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Trivariate Density Revisited
Steven E. ShreveCarnegie Mellon University
–Conference in Honor of Ioannis Karatzas
Columbia UniversityJune 4–8, 2012
June 1, 2012
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Outline
Rank-based diffusion
Bang-bang control
Trivariate density
Personal reflections
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Rank-based diffusion
E. R. Fernholz, T. Ichiba, I. Karatzas & V. Prokaj, Planardiffusions with rank-based characteristics and perturbed Tanakaequation, Probability Theory and Related Fields, to appear.
dX1 = −h dt + ρ dB1
dX2 = g dt + σ dB2
dX1 = g dt + σ dB1
dX2 = −h dt + ρ dB2
ρ2 + σ2 = 1g + h > 0
B1, B2 independent Br. motions
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Remarks
Karatzas, et. al. examine:
I Weak and strong existence and uniqueness in law;
I Properties of X1 ∨ X2 and X1 ∧ X2;
I The reversed dynamics of (X1,X2);
I Transition probabilities (X1,X2).
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Remarks
Karatzas, et. al. examine:
I Weak and strong existence and uniqueness in law;
I Properties of X1 ∨ X2 and X1 ∧ X2;
I The reversed dynamics of (X1,X2);
I Transition probabilities (X1,X2).
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The difference processDefine
Y (t) = X1(t)− X2(t).
Then
dY = lI{Y≤0}(g + h) dt − lI{Y>0}(g + h) dt
+lI{Y≤0}σ dB1 − lI{Y≤0}ρ dB2 + lI{Y>0}ρ dB1 − lI{Y>0}σ dB2
= −λ sgn(Y (t)
)dt + dW ,
where
λ = g + h > 0,
dW = σ(lI{Y≤0} dB1 − lI{Y>0} dB2
)︸ ︷︷ ︸dW1
+ρ(lI{Y>0} dB1 − lI{Y≤0} dB2
)︸ ︷︷ ︸dW2
.
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The difference processDefine
Y (t) = X1(t)− X2(t).
Then
dY = lI{Y≤0}(g + h) dt − lI{Y>0}(g + h) dt
+lI{Y≤0}σ dB1 − lI{Y≤0}ρ dB2 + lI{Y>0}ρ dB1 − lI{Y>0}σ dB2
= −λ sgn(Y (t)
)dt + dW ,
where
λ = g + h > 0,
dW = σ(lI{Y≤0} dB1 − lI{Y>0} dB2
)︸ ︷︷ ︸dW1
+ρ(lI{Y>0} dB1 − lI{Y≤0} dB2
)︸ ︷︷ ︸dW2
.
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The difference processDefine
Y (t) = X1(t)− X2(t).
Then
dY = lI{Y≤0}(g + h) dt − lI{Y>0}(g + h) dt
+lI{Y≤0}σ dB1 − lI{Y≤0}ρ dB2 + lI{Y>0}ρ dB1 − lI{Y>0}σ dB2
= −λ sgn(Y (t)
)dt + dW ,
where
λ = g + h > 0,
dW = σ(lI{Y≤0} dB1 − lI{Y>0} dB2
)︸ ︷︷ ︸dW1
+ρ(lI{Y>0} dB1 − lI{Y≤0} dB2
)︸ ︷︷ ︸dW2
.
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The difference processDefine
Y (t) = X1(t)− X2(t).
Then
dY = lI{Y≤0}(g + h) dt − lI{Y>0}(g + h) dt
+lI{Y≤0}σ dB1 − lI{Y≤0}ρ dB2 + lI{Y>0}ρ dB1 − lI{Y>0}σ dB2
= −λ sgn(Y (t)
)dt + dW ,
where
λ = g + h > 0,
dW = σ(lI{Y≤0} dB1 − lI{Y>0} dB2
)︸ ︷︷ ︸dW1
+ρ(lI{Y>0} dB1 − lI{Y≤0} dB2
)︸ ︷︷ ︸dW2
.
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The difference processDefine
Y (t) = X1(t)− X2(t).
Then
dY = lI{Y≤0}(g + h) dt − lI{Y>0}(g + h) dt
+lI{Y≤0}σ dB1 − lI{Y≤0}ρ dB2 + lI{Y>0}ρ dB1 − lI{Y>0}σ dB2
= −λ sgn(Y (t)
)dt + dW ,
where
λ = g + h > 0,
dW = σ(lI{Y≤0} dB1 − lI{Y>0} dB2
)︸ ︷︷ ︸dW1
+ρ(lI{Y>0} dB1 − lI{Y≤0} dB2
)︸ ︷︷ ︸dW2
.
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The sum processDefine
Z (t) = X1(t) + X2(t).
Then
dZ = (g − h) dt + lI{Y≤0}σ dB1 + lI{Y≤0}ρ dB2
+lI{Y>0}ρ dB1 + lI{Y>0}σ dB2
= ν dt + dV ,
where
ν = g − h,
dV = σ(lI{Y≤0} dB1 + lI{Y>0} dB2
)︸ ︷︷ ︸dV1
+ρ(lI{Y>0} dB1 + lI{Y≤0} dB2
)︸ ︷︷ ︸dV2
.
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The sum processDefine
Z (t) = X1(t) + X2(t).
Then
dZ = (g − h) dt + lI{Y≤0}σ dB1 + lI{Y≤0}ρ dB2
+lI{Y>0}ρ dB1 + lI{Y>0}σ dB2
= ν dt + dV ,
where
ν = g − h,
dV = σ(lI{Y≤0} dB1 + lI{Y>0} dB2
)︸ ︷︷ ︸dV1
+ρ(lI{Y>0} dB1 + lI{Y≤0} dB2
)︸ ︷︷ ︸dV2
.
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The sum processDefine
Z (t) = X1(t) + X2(t).
Then
dZ = (g − h) dt + lI{Y≤0}σ dB1 + lI{Y≤0}ρ dB2
+lI{Y>0}ρ dB1 + lI{Y>0}σ dB2
= ν dt + dV ,
where
ν = g − h,
dV = σ(lI{Y≤0} dB1 + lI{Y>0} dB2
)︸ ︷︷ ︸dV1
+ρ(lI{Y>0} dB1 + lI{Y≤0} dB2
)︸ ︷︷ ︸dV2
.
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The sum processDefine
Z (t) = X1(t) + X2(t).
Then
dZ = (g − h) dt + lI{Y≤0}σ dB1 + lI{Y≤0}ρ dB2
+lI{Y>0}ρ dB1 + lI{Y>0}σ dB2
= ν dt + dV ,
where
ν = g − h,
dV = σ(lI{Y≤0} dB1 + lI{Y>0} dB2
)︸ ︷︷ ︸dV1
+ρ(lI{Y>0} dB1 + lI{Y≤0} dB2
)︸ ︷︷ ︸dV2
.
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The sum processDefine
Z (t) = X1(t) + X2(t).
Then
dZ = (g − h) dt + lI{Y≤0}σ dB1 + lI{Y≤0}ρ dB2
+lI{Y>0}ρ dB1 + lI{Y>0}σ dB2
= ν dt + dV ,
where
ν = g − h,
dV = σ(lI{Y≤0} dB1 + lI{Y>0} dB2
)︸ ︷︷ ︸dV1
+ρ(lI{Y>0} dB1 + lI{Y≤0} dB2
)︸ ︷︷ ︸dV2
.
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Summary
X1(t) + X2(t) = Z (t)
= X1(0) + X2(0) + νt + V (t),
X1(t)− X2(t) = Y (t)
= X1(0) + X2(0)− λ∫ t
0sgn(Y (s)
)ds + W (t),
where V and W are correlated Brownian motions.
I To determine transition probabilities for (X1,X2), it suffices todetermine transition probabilites for (Z ,Y ).
I (Z ,W ) is a Gaussian process.I Y is determined by W by
dY (t) = −λ sgn(Y (t)
)dt + dW (t).
I It suffices to determine the distribution of (W (t),Y (t)).
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Summary
X1(t) + X2(t) = Z (t)
= X1(0) + X2(0) + νt + V (t),
X1(t)− X2(t) = Y (t)
= X1(0) + X2(0)− λ∫ t
0sgn(Y (s)
)ds + W (t),
where V and W are correlated Brownian motions.
I To determine transition probabilities for (X1,X2), it suffices todetermine transition probabilites for (Z ,Y ).
I (Z ,W ) is a Gaussian process.I Y is determined by W by
dY (t) = −λ sgn(Y (t)
)dt + dW (t).
I It suffices to determine the distribution of (W (t),Y (t)).
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Summary
X1(t) + X2(t) = Z (t)
= X1(0) + X2(0) + νt + V (t),
X1(t)− X2(t) = Y (t)
= X1(0) + X2(0)− λ∫ t
0sgn(Y (s)
)ds + W (t),
where V and W are correlated Brownian motions.
I To determine transition probabilities for (X1,X2), it suffices todetermine transition probabilites for (Z ,Y ).
I (Z ,W ) is a Gaussian process.I Y is determined by W by
dY (t) = −λ sgn(Y (t)
)dt + dW (t).
I It suffices to determine the distribution of (W (t),Y (t)).
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Summary
X1(t) + X2(t) = Z (t)
= X1(0) + X2(0) + νt + V (t),
X1(t)− X2(t) = Y (t)
= X1(0) + X2(0)− λ∫ t
0sgn(Y (s)
)ds + W (t),
where V and W are correlated Brownian motions.
I To determine transition probabilities for (X1,X2), it suffices todetermine transition probabilites for (Z ,Y ).
I (Z ,W ) is a Gaussian process.I Y is determined by W by
dY (t) = −λ sgn(Y (t)
)dt + dW (t).
I It suffices to determine the distribution of (W (t),Y (t)).
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Bang-bang control
Minimize
E∫ ∞
0e−tX 2
t dt,
Subject to
Xt = x +
∫ t
0us ds + Wt , a ≤ ut ≤ b, t ≥ 0.
Benes, Shepp & Witsenhausen (1980): Solution is ut = f (Xt),where
f (x) =
{b, if x < δ,a, if x ≥ δ,
and δ = (√
b2 + 2 + b)−1 − (√
a2 + 2− a)−1.What is the transition density for the controlled process X ?(Benes, et. al. computed its Laplace transform.)
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Bang-bang control
Minimize
E∫ ∞
0e−tX 2
t dt,
Subject to
Xt = x +
∫ t
0us ds + Wt , a ≤ ut ≤ b, t ≥ 0.
Benes, Shepp & Witsenhausen (1980): Solution is ut = f (Xt),where
f (x) =
{b, if x < δ,a, if x ≥ δ,
and δ = (√
b2 + 2 + b)−1 − (√
a2 + 2− a)−1.
What is the transition density for the controlled process X ?(Benes, et. al. computed its Laplace transform.)
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Bang-bang control
Minimize
E∫ ∞
0e−tX 2
t dt,
Subject to
Xt = x +
∫ t
0us ds + Wt , a ≤ ut ≤ b, t ≥ 0.
Benes, Shepp & Witsenhausen (1980): Solution is ut = f (Xt),where
f (x) =
{b, if x < δ,a, if x ≥ δ,
and δ = (√
b2 + 2 + b)−1 − (√
a2 + 2− a)−1.What is the transition density for the controlled process X ?(Benes, et. al. computed its Laplace transform.)
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Transition density by GirsanovCompute the transition density for X , where
Xt = x +
∫ t
0f (Xs) ds + Wt .
Start with a probability measure PX under which X is a Brownianmotion. Define W by
Wt = Xt − x −∫ t
0f (Xs) ds.
Change to a probability measure P under which W is a Brownianmotion. Compute the transition density for X under P.
P{Xt ∈ B} = EX
[lI{Xt∈B}
dPdPX
∣∣∣∣Ft
],
where
dPdPX
∣∣∣∣Ft
= exp
[∫ t
0f (Xs) dXs −
1
2
∫ t
0f 2(Xs) ds
].
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Transition density by GirsanovCompute the transition density for X , where
Xt = x +
∫ t
0f (Xs) ds + Wt .
Start with a probability measure PX under which X is a Brownianmotion. Define W by
Wt = Xt − x −∫ t
0f (Xs) ds.
Change to a probability measure P under which W is a Brownianmotion. Compute the transition density for X under P.
P{Xt ∈ B} = EX
[lI{Xt∈B}
dPdPX
∣∣∣∣Ft
],
where
dPdPX
∣∣∣∣Ft
= exp
[∫ t
0f (Xs) dXs −
1
2
∫ t
0f 2(Xs) ds
].
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Transition density by GirsanovCompute the transition density for X , where
Xt = x +
∫ t
0f (Xs) ds + Wt .
Start with a probability measure PX under which X is a Brownianmotion. Define W by
Wt = Xt − x −∫ t
0f (Xs) ds.
Change to a probability measure P under which W is a Brownianmotion. Compute the transition density for X under P.
P{Xt ∈ B} = EX
[lI{Xt∈B}
dPdPX
∣∣∣∣Ft
],
where
dPdPX
∣∣∣∣Ft
= exp
[∫ t
0f (Xs) dXs −
1
2
∫ t
0f 2(Xs) ds
].
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Transition density by GirsanovCompute the transition density for X , where
Xt = x +
∫ t
0f (Xs) ds + Wt .
Start with a probability measure PX under which X is a Brownianmotion. Define W by
Wt = Xt − x −∫ t
0f (Xs) ds.
Change to a probability measure P under which W is a Brownianmotion. Compute the transition density for X under P.
P{Xt ∈ B} = EX
[lI{Xt∈B}
dPdPX
∣∣∣∣Ft
],
where
dPdPX
∣∣∣∣Ft
= exp
[∫ t
0f (Xs) dXs −
1
2
∫ t
0f 2(Xs) ds
].
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Transition density by Girsanov (continued)To compute P{Xt ∈ B}, we need the joint distribution of
Xt ,
∫ t
0f (Xs) dXs ,
∫ t
0f 2(Xs) ds.
Assume without loss of generality that δ = 0. Define
F (x) =
∫ x
0f (ξ) dξ =
{bx , if x ≤ 0,ax , if x ≥ 0.
Tanaka’s formula implies
F (Xt) =
∫ t
0f (Xs) dXs +
1
2(a− b)LX
t
so ∫ t
0f (Xs) dXs = F (Xt)−
1
2(a− b)LX
t .
We need the joint distribution of
Xt , LXt ,
∫ t
0f 2(Xs) ds.
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Transition density by Girsanov (continued)To compute P{Xt ∈ B}, we need the joint distribution of
Xt ,
∫ t
0f (Xs) dXs ,
∫ t
0f 2(Xs) ds.
Assume without loss of generality that δ = 0. Define
F (x) =
∫ x
0f (ξ) dξ =
{bx , if x ≤ 0,ax , if x ≥ 0.
Tanaka’s formula implies
F (Xt) =
∫ t
0f (Xs) dXs +
1
2(a− b)LX
t
so ∫ t
0f (Xs) dXs = F (Xt)−
1
2(a− b)LX
t .
We need the joint distribution of
Xt , LXt ,
∫ t
0f 2(Xs) ds.
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Transition density by Girsanov (continued)To compute P{Xt ∈ B}, we need the joint distribution of
Xt ,
∫ t
0f (Xs) dXs ,
∫ t
0f 2(Xs) ds.
Assume without loss of generality that δ = 0. Define
F (x) =
∫ x
0f (ξ) dξ =
{bx , if x ≤ 0,ax , if x ≥ 0.
Tanaka’s formula implies
F (Xt) =
∫ t
0f (Xs) dXs +
1
2(a− b)LX
t
so ∫ t
0f (Xs) dXs = F (Xt)−
1
2(a− b)LX
t .
We need the joint distribution of
Xt , LXt ,
∫ t
0f 2(Xs) ds.
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Transition density by Girsanov (continued)
Define
Γ+(t) =
∫ t
0lI(0,∞)
(X (s)
)ds,
Γ−(t) =
∫ t
0lI(−∞,0)
(X (s)
)ds = t − Γ+(t).
Then∫ t
0f 2(Xs) ds = b2Γ−(t) + a2Γ+(t) = b2t + (a2 − b2)Γ+(t).
We need the joint distribution of
Xt , LXt , Γ+(t) =
∫ t
0lI(0,∞)(Xs) ds,
where X is a Brownian motion.
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Transition density by Girsanov (continued)
Define
Γ+(t) =
∫ t
0lI(0,∞)
(X (s)
)ds,
Γ−(t) =
∫ t
0lI(−∞,0)
(X (s)
)ds = t − Γ+(t).
Then∫ t
0f 2(Xs) ds = b2Γ−(t) + a2Γ+(t) = b2t + (a2 − b2)Γ+(t).
We need the joint distribution of
Xt , LXt , Γ+(t) =
∫ t
0lI(0,∞)(Xs) ds,
where X is a Brownian motion.
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Transition density by Girsanov (continued)
Define
Γ+(t) =
∫ t
0lI(0,∞)
(X (s)
)ds,
Γ−(t) =
∫ t
0lI(−∞,0)
(X (s)
)ds = t − Γ+(t).
Then∫ t
0f 2(Xs) ds = b2Γ−(t) + a2Γ+(t) = b2t + (a2 − b2)Γ+(t).
We need the joint distribution of
Xt , LXt , Γ+(t) =
∫ t
0lI(0,∞)(Xs) ds,
where X is a Brownian motion.
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Trivariate density
Theorem (Karatzas & Shreve (1984))
Let W be a Brownian motion, let L be its local time at zero, andlet
Γ+(t) ,∫ t
0lI(0,∞)(Ws) ds
denote the occupation time of the right half-line. Then for a ≤ 0,b ≥ 0, and 0 < t < T ,
P{WT ∈ da, LT ∈ db, Γ+(T ) ∈ dt}
=b(b − a)
π√
t3(T − t)3exp
[−b2
2t− (b − a)2
2(T − t)
]da db dt.
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RemarkBecause the occupation time of the left half-line is
Γ−(T ) ,∫ T
0lI(−∞,0)(Wt) dt = T − Γ+(T ),
we also know P{WT ∈ da, LT ∈ db, Γ−(T ) ∈ dt} for a ≤ 0, b ≥ 0,and 0 < t < T . Applying this to −W , we obtain a formula for
P{WT ∈ da, LT ∈ db, Γ+(T ) ∈ dt}, a ≥ 0, b ≥ 0, 0 < t < T .
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Classic trivariate density
Theorem (Levy (1948))
Let W be a Brownian motion, let MT = max0≤t≤T Wt , and let θTbe the (almost surely unique) time when W attains its maximumon [0,T ]. Then for a ∈ R, b ≥ max{a, 0}, and 0 < t < T ,
P{WT ∈ da,MT ∈ db, θT ∈ dt}
=b(b − a)
π√
t3(T − t)3exp
[−b2
2t− (b − a)2
2(T − t)
]da db dt.
The elementary proof uses the reflection principle and the Markovproperty.
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Positive part of Brownian motion
W
T t0
T t0
Γ+
Γ+(T ) t0
W+(t) , WΓ−1+ (t)
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Positive part of Brownian motion
W
T t0
T t0
Γ+
Γ+(T ) t0
W+(t) , WΓ−1+ (t)
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Positive part of Brownian motion
W
T t0
T t0
Γ+
Γ+(T ) t0
W+(t) , WΓ−1+ (t)
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Full decomposition
W
T t0
Γ+(T ) t0
W+(t) , WΓ−1+ (t)
t0
W−(t) , −WΓ−1− (t)
T t
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Full decomposition
W
T t0
Γ+(T ) t0
W+(t) , WΓ−1+ (t)
t0
W−(t) , −WΓ−1− (t)
T t
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Full decomposition
W
T t0
Γ+(T ) t0
W+(t) , WΓ−1+ (t)
t0
W−(t) , −WΓ−1− (t)
T t
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Full decomposition
W
T t0
Γ+(T ) t0
W+(t) , WΓ−1+ (t)
t0
W−(t) , −WΓ−1− (t)
T t
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Local time
Local time at zero of W :
Lt =1
2
∫ t
0δ0(Ws) ds = lim
ε↓0
1
4ε
∫ t
0lI(−ε,ε)(Ws) ds
= limε↓0
1
2ε
∫ t
0lI(0,ε)(Ws) ds = lim
ε↓0
1
2ε
∫ t
0lI(−ε,0)(Ws) ds.
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Tanaka’s formulaTanaka formula:
max{Wt , 0} =
∫ t
0lI(0,∞)(Ws) dWs +
1
2
∫ t
0δ0(Ws) ds
= −Bt + Lt ,
where
Bt = −∫ t
0lI(0,∞)(Ws) dWs , 〈B〉t = Γ+(t).
Then B+(t) , BΓ−1+ (t) is a Brownian motion.
Time-changed Tanaka formula:
W+(t) = −B+(t) + L+(t),
where L+(t) = LΓ−1+ (t).
Conclusion: W+ is a reflected Brownian motion. It is the Brownianmotion −B+ plus the nondecreasing process L+ that grows onlywhen W+ is at zero.
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Tanaka’s formulaTanaka formula:
max{Wt , 0} =
∫ t
0lI(0,∞)(Ws) dWs +
1
2
∫ t
0δ0(Ws) ds
= −Bt + Lt ,
where
Bt = −∫ t
0lI(0,∞)(Ws) dWs ,
〈B〉t = Γ+(t).
Then B+(t) , BΓ−1+ (t) is a Brownian motion.
Time-changed Tanaka formula:
W+(t) = −B+(t) + L+(t),
where L+(t) = LΓ−1+ (t).
Conclusion: W+ is a reflected Brownian motion. It is the Brownianmotion −B+ plus the nondecreasing process L+ that grows onlywhen W+ is at zero.
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Tanaka’s formulaTanaka formula:
max{Wt , 0} =
∫ t
0lI(0,∞)(Ws) dWs +
1
2
∫ t
0δ0(Ws) ds
= −Bt + Lt ,
where
Bt = −∫ t
0lI(0,∞)(Ws) dWs , 〈B〉t = Γ+(t).
Then B+(t) , BΓ−1+ (t) is a Brownian motion.
Time-changed Tanaka formula:
W+(t) = −B+(t) + L+(t),
where L+(t) = LΓ−1+ (t).
Conclusion: W+ is a reflected Brownian motion. It is the Brownianmotion −B+ plus the nondecreasing process L+ that grows onlywhen W+ is at zero.
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Tanaka’s formulaTanaka formula:
max{Wt , 0} =
∫ t
0lI(0,∞)(Ws) dWs +
1
2
∫ t
0δ0(Ws) ds
= −Bt + Lt ,
where
Bt = −∫ t
0lI(0,∞)(Ws) dWs , 〈B〉t = Γ+(t).
Then B+(t) , BΓ−1+ (t) is a Brownian motion.
Time-changed Tanaka formula:
W+(t) = −B+(t) + L+(t),
where L+(t) = LΓ−1+ (t).
Conclusion: W+ is a reflected Brownian motion. It is the Brownianmotion −B+ plus the nondecreasing process L+ that grows onlywhen W+ is at zero.
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Tanaka’s formulaTanaka formula:
max{Wt , 0} =
∫ t
0lI(0,∞)(Ws) dWs +
1
2
∫ t
0δ0(Ws) ds
= −Bt + Lt ,
where
Bt = −∫ t
0lI(0,∞)(Ws) dWs , 〈B〉t = Γ+(t).
Then B+(t) , BΓ−1+ (t) is a Brownian motion.
Time-changed Tanaka formula:
W+(t) = −B+(t) + L+(t),
where L+(t) = LΓ−1+ (t).
Conclusion: W+ is a reflected Brownian motion. It is the Brownianmotion −B+ plus the nondecreasing process L+ that grows onlywhen W+ is at zero.
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Tanaka’s formulaTanaka formula:
max{Wt , 0} =
∫ t
0lI(0,∞)(Ws) dWs +
1
2
∫ t
0δ0(Ws) ds
= −Bt + Lt ,
where
Bt = −∫ t
0lI(0,∞)(Ws) dWs , 〈B〉t = Γ+(t).
Then B+(t) , BΓ−1+ (t) is a Brownian motion.
Time-changed Tanaka formula:
W+(t) = −B+(t) + L+(t),
where L+(t) = LΓ−1+ (t).
Conclusion: W+ is a reflected Brownian motion. It is the Brownianmotion −B+ plus the nondecreasing process L+ that grows onlywhen W+ is at zero.
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Skorohod representation
Time-changed Tanaka formula:
W+(t) = −B+(t) + L+(t).
Skorohod representation:The nondecreasing process added to −B+ that grows only whenW+ = 0 is
L+(t) = max0≤s≤t
B+(s).
In particular,
B+(t) = −W+(t) + L+(t) = −W+(t) + max0≤s≤t
B+(s).
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W+ and B+
B+(t) = −W+(t) + L+(t) = −W+(t) + max0≤s≤t
B+(s).
Γ+(T ) t0
W+(t) , WΓ−1+ (t)
t0
−W+
B+ L+(T ) = max0≤t≤Γ+(t) B+(t)
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W+ and B+
B+(t) = −W+(t) + L+(t) = −W+(t) + max0≤s≤t
B+(s).
Γ+(T ) t0
W+(t) , WΓ−1+ (t)
t0
−W+
B+ L+(T ) = max0≤t≤Γ+(t) B+(t)
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W+ and B+
B+(t) = −W+(t) + L+(t) = −W+(t) + max0≤s≤t
B+(s).
Γ+(T ) t0
W+(t) , WΓ−1+ (t)
t0
−W+
B+
L+(T ) = max0≤t≤Γ+(t) B+(t)
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W+ and B+
B+(t) = −W+(t) + L+(t) = −W+(t) + max0≤s≤t
B+(s).
Γ+(T ) t0
W+(t) , WΓ−1+ (t)
t0
−W+
B+ L+(T ) = max0≤t≤Γ+(t) B+(t)
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W+ and B+. W− and B−.B+(t) = −W+(t) + L+(t) = −W+(t) + max
0≤s≤tB+(s)
B−(t) = −W−(t) + L−(t) = −W−(t) + max0≤s≤t
B−(s).
Γ+(T ) tT0
W+
W− run backwards
T t0
−W+
−W− run backwards
B+ B− run backwards
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W+ and B+. W− and B−.B+(t) = −W+(t) + L+(t) = −W+(t) + max
0≤s≤tB+(s)
B−(t) = −W−(t) + L−(t) = −W−(t) + max0≤s≤t
B−(s).
Γ+(T ) tT0
W+
W− run backwards
T t0
−W+
−W− run backwards
B+ B− run backwards
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W+ and B+. W− and B−.B+(t) = −W+(t) + L+(t) = −W+(t) + max
0≤s≤tB+(s)
B−(t) = −W−(t) + L−(t) = −W−(t) + max0≤s≤t
B−(s).
Γ+(T ) tT0
W+
W− run backwards
T t0
−W+
−W− run backwards
B+ B− run backwards
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W+ and B+. W− and B−.B+(t) = −W+(t) + L+(t) = −W+(t) + max
0≤s≤tB+(s)
B−(t) = −W−(t) + L−(t) = −W−(t) + max0≤s≤t
B−(s).
Γ+(T ) tT0
W+
W− run backwards
T t0
−W+
−W− run backwards
B+ B− run backwards
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W+ and B+. W− and B−.B+(t) = −W+(t) + L+(t) = −W+(t) + max
0≤s≤tB+(s)
B−(t) = −W−(t) + L−(t) = −W−(t) + max0≤s≤t
B−(s).
Γ+(T ) tT0
W+
W− run backwards
T t0
−W+
−W− run backwards
B+ B− run backwards
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W+ and B+. W− and B−.B+(t) = −W+(t) + L+(t) = −W+(t) + max
0≤s≤tB+(s)
B−(t) = −W−(t) + L−(t) = −W−(t) + max0≤s≤t
B−(s).
Γ+(T ) tT0
W+
W− run backwards
T t0
−W+
−W− run backwards
B+
B− run backwards
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W+ and B+. W− and B−.B+(t) = −W+(t) + L+(t) = −W+(t) + max
0≤s≤tB+(s)
B−(t) = −W−(t) + L−(t) = −W−(t) + max0≤s≤t
B−(s).
Γ+(T ) tT0
W+
W− run backwards
T t0
−W+
−W− run backwards
B+ B− run backwards
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W+ and B+. W− and B−.B+(t) = −W+(t) + L+(t) = −W+(t) + max
0≤s≤tB+(s)
B−(t) = −W−(t) + L−(t) = −W−(t) + max0≤s≤t
B−(s).
Γ+(T ) tT0
W+
W− run backwards
T t0
−W+
−W− run backwards
B+ B− run backwards
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W+ and B+. W− and B−.B+(t) = −W+(t) + L+(t) = −W+(t) + max
0≤s≤tB+(s)
B−(t) = −W−(t) + L−(t) = −W−(t) + max0≤s≤t
B−(s).
Γ+(T ) tT0
W+
W− run backwards
T t0
−W+
−W− run backwards
B+ B− run backwards
Local time L+(T ) = LT time has become the maximum.Γ+(T ) has become the time of the maximum.
63 / 94
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Personal reflections
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In the beginning....S. Shreve, Reflected Brownian motion in the “bang-bang” controlof Browian drift, SIAM J. Control Optimization 19, 469–478,(1981).
Acknowledgment in the paper
The author wishes to acknowledge the aid of V. E.Benes, who found an error in some preliminary work onthis subject and suggested the applicability of Tanaka’sformula. He also wishes to thank the referee for pointingout the uniqueness of the transition densitycorresponding to the weak solution in Section 5.
I Work on stochastic control (monotone follower, boundedvariation follower, finite-fuel, ....)
I Work on optimal investment, consumption and duality withJohn Lehoczky, Suresh Sethi, Gan-Lin Xu, Jaksa Cvitanic ....
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In the beginning....S. Shreve, Reflected Brownian motion in the “bang-bang” controlof Browian drift, SIAM J. Control Optimization 19, 469–478,(1981).
Acknowledgment in the paper
The author wishes to acknowledge the aid of V. E.Benes, who found an error in some preliminary work onthis subject and suggested the applicability of Tanaka’sformula. He also wishes to thank the referee for pointingout the uniqueness of the transition densitycorresponding to the weak solution in Section 5.
I Work on stochastic control (monotone follower, boundedvariation follower, finite-fuel, ....)
I Work on optimal investment, consumption and duality withJohn Lehoczky, Suresh Sethi, Gan-Lin Xu, Jaksa Cvitanic ....
66 / 94
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In the beginning....S. Shreve, Reflected Brownian motion in the “bang-bang” controlof Browian drift, SIAM J. Control Optimization 19, 469–478,(1981).
Acknowledgment in the paper
The author wishes to acknowledge the aid of V. E.Benes, who found an error in some preliminary work onthis subject and suggested the applicability of Tanaka’sformula. He also wishes to thank the referee for pointingout the uniqueness of the transition densitycorresponding to the weak solution in Section 5.
I Work on stochastic control (monotone follower, boundedvariation follower, finite-fuel, ....)
I Work on optimal investment, consumption and duality withJohn Lehoczky, Suresh Sethi, Gan-Lin Xu, Jaksa Cvitanic ....
67 / 94
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In the beginning....S. Shreve, Reflected Brownian motion in the “bang-bang” controlof Browian drift, SIAM J. Control Optimization 19, 469–478,(1981).
Acknowledgment in the paper
The author wishes to acknowledge the aid of V. E.Benes, who found an error in some preliminary work onthis subject and suggested the applicability of Tanaka’sformula. He also wishes to thank the referee for pointingout the uniqueness of the transition densitycorresponding to the weak solution in Section 5.
I Work on stochastic control (monotone follower, boundedvariation follower, finite-fuel, ....)
I Work on optimal investment, consumption and duality withJohn Lehoczky, Suresh Sethi, Gan-Lin Xu, Jaksa Cvitanic ....
68 / 94
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....and then THE BOOK,
69 / 94
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....and then THE BOOK,
70 / 94
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and mathematical finance took off.
71 / 94
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and mathematical finance took off.
72 / 94
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Ten things I learned from Ioannis Karatzas
I Greek culture10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunityto do so actually arises.
I Spelling8. Lemmata (From the Greek ληµµατα; not commonly used in
West Virginia dialect.)7. Coordinate (Diaeresis: [Ancient Greek] The separate
pronunciation of two vowels in a diphthong.)I Scholarship
6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
73 / 94
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Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata (From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate (Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
74 / 94
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Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.
9. All Greeks plan to return home some day, until the opportunityto do so actually arises.
I Spelling8. Lemmata (From the Greek ληµµατα; not commonly used in
West Virginia dialect.)7. Coordinate (Diaeresis: [Ancient Greek] The separate
pronunciation of two vowels in a diphthong.)I Scholarship
6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
75 / 94
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Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day,
until the opportunityto do so actually arises.
I Spelling8. Lemmata (From the Greek ληµµατα; not commonly used in
West Virginia dialect.)7. Coordinate (Diaeresis: [Ancient Greek] The separate
pronunciation of two vowels in a diphthong.)I Scholarship
6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
76 / 94
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Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.
I Spelling8. Lemmata (From the Greek ληµµατα; not commonly used in
West Virginia dialect.)7. Coordinate (Diaeresis: [Ancient Greek] The separate
pronunciation of two vowels in a diphthong.)I Scholarship
6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
77 / 94
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Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata (From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate (Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
78 / 94
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Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata
(From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate (Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
79 / 94
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Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata (From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate (Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
80 / 94
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Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata (From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate
(Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
81 / 94
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Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata (From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate (Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
82 / 94
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Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata (From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate (Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship
6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
83 / 94
![Page 84: Trivariate Density Revisited - Columbia University · Trivariate Density Revisited Steven E. Shreve Carnegie Mellon University {Conference in Honor of Ioannis Karatzas Columbia University](https://reader030.vdocuments.mx/reader030/viewer/2022040810/5e51afc709cbf337a32c9df3/html5/thumbnails/84.jpg)
Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata (From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate (Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship6. Appreciate the work of others.
5. Revise, revise, revise.I Advising
4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
84 / 94
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Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata (From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate (Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
85 / 94
![Page 86: Trivariate Density Revisited - Columbia University · Trivariate Density Revisited Steven E. Shreve Carnegie Mellon University {Conference in Honor of Ioannis Karatzas Columbia University](https://reader030.vdocuments.mx/reader030/viewer/2022040810/5e51afc709cbf337a32c9df3/html5/thumbnails/86.jpg)
Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata (From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate (Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship6. Appreciate the work of others.5. Revise, revise, revise.
I Advising
4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
86 / 94
![Page 87: Trivariate Density Revisited - Columbia University · Trivariate Density Revisited Steven E. Shreve Carnegie Mellon University {Conference in Honor of Ioannis Karatzas Columbia University](https://reader030.vdocuments.mx/reader030/viewer/2022040810/5e51afc709cbf337a32c9df3/html5/thumbnails/87.jpg)
Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata (From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate (Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.
3. Teach students to appreciate the work of others and to revise,revise, revise.
I Administration2. Avoid it.
87 / 94
![Page 88: Trivariate Density Revisited - Columbia University · Trivariate Density Revisited Steven E. Shreve Carnegie Mellon University {Conference in Honor of Ioannis Karatzas Columbia University](https://reader030.vdocuments.mx/reader030/viewer/2022040810/5e51afc709cbf337a32c9df3/html5/thumbnails/88.jpg)
Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata (From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate (Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.
I Administration2. Avoid it.
88 / 94
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Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata (From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate (Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
89 / 94
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Ten things I learned from Ioannis KaratzasI Greek culture
10. Macedonia is a province in Greece.9. All Greeks plan to return home some day, until the opportunity
to do so actually arises.I Spelling
8. Lemmata (From the Greek ληµµατα; not commonly used inWest Virginia dialect.)
7. Coordinate (Diaeresis: [Ancient Greek] The separatepronunciation of two vowels in a diphthong.)
I Scholarship6. Appreciate the work of others.5. Revise, revise, revise.
I Advising4. Choose excellent students.3. Teach students to appreciate the work of others and to revise,
revise, revise.I Administration
2. Avoid it.
90 / 94
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Number one
1. A professional colleague who is also a friend is more preciousthan Euros/Drachmae.
Thank you, Yannis,
for all you have done
I for your students,
I for your colleagues,
I for science,
I and for me personally.
91 / 94
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Number one
1. A professional colleague who is also a friend is more preciousthan Euros/Drachmae.
Thank you, Yannis,
for all you have done
I for your students,
I for your colleagues,
I for science,
I and for me personally.
92 / 94
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Number one
1. A professional colleague who is also a friend is more preciousthan Euros/Drachmae.
Thank you, Yannis,
for all you have done
I for your students,
I for your colleagues,
I for science,
I and for me personally.
93 / 94
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Number one
1. A professional colleague who is also a friend is more preciousthan Euros/Drachmae.
Thank you, Yannis,
for all you have done
I for your students,
I for your colleagues,
I for science,
I and for me personally.
94 / 94