X (t − δ)) 1[0,δ)(t)dψ(t − δ)

+∂φ

∂x2(X (t),X (t − δ))1[δ,∞)(t)

[F

(X (t − δ),X (t − τ2 − δ)

)dt

+G(X (t − δ),X (t − σ2 − δ)

)dW (t − δ)

]+

∂2φ

∂x1∂x2

(X (t),X (t − δ)

)G

(X (t − δ),X (t − σ2 − δ)

)1[δ,∞)(t)Dt−δX (t)dt

+1

2

∂2φ

∂x22

(X (t),X (t − δ)

)G

(X (t − δ),X (t − σ2 − δ)

)21[δ,∞)(t)dt

+∂φ

∂x1

(X (t),X (t − δ)

) [F

(X (t),X (t − τ2)

)dt + G

(X (t),X (t − σ2)

)dW (t)

]+

1

2

∂2φ

∂x21

(X (t),X (t − δ)

)G

(X (t),X (t − σ2)

)2dt,


Fokker-Planck equation etc. :

Problems arising due to: infinite dimensional nature of the segmentprocess, the segment process is in general not a semi-martingale,non-Markovian structure of the solution of an SFDE.

• an approximate Fokker-Planck equation using the method ofsteps is presented by A. Longtin et al, Physical Review E, 1999

• weak infinitesimal generator of SFDEs and a Feynman-Kacformula are derived in SEA Mohammed and F Yan, StochasticAnal. Appl. 2005, using Malliavin calculus


Oscillatory behaviour


Oscillatory Behaviour

Deterministic theory (math.):

Well-known: DDEs display a rich variety of dynamical behaviour,already in the scalar case:Oscillations, bifurcations, multi-stability, etc...Books by Gopalsamy (1992) and Ladde, Lakshmikantham &Zhang (1987) and lots of articles on the subject exist.


Stochastic theory

Physics literature: many references (Garcia-Ojalvo and Roy,Mackey and Glass, Longtin, etc.)Mathematical literature: SODEs (e.g., Mao 1997, Baxendale,Namachchivaya,..)

SDDEs????

What does a mathematician do? Take a simple example.....


What is the effect of the delays?

Example:

y ′(t) = λy(t), t ≥ 0, λ ∈ R, y(0) = 1

versus

x ′(t) = λx(t) + µx(t − 1), t ≥ 0, λ, µ ∈ R, x(t) = 1 + t, t ≤ 0


Results for a simple example, joint work with J. Appleby

dX (t) = (aX (t) + bX (t − τ)) dt + σX (t) dW (t)

X (t) = ψ(t), −τ ≤ t ≤ 0,

I With (Φ(t))t≥−τ given by Φ(t) = 1 for t ∈ [−τ, 0] andΦ(t) = exp((a− σ2/2)t + σW (t)) for t ≥ 0

(Φ(t) solves dΦ(t) = aΦ(t)dt + σΦ(t) dW (t))

set Y (t) = X (t)/Φ(t)

then Y (t) = Y (0) +∫ t

0b Y (s − τ) Φ(s − τ) Φ(s)−1 ds, t ≥ 0,

(X (t))t≥0 satisfies

X (t) = Φ(t)(ψ(0) +

∫ t

0b X (s − τ) Φ(s)−1 ds

).

Now Y ∈ C 1((0,∞); R), thusY ′(t) = b Φ(t − τ) Φ(t)−1Y (t − τ)


Now, for Y ′(t) = b Φ(t − τ) Φ(t)−1Y (t − τ)use path-wise results for deterministic DDE:I b > 0, ψ(t) ≥ 0,⇒ solutions of SDDE are always positive and

non-oscillatoryI b < 0, any ψ,⇒ solutions of the SDDE are always oscillatory

for σ 6= 0.For σ = 0 there are parameter ranges where non-oscillatorysolutions exist!Further, in contrast to the zeros of the Wiener process andadditive noise SODEs and SDDEs, the zeros of the solution X areat least a τ apart.more general approach to SDDE→ RDDE: H Lisei, Conjugation of Flows for Stochastic and Random Functional

Differential Equations. Stochastics and Dynamics 1(2), 283-298 (2001)


Stability issues


Lyapunov stability of equilibrium solutions of ODEs

Consider ODE (1) x ′(t) = f (x(t)), x(0) = x0,and assume (the constant function) xE is an equilibrium solution of(1)I then the equilibrium xE is said to be Lyapunov stable, if for

every ε > 0, there exists a δ = δ(ε), such that for some norm ‖.‖

if ‖x(0)− xE‖ < δ, then ‖x(t)− xE‖ < ε for all t ≥ 0;

I the equilibrium is said to be asymptotically Lyapunov stable, ifit is Lyapunov stable and there exists a δ > 0 such that

if ‖x(0)− xE‖ < δ, then limt→∞

‖x(t)− xE‖ = 0 .


What do we consider as equilibrium solutions of SDDEs?

dX (t) = F (X (t),X (t − τ))dt + G (X (t),X (t − τ))dW (t)

Obvious choice: a constant function XE such thatF (XE ,XE ) = G (XE ,XE ) = 0.

Look at some example models......


Time-delayed feedback in neurosystems Scholl et al, 2009

FitzHugh-Nagumo systemε1x′1(t) = x1(t)− x3

1 (t)/3− y1(t) + C [x2(t − τ)− x1(t)]

y ′1(t) = x1(t) + a + D1ξ1(t)

ε2x′2(t) = x2(t)− x3

2 (t)/3− y2(t) + C [x1(t − τ)− x2(t)]

y ′2(t) = x2(t) + a + D2ξ2(t)

x1, y1, x2, y2 correspond to single neurons or populations, linearlycoupled with coupling strength Cx1, x2 are related to transmembrane voltage, y1, y2 are connected toelectrical conductance of ion currentsa excitability parameter, a > 1 excitable, a < 1 self-sustained periodicfiringεi time-scale parameters 1, fast activator variables xi , y1 slow inhibitorvariables

ξ1, ξ2 Gaussian white noise, mean zero, unit variance, noise intensities

D1, D2


Time-delayed feedback in neurosystems Scholl et al, 2009

FitzHugh-Nagumo systemε1x′1(t) = x1(t)− x3

1 (t)/3− y1(t) + C [x2(t − τ)− x1(t)]

y ′1(t) = x1(t) + a + D1ξ1(t)

ε2x′2(t) = x2(t)− x3

2 (t)/3− y2(t) + C [x1(t − τ)− x2(t)]

y ′2(t) = x2(t) + a + D2ξ2(t)

Modelling approach: deterministic system + extrinsic additive noise

Equilibrium solution of deterministic system(fix ε1 = ε2 = 0.01 and a = 1.05):

xE = (xE ,1, yE ,1, xE ,2, yE ,2) with xE ,i = −a and yE ,i = a3/3− a

. . . which is not an equilibrium solution of the stochastic system


Infectious Diseases Model Beretta et al, 1998

dS(t)

dt= (−βS(t)

∫ h

0

f (s)I (t − s)ds − µ1S(t) + b)dt

+σ1(S(t)− SE )dW1(t)

dI (t)

dt= (βS(t)

∫ h

0

f (s)I (t − s)ds − (µ2 + λ)I (t))dt

+σ2(I (t)− IE )dW2(t)

dR(t)

dt= (λI (t)− µ3R(t))dt + σ3(R(t)− RE )dW3(t) .

Modelling approach: deterministic system + white noiseperturbations proportional to distances S(t)− SE , etc.


Example: Hematopoietic Stem Cell RegulationSystem Mackey et al, 2007

dQ(t)

dt= (− b1Q(t)

1 + Q4(t)− δQ(t) +

b1µ1Q(t − 1)

1 + Q4(t − 1))dt

+σQ(t)dW (t) .

Non-dimensional form with Q modelling quiescent stem cells withmaturation delayModelling approach: deterministic system + white noiseperturbations around mean of parameter δ

Equilibrium solution of deterministic system:

QE =

(b1(µ1 − 1)

δ− 1

)4

. . . which is not an equilibrium solution of the stochastic system


Chemical Kinetics (genetic regulatory network)Burrage et al, 1998

Chemical Langevin Equation with Delay

dX (t) =M1∑j=1

υjaj (X (t − Tj )dt +M∑

j=Mj+1

υjaj (X (t)dt +N∑

j=1

bj (X (t))dWj (t) .

aj propensity functions, υj state change vector, bj is a column vector of aprincipal root of a certain matrix

Has deterministic equilibrium solution XE ≡ 0

Modeling approach: As limit of a diffusion approximation of a discretestochastic system

See Diffusion approximation of birth-death processes: Comparison in terms of large deviations and exit points, K

Pakdaman, M Thieullen, G Wainrib, Statistics and Probability Letters 80 (2010) 1121-1127, for a discussion


What do we consider as equilibrium solutions ofSDDEs?

I Observation: Transition from deterministic to stochastic systemmay be equilibrium preserving or not!

I Possibility: Consider stability of deterministic equilibrium XE

(Mackey et al, 2007)⇒ NOT the same concept as standard Lyapunov stability as XE

not even a solution of the stochastic system! Practically nomathematical theory developed.

I However: A stochastic system may have a ’stochasticequilibrium’....


Stationary solutions as stochastic equilibrium

General additive noise equation:

dX (t) = f (X (t)) dt + σ dW (t) . (1)

A special case:

dX (t) = −λX (t)dt + σdW (t), (2)


The Ornstein-Uhlenbeck stochastic stationaryprocess

The linear, scalar SDE (2) has the explicit solution

X (t) = e−λ(t−t0)X0 + σe−λt

∫ t

t0

eλsdW (s) (3)

for all t ≥ t0 and the initial value X (t0) = X0. This expression has no forwardlimit but the pathwise pullback limit (i.e., as t0 → −∞ with t fixed) exists andis given by

O(t) := σe−λt

∫ t

−∞eλsdW (s), (4)

which is known as the scalar Ornstein-Uhlenbeck stochastic stationary process.

It is Gaussian with zero mean and variance σ2/2λ. (This requires the use of a

two-sided Wiener process, i.e., with W (t) defined for all t ∈ R rather than just

for t ∈ R+).


The Ornstein-Uhlenbeck stochastic stationaryprocess

The Ornstein-Uhlenbeck process (4) is a stochastic stationary solution of thelinear SDE (2). Moreover, it attracts all other solutions of this SDE forwards intime in the pathwise sense. To see this simply subtract one solution of (2)from another to obtain

X 1(t)− X 2(t) = e−λ(t−t0)(X 1

0 − X 20

),

and then replace X 2(t) by the Ornstein-Uhlenbeck process O(t).


Stationary solutions of SDDEs

• Ito, K.; Nisio, M., On stationary solutions of a stochastic differentialequation. J. Math. Kyoto Univ. 4, 1–75 (1964).

• Kuchler, U.; Mensch, B., Langevins stochastic differential equationextended by a time-delayed term. Stochastics 40(1), 23–42 (1992).

• Mohammed, Salah EA , book and articles

• Scheutzow, M, several articles

• Bakhtin, Yu; Mattingly, JC., Stationary solutions of stochasticdifferential equations with memory and stochastic partial differentialequations. Commun. Contemp. Math. 7, No. 5, 553-582 (2005)


Example: stochastic logistic DDE Scheutzow, 1984

The stochastic logistic DDE

dX (t) = (k1 − k2Xk3(t − 1))X (t)dt + k4 X (t) dW (t)

has a unique, positive-valued stationary solution, ifk1 − k2

4/2, k2, k3, k4 are all positive. All its moments are finite.


Next choice: Norm

Definition

1. The equilibrium solution XS of an SDDE is mean-squarestable/a.s. stable if and only if, for each ε > 0, there exists aδ ≥ 0 such that

E|X (t)− XS |p < ε, t ≥ 0, / |X (t)− XS | < ε, t ≥ 0, a.s.

whenever E|X (0)− XS |p < δ / |X (0)− XS | < δ;

2. The equilibrium solution is asymptotically mean-squarestable/a.s. stable if and only if it is mean-squarestable/a.s. stable, and for all X (0)− XS ∈ R,

limt→∞

E|X (t)− XS |p = 0 / limt→∞

X (t)− XS = 0 a.s.


Mean-square vs. Almost sure stability

d(

X1(t)X2(t)

)=

(λ 00 λ

)(X1(t)X2(t)

)dt +

(σ 00 σ

)(X1(t)X2(t)

)dW (t)

Stability conditions: MS-stab λ+ 12σ

2 < 0, a.s. stab λ− 12σ

2 < 0Choose λ = 0.1, σ = 0.5, we see that

λ+1

2σ2 = 0.225 > 0, λ− 1

2σ2 = −0.025 < 0,

and therefore the equilibrium solution XE ≡ 0 is simultaneouslymean-square unstable and a.s. asymptotically stable.


Figure: Simulations of System above with λ = 0.1, σ = 0.5. X1 and X2

represent the components of a single path of the simulation, whereasMS-X1 and MS-X2 represent the estimated mean-square norm of eachsolution component.


Existing techniques

I Lyapunov functional techniques, mostly MS-stab: VBKolmanovskii, L ShaikhetI Razumikhin techniques, LaSalle principle, MS-stab & a.s. stab:

X MaoI Random dynamical systems techniques (i.e., pathwise results)

giving Lyapunov exponents: SEA Mohammed, M ScheutzowI Random dynamical systems techniques for SPDEs with

memory, also considering ms and a.s. stab of stationary solutions:T Caraballo, M Garrido-Atienza, B SchmalfussI Halanay inequalities, MS stab.: CTH Baker & EB


Open problems

• Linear and nonlinear stability analysis for SDDEs arising, e.g.,in the models above

• Investigating oscillatory and other dynamics for SDDEs

• Averaging techniques

• Amplitude equations

• Large deviations

stochastic functional differential equations · 2011-08-31 · stochastic functional di erential...

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