from martingale optimal transport to mckean-vlasov control ... · from mot to mckean-vlasov control...

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IntroductionFrom MOT to McKean-Vlasov Control Problems

Solving the McKean-Vlasov Control Problems

From Martingale Optimal Transportto McKean-Vlasov Control Problems

Xiaolu Tan

University of Paris-Dauphine, PSL Universitymoving to the Chinese University of Hong Kong

based on joint works withB. Bouchard, P. Henry-Labordère, F. Dejete, D. Possamaï

26 July 2019Workshop “Stochastic Control in Finance”

@NUS

Xiaolu Tan MOT and McKean-Vlasov Control

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Mathematical Finance ModellingStochastic Optimal Control

Outline

1 IntroductionMathematical Finance ModellingStochastic Optimal Control

2 From MOT to McKean-Vlasov Control ProblemsThe McKean-Vlasov Control ProblemMartingale Optimal Transport (MOT)From MOT to McKean-Vlasov Control

3 Solving the McKean-Vlasov Control Problems


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Financial market

• Financial market modelling : Let (Ω,F ,P) be a probability spaceequipped with filtration F = (Ft)t∈T, prices of some financial assetsis given by S = (St)t∈T, the payoff at maturity T of a derivativeoption is a random variable ξ : Ω→ R (e.g. ξ=(ST−K )+).

Discrete time market : T = 0, 1, · · · ,T.Continuous time market : T = [0,T ].

• Trading : A trading strategy is predictable process H =(Ht)0≤t≤T,whose P&L given by

(H S)T :=T∑t=1

Ht(St − St−1).


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Fundamental Theorem of Asset Pricing (FTAP)

For simplicity T = 0, 1, interest rate r = 0.

No arbitrage condition (NA) : there is no H s.t.

H(S1 − S0) ≥ 0, and P[H(S1 − S0) > 0

]> 0.

Equiv. martingale measures :M :=Q ∼ P : EQ[S1|F0] = S0.The market is complete if

∀ξ ∈ F , ∃(y ,H) s.t. y + H(S1 − S0) = ξ, a.s.

Theorem (FTAP (discrete time))

(i) (NA) is equivalent to the existence of equivalent martingalemeasure, i.e.

(NA) ⇔ M 6= ∅.

(ii) The market is complete iffM is a singleton.


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Basic problems in mathematical finance

• Pricing and hedging for a derivative option ξ : Ω→ R :Complet market : Price of derivative option equals to itsreplication cost.Incomplete market : Every equivalent martingale measureQ ∈M provides a no-arbitrage price : EQ[ξ].A pricing-hedging duality :

supQ∈M

EQ[ξ] = infy ∈ R : y + H(S1 − S0) ≥ ξ, a.s.

,

(see e.g. El Karoui and Quenez (1995)).• Utility maximization Let U : R→ R ∪ −∞ be a utilityfunction, we consider

maxH

E[U(x + (H S)T

)].


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Concrete market models

• The simplest martingales :Random walk : ∆Xt := Xt − Xt−1⊥⊥(X0, · · · ,Xt−1) and

P[∆Xt = 1] = P[∆Xt = −1] =12.

Brownian motion : W = (Wt)t≥0 is a continuous process withindependent and centred stationary increment.

• Two basic models :Binomial tree model : for 0 < d < 1 < u,

P[St = dSt−1] = p, P[St = uSt−1] = 1− p.

Black-Scholes model : St = S0 exp(−12σ

2t + σWt), orequivalently

dSt = σStdBt .


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Diffusion process, SDE and PDE

• Diffusion process defined by SDE (stochastic differential equation)

dXt = b(t,Xt)dt + σ(t,Xt)dWt .

• Kolmogorov forward equation (Fokker-Planck equation) on thedensity p(t, x) of Xt :

∂tp(t, x) + ∂x(b(t, x)p(t, x)

)− 1

2∂2xx

(σ2(t, x)p(t, x)

)= 0.

• Kolmogorov backward equation (Feynman-Kac formula) onu(t, x) := E[g(X t,x

T )] :

∂tu(t, x) + b(t, x)∂xu(t, x) +12σ2(t, x)∂2

xxu(t, x) = 0.


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Calibration from market data, local volatility model

• Assume that the market data (price), C (T ,K ) = E[(ST − K )+],is known for all K ∈ R, then one can obtain the distributionp(T ,K ) of ST by

∂KC (T ,K ) = −E[1IST≥K ], ∂2KKC (T ,K ) = E[δK (ST )] = p(T ,K ).

• Dupire’s local volatility model : dSt = σloc(t, St)dWt , then theFokker-Planck equation leads to

∂TC (T ,K )− 12σ2loc(T ,K )∂2

KKC (T ,K ) = 0.

• Markovian projection (Gyöngy, 1986) : let dSt = σtdWt has themarginal distribution p(t, x), then

E[σ2t |St ] = σ2

loc(t, St), a.s. ∀t.


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Controlled diffusion processes problem

• Let Xα· be a controlled process, with the control process α,

defined by

dXαt = b(t,Xα

t , αt)dt + σ(t,Xαt , αt)dWt ,

a standard stochastic control problem is given by

V (0,X0) := supα

E[ ∫ T

0f (t,Xα

t , αt)dt + g(XαT )].

• Applications in finance : utility maximization, pricing under modeluncertainty, American option pricing, etc.• A very large literature on the subject :

A big community on deterministic control.On stochastic control : Fleming-Rishel (1975),Lions-Bensoussan (1978), Krylov (1980), El Karoui (1981),Borkar (1989), Fleming-Soner(1993), Yong-Zhou (1999),Pham (2009), Touzi (2012), etc.


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Different approaches

• Approach 1 : Pontryagin’s maximum principle : if (α∗,X ∗) is anoptimal control, the first order necessary condition leads to aforward-backward system.

• Approach 2 : Dynamic Programming (Bellman) Principle :

V (0,X0) = supα

E[ ∫ t

0L(s,Xα

s , αs)ds + V (t,Xαt )].

PDE (HJB equation) characterization for the value function,numerical methods, etc.

Numerical computation by a backward scheme,Supermartingale characterization of V (t,Xα

t ),Optional decomposition, duality, etc.

• BSDE Approach, etc.Xiaolu Tan MOT and McKean-Vlasov Control

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The McKean-Vlasov Control ProblemMartingale Optimal Transport (MOT)From MOT to McKean-Vlasov Control

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The McKean-Vlasov stochastic equation

• A large population (i = 1, · · · ,N) system with interaction :

dX i ,Nt = b

(t,X i ,N

t ,1N

∑δX j,Nt

)dt + σ

(t,X i ,N

t ,1N

∑δX j,Nt

)dW i

t .

• McKean-Vlasov equation : let N →∞, using the limit theory (lawof large numbers) and considering a representative agent, it leads to

dXt = b(t,Xt ,L(Xt)

)dt + σ

(t,Xt ,L(Xt)

)dWt .


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The McKean-Vlasov control problem

• The McKean-Vlasov (mean-field) control problem :

supα

E[ ∫ T

0f (t,Xα

t ,L(Xαt ), αt)dt + g(Xα

T ,L(XαT ))],

where

dXαt = b

(t,Xα

t ,L(Xαt ), αt

)dt + σ

(t,Xα

t ,L(Xαt ), αt

)dWt .

• Main questions :The limit theoryPontryagin’s maximum principleThe dynamic programming principleNumerical approximationApplications in Economy, Finance, etc.See e.g. Book of Bensoussan-Frehse-Yam (2013),Carmona-Delarue (2018), etc.


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An optimal control problem under marginal constraint

• An optimal control problem under marginal constraint (Tan-Touzi(AOP, 13)) :

dXαt = b(t,Xα

t , αt)dt + σ(t,Xαt , αt)dWt ,

and for some given marginal distribution µ,

supα : XαT ∼µ

E[ ∫ T

0L(t,Xα

t∧·, αt)dt + Φ(Xα· )].

• Motivation from finance : model-free no-arbitrage pricing forexotic options,

When b ≡ 0, Xα is a martingale, i.e. no-arbitrage pricing.Marginal law µ is recovered from the price of call options.


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Martingale Optimal Transport (MOT)

• Numerous variated formulations :Discrete time vs Continuous time,Continuous path vs Càdlàd path,One marginal vs Multiple marginals.

• Some pioneering works : Skrokohod Embedding : Hobson (98) ;MOT problem : Beiglböck, Henry-Labordère, Penkner (13),Galichon, Henry-Labordère, Touzi (14), Dolinsky-Soner(14), etc.

Theorem (Kellerer, 1972)

Given a family of marginals µ = (µt)t≥0, there is martingale Msuch that Mt ∼ µt for all t ≥ 0 iff µt has finite first order andt 7→ µt(φ) is increasing for all convex functions φ.

• Process in convex ordering (PCOC : “Processus Croissant pourl’Ordre Convex” in French).


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Peacocks

Figure – The bookXiaolu Tan MOT and McKean-Vlasov Control

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Peacocks

Haïku by Pf. Y. Takahashi

Figure – Peacocks


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Optimal martingales given full marginals

• Fake Brownian motion : Albin (08), Oleszkiewicz (08), etc.• More recent constructions : Pagès (2013), Jourdain-Margheriti(2018), etc.

• Explicit constructions of optimal martingales :Madan-Yor (02) : maximizing the expected value of therunning maximum.Hobson (17) : minimizing the expected total variation.Henry-Labordère-Tan-Touzi (16, SPA), maximizing theexpected quadratic variation.

• Duality and existence of optimal martingales :Guo-Tan-Touzi (16, SICON) : S-topology.Källblad-Tan-Touzi (17, AAP) : Skorokhod embeddingapproach.


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A MOT problem givne full marginals

• Let µ = (µt)0≤t≤T a peacock, we consider the MOT problem

VMOT := supσ

E[Φ(X σ·)], dX σ

t = σtdWt , X σt ∼ µt , ∀t ∈ [0,T ].

• Remark 1 : There exists a unique Markovian diffusion process X ,defined by

dXt = σloc(t,Xt)dWt ,

satisfying the marginal constraints.

• Remark 2 : Let X σt = X σ

0 +∫ t0 σsdWs be a diffusion process such

that X σt ∼ µt , for all t ∈ [0, 1], then, by Markovian projection,

E[σ2t |X σ

t ] = σ2loc(t,X σ

t ), a.s. ∀t ∈ [0,T ],

and it follows that

dX σt =

σt√E[σ2

t |X σt ]σloc(t,X σ

t )dWt .


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A McKean-Vlasov control problem

• Let us consider all processes (σ, X σ) satisfying

dX σt =

σt√E[σ2

t |X σt ]σloc(t, X σ

t )dWt ,

and letVMKV := sup

(σ,Xσ)

E[Φ(X σ·)].

• Remark (Guyon-Henry-Labordère, 2011) :

σt :=σt√

E[σ2t |X σ

t ]σloc(t, X σ

t ) = σ(t, X σ

t ,L(σt , Xσt ), σt

)satisfies

E[σ2t

∣∣X σt

]= σ2

loc(t, X σt ).


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A McKean-Vlasov control problem

TheoremUnder technical conditions, one has

VMOT = VMKV.

An optimal solution to VMOT induces an optimal solution to VMKV,and vice versa.


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The dynamic programming principle

• Let us consider a general McKean-Vlasov control problem

V (0,L(X0)) :=supαE[∫ T

0f(· · · , αt

)dt + g

(XαT ,L(Xα

T |FBT ))],

where

dXαt = b(·)dt + σ(·)dWt + σ0

(t,Xα

t ,L((αt ,Xαt )|FB

t ), αt

)dBt .

Theorem (Dejete-Possamaï-Tan, 19)

Assume that f , g are Borel measurable, then one has the DPP :

V (0,L(X0)) :=supα

E[ ∫ τ

0f (·)ds + V

(τ,L(Xα

t |FBt )t=τ

)].

• Strong formulation, weak formulation, etc.• Using the measurable selection arguments, no requirement oncontinuity of coefficients.


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The limit thoery

• A large population control problem, i = 1, · · · ,N,

VN := supα

1N

N∑i=1

E[ ∫ T

0f (·)ds + g

(X i ,N,αT , ϕN,α

T

)],

where ϕN,αt := 1

N

∑Nj=1 δXN,j,α

tand

dXN,i ,αt = b(·)dt + σ(·)dW i

t + σ0(t,XN,i ,α

t , ϕN,αt , αi

t

)dBt .

Theorem (Dejete-Possamaï-Tan, 19)

Under some continuity conditions, one has the convergence result :

VN → V as N →∞.


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Numerical approximation

• Let ∆=(0= t0< · · ·< tn =T ) be a discrete time grid, we consider

V∆N := sup

α

1N

N∑i=1

E[ n∑k=1

f (·)∆t + g(X i ,N,∆,αtn , ϕN,∆,α

tn

)],

where ϕN,∆,αtk := 1

N

∑Nj=1 δXN,j,∆,α

tk

and

XN,i ,αtk+1

= XN,i ,αtk + T∆

(t,XN,i ,∆,α

tk+1, ϕN,∆,α

tk , αitk

).

Theorem (Dejete-Possamaï-Tan, 2019 ?)

With good choice of functions H∆, one has

V∆N → V as N →∞ and ∆→ 0.

• Kushner-Dupuis’s weak convergence technique.Xiaolu Tan MOT and McKean-Vlasov Control

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