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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
Logo
IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
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
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
Mathematical Finance ModellingStochastic Optimal Control
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).
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
Mathematical Finance ModellingStochastic Optimal Control
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.
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
Mathematical Finance ModellingStochastic Optimal Control
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
)].
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
Mathematical Finance ModellingStochastic Optimal Control
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 .
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
Mathematical Finance ModellingStochastic Optimal Control
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.
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
Mathematical Finance ModellingStochastic Optimal Control
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.
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
Mathematical Finance ModellingStochastic Optimal Control
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.
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
Mathematical Finance ModellingStochastic Optimal Control
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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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
The McKean-Vlasov Control ProblemMartingale Optimal Transport (MOT)From MOT to McKean-Vlasov 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
Xiaolu Tan MOT and McKean-Vlasov Control
Logo
IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
The McKean-Vlasov Control ProblemMartingale Optimal Transport (MOT)From MOT to McKean-Vlasov Control
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 .
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
The McKean-Vlasov Control ProblemMartingale Optimal Transport (MOT)From MOT to McKean-Vlasov Control
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.
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
The McKean-Vlasov Control ProblemMartingale Optimal Transport (MOT)From MOT to McKean-Vlasov Control
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.
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
The McKean-Vlasov Control ProblemMartingale Optimal Transport (MOT)From MOT to McKean-Vlasov Control
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).
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
The McKean-Vlasov Control ProblemMartingale Optimal Transport (MOT)From MOT to McKean-Vlasov Control
Peacocks
Figure – The bookXiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
The McKean-Vlasov Control ProblemMartingale Optimal Transport (MOT)From MOT to McKean-Vlasov Control
Peacocks
Haïku by Pf. Y. Takahashi
Figure – Peacocks
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
The McKean-Vlasov Control ProblemMartingale Optimal Transport (MOT)From MOT to McKean-Vlasov Control
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.
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
The McKean-Vlasov Control ProblemMartingale Optimal Transport (MOT)From MOT to McKean-Vlasov Control
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 .
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
The McKean-Vlasov Control ProblemMartingale Optimal Transport (MOT)From MOT to McKean-Vlasov Control
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 ).
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
The McKean-Vlasov Control ProblemMartingale Optimal Transport (MOT)From MOT to McKean-Vlasov Control
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.
Xiaolu Tan MOT and McKean-Vlasov Control
Logo
IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
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
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
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.
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
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 →∞.
Xiaolu Tan MOT and McKean-Vlasov Control
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IntroductionFrom MOT to McKean-Vlasov Control Problems
Solving the McKean-Vlasov Control Problems
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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