Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Quenched Mass Transport of Particles Towards a Target Idris Kharroubi LPSM Sorbonne University (Ex-Pierre et Marie Curie-Paris 6 University) International Conference on Control, Games and Stochastic Analysis Hammamet, Tunisia, Oct.-Nov. 2018 Joint work with B. Bouchard (Universit´ e Paris Dauphine) and B. Djehiche (KTH Stockholm) Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Outline Introduction 1 Quenched mean-field SDE 2 Stochastic target problem 3 The dynamic programming PDE 4 Conclusion 5 Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Stochastic target problems • Consider a random dynamical system controlled by ν and starting at time t with the value x . • Suppose that this system is described by a process X t , x ,ν . Look for the values x such that the system reaches a set K at a terminal time T by choosing an appropriate control ν . Namely, the objective is to characterize the reachability sets � � x ∈ R d : X t , x ,ν V ( t ) = ∈ K for some admissible control ν T for t ∈ [0 , T ]. Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Brownian diffusions and HJB equations • In the Brownian diffusion case, v ( t , . ) = 1 − 1 V ( t ) is shown to be solution to an HJB PDE (Soner & Touzi 2001). • The main motivation is the super-replication problem in finance: find initial endowments such that there exists an investment strategy allowing the terminal wealth to be greater than a given pay-off (see e.g. El Karoui & Quenez 95). Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Extension of the stochastic target problem • In general the super-replication price is too high. • Possible generalization: investment under terminal loss constraint: � � x ∈ R d : E [ ℓ ( X t , x ,ν V ℓ ( t ) = )] � 0 for some control ν . T Motivation: relaxing the a.s. super-hedging constraint to get a lower price. In this case, we take ℓ ( x ) = 1 K ( x ) − p with p ∈ [0 , 1]. Approach introduced in F¨ olmer and Leuckert 99. Then developed in Bouchard et al. 10. Main idea of this last paper: use martingale representation to express the expectation constraint as an a.s. constraint on an extended process. Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Our motivation Study the stochastic target problem for non-linear controlled diffusions: � s � s X t ,χ,ν b u ( X t ,χ,ν σ u ( X t ,χ,ν = χ + , P X t ,χ,ν , ν u ) du + , P X t ,χ,ν , ν u ) dB u , s u u u u t t where B is a standard Brownian motion, χ an independent random variable whose distribution can be interpreted as the initial repartition of a population. Still with constraint E [ ℓ ( X t ,χ,ν )] � 0. T Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Extended problem: conditional law We consider a constraint on the condition law of X t ,χ,ν given B : T � � x ∈ R d : P B V ( t ) = ∈ K for some control ν , X t ,χ,ν T This includes the previous problem. Indeed, from Mart. Rep. Thm � � T E [ ℓ ( X t ,χ,ν ℓ ( x ) d P B )] = ( x ) − α s dB s . X t ,χ,ν T T 0 Hence the constraint E [ ℓ ( X t ,χ,ν )] � 0 can be rewritten T L ( P B ) 0 � X t , ¯ χ, ¯ ¯ ν T � . ν = ( X t ,χ,ν , η + χ = ( χ, η ), ¯ X t , ¯ χ, ¯ with ¯ ν = ( ν, α ), ¯ t α s dB s ) and � L ( µ ) = ( ℓ ( x ) − y ) µ ( dx , dy ) . Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Extended problem We can also extend the dynamics of X t ,χ,ν as follows: T � s � s X t ,χ,ν b u ( X t ,χ,ν , P B σ u ( X t ,χ,ν , P B = χ + , ν u ) du + , ν u ) dB u . X t ,χ,ν X t ,χ,ν s u u u u t t Such general formulation is related to the probabilistic analysis of large scale particle systems. In those systems, one is interested in the behavior of particles conditionally to the environment (‘quenched’ behaviors/properties). Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Interpretation One can also identify the initial condition χ as a law µ . Then, our problem can be interpreted as a transport problem: what is the collection of initial distributions µ of a population of particles, such that the terminal repartition P B , given the X t ,χ,ν T environment (modeled by B ) satisfies the constraint ? � � µ : ∃ ( χ, ν ) s.t. P B χ = µ and P B V ( t ) = ∈ G . X t ,χ,ν T Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Outline Introduction 1 Quenched mean-field SDE 2 Stochastic target problem 3 The dynamic programming PDE 4 Conclusion 5 Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Outline Introduction 1 Quenched mean-field SDE 2 Stochastic target problem 3 The dynamic programming PDE 4 Conclusion 5 Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Probabilistic setting T > 0 fixed time horizon. Ω ◦ = { ω ◦ ∈ C ([0 , T ] , R d ) : ω ◦ 0 = 0 } F ◦ = ( F ◦ t ) t ≤ T filtration generated by the canonical process B ( ω ◦ ) := ω ◦ , ω ◦ ∈ Ω ◦ . P ◦ Wiener measure on (Ω ◦ , F ◦ T ). F ◦ = ( ¯ ¯ F ◦ t ) t ≤ T the P ◦ -completion of F ◦ . Ω ı := [0 , 1] d endowed with σ -algebra F ı := B ([0 , 1] d ) Lebesgue measure P ı . Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Probability space We then define the product filtered space (Ω , F , F , P ) by Ω := Ω ◦ × Ω ı , P = P ◦ ⊗ P ı , F = F T where F = ( F t ) t ≤ T is the completion of ( F ◦ t ⊗ F ı ) t ≤ T . We canonically extend the random variable ξ and the process B on Ω by setting ξ ( ω ) = ξ ( ω ı ) and B ( ω ) = B ( ω ◦ ) for any ω = ( ω ◦ , ω ı ) ∈ Ω. Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Wasserstein space We define � � � µ probability measure on ( R d , B ( R d )) s.t. R d | x | 2 µ ( dx ) < + ∞ P 2 := . This space is endowed with the 2-Wasserstein distance defined by � � W 2 ( µ, µ ′ ) := inf R d × R d | x − y | 2 π ( dy , dy ) : s.t. π ( · × R d ) = µ and π ( R d × · ) = µ ′ � 1 2 , for µ, µ ′ ∈ P 2 . ( P 2 , W 2 ) is then Polish. Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Controlled diffusion Let U be a closed subset of R q for some q � 1 and U the set of U -valued F -progressive processes. Given T ◦ (the set of [0 , T ]-valued ¯ θ ∈ ¯ F ◦ -stopping times), χ ∈ L 2 (Ω , F θ , P ; R d ), ν ∈ U , we let X θ,χ,ν denote the solution of � · � · � � � � X s , P B X s , P B = χ + ds + X b s X s , ν s a s X s , ν s dB s , θ θ on [ θ, T ]. Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Existence and stability for SDEs We suppose that b , a are continuous, bounded and Lipschitz: there exists a constant L such that � � | b t ( x , µ, · ) − b t ( x ′ , µ ′ , · ) | + | a t ( x , µ, · ) − a t ( x ′ , µ ′ , · ) | � L | x − x ′ | + W 2 ( µ, µ ′ ) for all t ∈ [0 , T ], x , x ′ ∈ R d and µ, µ ′ ∈ P 2 . Proposition For all θ ∈ ¯ T ◦ , ν ∈ U and χ ∈ L 2 ( F θ ) , the SDE admits a unique strong solution X θ,χ,ν , and it satisfies � | X θ,χ,ν | 2 � sup < + ∞ . E [0 , T ] Idris Kharroubi Mass Transport Towards a Target
Introduction Quenched mean-field SDE Stochastic target problem The dynamic programming PDE Conclusion Existence and stability for SDEs We suppose that b , a are continuous, bounded and Lipschitz: there exists a constant L such that � � | b t ( x , µ, · ) − b t ( x ′ , µ ′ , · ) | + | a t ( x , µ, · ) − a t ( x ′ , µ ′ , · ) | � L | x − x ′ | + W 2 ( µ, µ ′ ) for all t ∈ [0 , T ], x , x ′ ∈ R d and µ, µ ′ ∈ P 2 . Proposition For all θ ∈ ¯ T ◦ , ν ∈ U and χ ∈ L 2 ( F θ ) , the SDE admits a unique strong solution X θ,χ,ν , and it satisfies � | X θ,χ,ν | 2 � sup < + ∞ . E [0 , T ] Idris Kharroubi Mass Transport Towards a Target
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