Trivariate Density Revisited Steven E. Shreve Carnegie Mellon - - PowerPoint PPT Presentation

Trivariate Density Revisited

Steven E. Shreve Carnegie Mellon University – Conference in Honor of Ioannis Karatzas Columbia University June 4–8, 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, Planar

diffusions with rank-based characteristics and perturbed Tanaka equation, 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 = 1 g + h > 0 B1, B2 independent Br. motions

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Remarks

Karatzas, et. al. examine:

◮ Weak and strong existence and uniqueness in law; ◮ Properties of X1 ∨ X2 and X1 ∧ X2; ◮ The reversed dynamics of (X1, X2);

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Remarks

Karatzas, et. al. examine:

◮ Weak and strong existence and uniqueness in law; ◮ Properties of X1 ∨ X2 and X1 ∧ X2; ◮ The reversed dynamics of (X1, X2); ◮ Transition probabilities (X1, X2).

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The difference process

Define Y (t) = X1(t) − X2(t).

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The difference process

Define Y (t) = X1(t) − X2(t). Then dY = l I{Y ≤0}(g + h) dt − l I{Y >0}(g + h) dt +l I{Y ≤0}σ dB1 − l I{Y ≤0}ρ dB2 + l I{Y >0}ρ dB1 − l I{Y >0}σ dB2

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The difference process

Define Y (t) = X1(t) − X2(t). Then dY = l I{Y ≤0}(g + h) dt − l I{Y >0}(g + h) dt +l I{Y ≤0}σ dB1 − l I{Y ≤0}ρ dB2 + l I{Y >0}ρ dB1 − l I{Y >0}σ dB2 = −λ sgn

• Y (t)
• dt + dW ,

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The difference process

Define Y (t) = X1(t) − X2(t). Then dY = l I{Y ≤0}(g + h) dt − l I{Y >0}(g + h) dt +l I{Y ≤0}σ dB1 − l I{Y ≤0}ρ dB2 + l I{Y >0}ρ dB1 − l I{Y >0}σ dB2 = −λ sgn

• Y (t)
• dt + dW ,

where λ = g + h > 0,

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The difference process

Define Y (t) = X1(t) − X2(t). Then dY = l I{Y ≤0}(g + h) dt − l I{Y >0}(g + h) dt +l I{Y ≤0}σ dB1 − l I{Y ≤0}ρ dB2 + l I{Y >0}ρ dB1 − l I{Y >0}σ dB2 = −λ sgn

• Y (t)
• dt + dW ,

where λ = g + h > 0, dW = σ

• l

I{Y ≤0} dB1 − l I{Y >0} dB2

• dW1

+ρ

• l

I{Y >0} dB1 − l I{Y ≤0} dB2

• dW2

.

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The sum process

Define Z(t) = X1(t) + X2(t).

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The sum process

Define Z(t) = X1(t) + X2(t). Then dZ = (g − h) dt + l I{Y ≤0}σ dB1 + l I{Y ≤0}ρ dB2 +l I{Y >0}ρ dB1 + l I{Y >0}σ dB2

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The sum process

Define Z(t) = X1(t) + X2(t). Then dZ = (g − h) dt + l I{Y ≤0}σ dB1 + l I{Y ≤0}ρ dB2 +l I{Y >0}ρ dB1 + l I{Y >0}σ dB2 = ν dt + dV ,

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The sum process

Define Z(t) = X1(t) + X2(t). Then dZ = (g − h) dt + l I{Y ≤0}σ dB1 + l I{Y ≤0}ρ dB2 +l I{Y >0}ρ dB1 + l I{Y >0}σ dB2 = ν dt + dV , where ν = g − h,

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The sum process

Define Z(t) = X1(t) + X2(t). Then dZ = (g − h) dt + l I{Y ≤0}σ dB1 + l I{Y ≤0}ρ dB2 +l I{Y >0}ρ dB1 + l I{Y >0}σ dB2 = ν dt + dV , where ν = g − h, dV = σ

• l

I{Y ≤0} dB1 + l I{Y >0} dB2

• dV1

+ρ

• l

I{Y >0} dB1 + l I{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 sgn

• Y (s)
• ds + W (t),

where V and W are correlated Brownian motions.

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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 sgn

• Y (s)
• ds + W (t),

where V and W are correlated Brownian motions.

◮ To determine transition probabilities for (X1, X2), it suffices to

determine transition probabilites for (Z, Y ).

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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 sgn

• Y (s)
• ds + W (t),

where V and W are correlated Brownian motions.

◮ To determine transition probabilities for (X1, X2), it suffices to

determine transition probabilites for (Z, Y ).

◮ (Z, W ) is a Gaussian process. ◮ Y is determined by W by

dY (t) = −λ sgn

• Y (t)
• dt + dW (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 sgn

• Y (s)
• ds + W (t),

where V and W are correlated Brownian motions.

◮ To determine transition probabilities for (X1, X2), it suffices to

determine transition probabilites for (Z, Y ).

◮ (Z, W ) is a Gaussian process. ◮ Y is determined by W by

dY (t) = −λ sgn

• Y (t)
• dt + dW (t).

◮ It suffices to determine the distribution of (W (t), Y (t)).

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Bang-bang control

Minimize E ∞ e−tX 2

t dt,

Subject to Xt = x + t us ds + Wt, a ≤ ut ≤ b, t ≥ 0.

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Bang-bang control

Minimize E ∞ e−tX 2

t dt,

Subject to Xt = x + t us ds + Wt, a ≤ ut ≤ b, t ≥ 0. Beneˇ s, 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.

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Bang-bang control

Minimize E ∞ e−tX 2

t dt,

Subject to Xt = x + t us ds + Wt, a ≤ ut ≤ b, t ≥ 0. Beneˇ s, 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? (Beneˇ s, et. al. computed its Laplace transform.)

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Transition density by Girsanov

Compute the transition density for X, where Xt = x + t f (Xs) ds + Wt.

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Transition density by Girsanov

Compute the transition density for X, where Xt = x + t f (Xs) ds + Wt. Start with a probability measure PX under which X is a Brownian

• motion. Define W by

Wt = Xt − x − t f (Xs) ds.

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Transition density by Girsanov

Compute the transition density for X, where Xt = x + t f (Xs) ds + Wt. Start with a probability measure PX under which X is a Brownian

• motion. Define W by

Wt = Xt − x − t f (Xs) ds. Change to a probability measure P under which W is a Brownian

• motion. Compute the transition density for X under P.

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Transition density by Girsanov

Compute the transition density for X, where Xt = x + t f (Xs) ds + Wt. Start with a probability measure PX under which X is a Brownian

• motion. Define W by

Wt = Xt − x − t f (Xs) ds. Change to a probability measure P under which W is a Brownian

• motion. Compute the transition density for X under P.

P{Xt ∈ B} = EX

• l

I{Xt∈B} dP dPX

• Ft
• ,

where dP dPX

• Ft

= exp t f (Xs) dXs − 1 2 t f 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 f (Xs) dXs, t f 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 f (Xs) dXs, t f 2(Xs) ds. Assume without loss of generality that δ = 0. Define F(x) = x f (ξ) dξ = bx, if x ≤ 0, ax, if x ≥ 0. Tanaka’s formula implies F(Xt) = t f (Xs) dXs + 1 2(a − b)LX

t

so t f (Xs) dXs = F(Xt) − 1 2(a − b)LX

t .

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Transition density by Girsanov (continued)

To compute P{Xt ∈ B}, we need the joint distribution of Xt, t f (Xs) dXs, t f 2(Xs) ds. Assume without loss of generality that δ = 0. Define F(x) = x f (ξ) dξ = bx, if x ≤ 0, ax, if x ≥ 0. Tanaka’s formula implies F(Xt) = t f (Xs) dXs + 1 2(a − b)LX

t

so t f (Xs) dXs = F(Xt) − 1 2(a − b)LX

t .

We need the joint distribution of Xt, LX

t ,

t f 2(Xs) ds.

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Transition density by Girsanov (continued)

Define Γ+(t) = t l I(0,∞)

• X(s)
• ds,

Γ−(t) = t l I(−∞,0)

• X(s)
• ds = t − Γ+(t).

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Transition density by Girsanov (continued)

Define Γ+(t) = t l I(0,∞)

• X(s)
• ds,

Γ−(t) = t l I(−∞,0)

• X(s)
• ds = t − Γ+(t).

Then t f 2(Xs) ds = b2Γ−(t) + a2Γ+(t) = b2t + (a2 − b2)Γ+(t).

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Transition density by Girsanov (continued)

Define Γ+(t) = t l I(0,∞)

• X(s)
• ds,

Γ−(t) = t l I(−∞,0)

• X(s)
• ds = t − Γ+(t).

Then t f 2(Xs) ds = b2Γ−(t) + a2Γ+(t) = b2t + (a2 − b2)Γ+(t). We need the joint distribution of Xt, LX

t ,

Γ+(t) = t l I(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, and let Γ+(t) t l I(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)3 exp
• −b2

2t − (b − a)2 2(T − t)

• da db dt.

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Remark

Because the occupation time of the left half-line is Γ−(T) T l I(−∞,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 (L´ evy (1948))

Let W be a Brownian motion, let MT = max0≤t≤T Wt, and let θT be the (almost surely unique) time when W attains its maximum

• n [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)3 exp
• −b2

2t − (b − a)2 2(T − t)

• da db dt.

The elementary proof uses the reflection principle and the Markov property.

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Positive part of Brownian motion

W T t

Positive part of Brownian motion

W T t T t Γ+

Positive part of Brownian motion

W T t T t Γ+ Γ+(T) t W+(t) WΓ−1

+ (t) 38 / 94

Full decomposition

W T t

Full decomposition

W T t Γ+(T) t W+(t) WΓ−1

+ (t)

Full decomposition

W T t Γ+(T) t W+(t) WΓ−1

+ (t)

t W−(t) −WΓ−1

− (t)

Full decomposition

W T t Γ+(T) t W+(t) WΓ−1

+ (t)

t W−(t) −WΓ−1

− (t)

T t

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Local time

Local time at zero of W : Lt = 1 2 t δ0(Ws) ds = lim

ǫ↓0

1 4ǫ t l I(−ǫ,ǫ)(Ws) ds = lim

ǫ↓0

1 2ǫ t l I(0,ǫ)(Ws) ds = lim

ǫ↓0

1 2ǫ t l I(−ǫ,0)(Ws) ds.

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Tanaka’s formula

Tanaka formula: max{Wt, 0} = t l I(0,∞)(Ws) dWs + 1 2 t δ0(Ws) ds

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Tanaka’s formula

Tanaka formula: max{Wt, 0} = t l I(0,∞)(Ws) dWs + 1 2 t δ0(Ws) ds = −Bt + Lt, where Bt = − t l I(0,∞)(Ws) dWs,

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Tanaka’s formula

Tanaka formula: max{Wt, 0} = t l I(0,∞)(Ws) dWs + 1 2 t δ0(Ws) ds = −Bt + Lt, where Bt = − t l I(0,∞)(Ws) dWs, Bt = Γ+(t).

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Tanaka’s formula

Tanaka formula: max{Wt, 0} = t l I(0,∞)(Ws) dWs + 1 2 t δ0(Ws) ds = −Bt + Lt, where Bt = − t l I(0,∞)(Ws) dWs, Bt = Γ+(t). Then B+(t) BΓ−1

+ (t) is a Brownian motion. 47 / 94

Tanaka’s formula

Tanaka formula: max{Wt, 0} = t l I(0,∞)(Ws) dWs + 1 2 t δ0(Ws) ds = −Bt + Lt, where Bt = − t l I(0,∞)(Ws) dWs, Bt = 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). 48 / 94

Tanaka’s formula

Tanaka formula: max{Wt, 0} = t l I(0,∞)(Ws) dWs + 1 2 t δ0(Ws) ds = −Bt + Lt, where Bt = − t l I(0,∞)(Ws) dWs, Bt = 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 Brownian motion −B+ plus the nondecreasing process L+ that grows only when 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 when W+ = 0 is L+(t) = max

0≤s≤t B+(s).

In particular, B+(t) = −W+(t) + L+(t) = −W+(t) + max

0≤s≤t B+(s).

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W+ and B+

B+(t) = −W+(t) + L+(t) = −W+(t) + max

0≤s≤t B+(s).

Γ+(T) t W+(t) WΓ−1

+ (t)

W+ and B+

B+(t) = −W+(t) + L+(t) = −W+(t) + max

0≤s≤t B+(s).

Γ+(T) t W+(t) WΓ−1

+ (t)

t −W+

W+ and B+

B+(t) = −W+(t) + L+(t) = −W+(t) + max

0≤s≤t B+(s).

Γ+(T) t W+(t) WΓ−1

+ (t)

t −W+ B+

W+ and B+

B+(t) = −W+(t) + L+(t) = −W+(t) + max

0≤s≤t B+(s).

Γ+(T) t W+(t) WΓ−1

+ (t)

t −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≤t B+(s)

B−(t) = −W−(t) + L−(t)= −W−(t) + max

0≤s≤t B−(s).

Γ+(T) t T W+

W+ and B+. W− and B−.

B+(t) = −W+(t) + L+(t) = −W+(t) + max

0≤s≤t B+(s)

B−(t) = −W−(t) + L−(t)= −W−(t) + max

0≤s≤t B−(s).

Γ+(T) t T W+

W+ and B+. W− and B−.

B+(t) = −W+(t) + L+(t) = −W+(t) + max

0≤s≤t B+(s)

B−(t) = −W−(t) + L−(t)= −W−(t) + max

0≤s≤t B−(s).

Γ+(T) t T W+ W− run backwards

W+ and B+. W− and B−.

B+(t) = −W+(t) + L+(t) = −W+(t) + max

0≤s≤t B+(s)

B−(t) = −W−(t) + L−(t)= −W−(t) + max

0≤s≤t B−(s).

Γ+(T) t T W+ W− run backwards T t −W+

W+ and B+. W− and B−.

B+(t) = −W+(t) + L+(t) = −W+(t) + max

0≤s≤t B+(s)

B−(t) = −W−(t) + L−(t)= −W−(t) + max

0≤s≤t B−(s).

Γ+(T) t T W+ W− run backwards T t −W+ −W− run backwards

W+ and B+. W− and B−.

B+(t) = −W+(t) + L+(t) = −W+(t) + max

0≤s≤t B+(s)

B−(t) = −W−(t) + L−(t)= −W−(t) + max

0≤s≤t B−(s).

Γ+(T) t T W+ W− run backwards T t −W+ −W− run backwards B+

W+ and B+. W− and B−.

B+(t) = −W+(t) + L+(t) = −W+(t) + max

0≤s≤t B+(s)

B−(t) = −W−(t) + L−(t)= −W−(t) + max

0≤s≤t B−(s).

Γ+(T) t T W+ W− run backwards T t −W+ −W− run backwards B+ B− run backwards

W+ and B+. W− and B−.

B+(t) = −W+(t) + L+(t) = −W+(t) + max

0≤s≤t B+(s)

B−(t) = −W−(t) + L−(t)= −W−(t) + max

0≤s≤t B−(s).

Γ+(T) t T W+ W− run backwards T t −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≤t B+(s)

B−(t) = −W−(t) + L−(t)= −W−(t) + max

0≤s≤t B−(s).

Γ+(T) t T W+ W− run backwards T t −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.

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Personal reflections

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In the beginning....

• S. Shreve, Reflected Brownian motion in the “bang-bang” control
• f Browian drift, SIAM J. Control Optimization 19, 469–478,

(1981).

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In the beginning....

• S. Shreve, Reflected Brownian motion in the “bang-bang” control
• f Browian drift, SIAM J. Control Optimization 19, 469–478,

(1981). Acknowledgment in the paper The author wishes to acknowledge the aid of V. E. Beneˇ s, who found an error in some preliminary work on this subject and suggested the applicability of Tanaka’s

• formula. He also wishes to thank the referee for pointing
• ut the uniqueness of the transition density

corresponding to the weak solution in Section 5.

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In the beginning....

• S. Shreve, Reflected Brownian motion in the “bang-bang” control
• f Browian drift, SIAM J. Control Optimization 19, 469–478,

(1981). Acknowledgment in the paper The author wishes to acknowledge the aid of V. E. Beneˇ s, who found an error in some preliminary work on this subject and suggested the applicability of Tanaka’s

• formula. He also wishes to thank the referee for pointing
• ut the uniqueness of the transition density

corresponding to the weak solution in Section 5.

◮ Work on stochastic control (monotone follower, bounded

variation follower, finite-fuel, ....)

67 / 94

In the beginning....

• S. Shreve, Reflected Brownian motion in the “bang-bang” control
• f Browian drift, SIAM J. Control Optimization 19, 469–478,

(1981). Acknowledgment in the paper The author wishes to acknowledge the aid of V. E. Beneˇ s, who found an error in some preliminary work on this subject and suggested the applicability of Tanaka’s

• formula. He also wishes to thank the referee for pointing
• ut the uniqueness of the transition density

corresponding to the weak solution in Section 5.

◮ Work on stochastic control (monotone follower, bounded

variation follower, finite-fuel, ....)

◮ Work on optimal investment, consumption and duality with

John Lehoczky, Suresh Sethi, Gan-Lin Xu, Jaksa Cvitaniˇ c ....

68 / 94

....and then THE BOOK,

69 / 94

....and then THE BOOK,

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and mathematical finance took off.

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and mathematical finance took off.

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Ten things I learned from Ioannis Karatzas

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Ten things I learned from Ioannis Karatzas

◮ Greek culture

74 / 94

Ten things I learned from Ioannis Karatzas

◮ Greek culture

• 10. Macedonia is a province in Greece.

75 / 94

Ten things I learned from Ioannis Karatzas

◮ Greek culture

• 10. Macedonia is a province in Greece.
• 9. All Greeks plan to return home some day,

76 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

77 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

◮ Spelling

78 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

◮ Spelling

• 8. Lemmata

79 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

◮ Spelling

• 8. Lemmata (From the Greek ληµµατα; not commonly used in

West Virginia dialect.)

80 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

◮ Spelling

• 8. Lemmata (From the Greek ληµµατα; not commonly used in

West Virginia dialect.)

• 7. Co¨
• rdinate

81 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

◮ Spelling

• 8. Lemmata (From the Greek ληµµατα; not commonly used in

West Virginia dialect.)

• 7. Co¨
• rdinate (Diaeresis: [Ancient Greek] The separate

pronunciation of two vowels in a diphthong.)

82 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

◮ Spelling

• 8. Lemmata (From the Greek ληµµατα; not commonly used in

West Virginia dialect.)

• 7. Co¨
• rdinate (Diaeresis: [Ancient Greek] The separate

pronunciation of two vowels in a diphthong.)

◮ Scholarship

83 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

◮ Spelling

• 8. Lemmata (From the Greek ληµµατα; not commonly used in

West Virginia dialect.)

• 7. Co¨
• rdinate (Diaeresis: [Ancient Greek] The separate

pronunciation of two vowels in a diphthong.)

◮ Scholarship

• 6. Appreciate the work of others.

84 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

◮ Spelling

• 8. Lemmata (From the Greek ληµµατα; not commonly used in

West Virginia dialect.)

• 7. Co¨
• rdinate (Diaeresis: [Ancient Greek] The separate

pronunciation of two vowels in a diphthong.)

◮ Scholarship

• 6. Appreciate the work of others.
• 5. Revise, revise, revise.

85 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

◮ Spelling

• 8. Lemmata (From the Greek ληµµατα; not commonly used in

West Virginia dialect.)

• 7. Co¨
• rdinate (Diaeresis: [Ancient Greek] The separate

pronunciation of two vowels in a diphthong.)

◮ Scholarship

• 6. Appreciate the work of others.
• 5. Revise, revise, revise.

◮ Advising

86 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

◮ Spelling

• 8. Lemmata (From the Greek ληµµατα; not commonly used in

West Virginia dialect.)

• 7. Co¨
• rdinate (Diaeresis: [Ancient Greek] The separate

pronunciation of two vowels in a diphthong.)

◮ Scholarship

• 6. Appreciate the work of others.
• 5. Revise, revise, revise.

◮ Advising

• 4. Choose excellent students.

87 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

◮ Spelling

• 8. Lemmata (From the Greek ληµµατα; not commonly used in

West Virginia dialect.)

• 7. Co¨
• rdinate (Diaeresis: [Ancient Greek] The separate

pronunciation of two vowels in a diphthong.)

◮ Scholarship

• 6. Appreciate the work of others.
• 5. Revise, revise, revise.

◮ Advising

• 4. Choose excellent students.
• 3. Teach students to appreciate the work of others and to revise,

revise, revise.

88 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

◮ Spelling

• 8. Lemmata (From the Greek ληµµατα; not commonly used in

West Virginia dialect.)

• 7. Co¨
• rdinate (Diaeresis: [Ancient Greek] The separate

pronunciation of two vowels in a diphthong.)

◮ Scholarship

• 6. Appreciate the work of others.
• 5. Revise, revise, revise.

◮ Advising

• 4. Choose excellent students.
• 3. Teach students to appreciate the work of others and to revise,

revise, revise.

◮ Administration

89 / 94

Ten things I learned from Ioannis Karatzas

◮ 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.

◮ Spelling

• 8. Lemmata (From the Greek ληµµατα; not commonly used in

West Virginia dialect.)

• 7. Co¨
• rdinate (Diaeresis: [Ancient Greek] The separate

pronunciation of two vowels in a diphthong.)

◮ Scholarship

• 6. Appreciate the work of others.
• 5. Revise, revise, revise.

◮ Advising

• 4. Choose excellent students.
• 3. Teach students to appreciate the work of others and to revise,

revise, revise.

◮ Administration

• 2. Avoid it.

90 / 94

Number one

91 / 94

Number one

• 1. A professional colleague who is also a friend is more precious

than Euros/Drachmae.

92 / 94

Number one

• 1. A professional colleague who is also a friend is more precious

than Euros/Drachmae.

Thank you, Yannis,

93 / 94

Number one

• 1. A professional colleague who is also a friend is more precious

than Euros/Drachmae.

Thank you, Yannis,

for all you have done

◮ for your students, ◮ for your colleagues, ◮ for science, ◮ and for me personally.

94 / 94