Mean-field SDE driven by a fractional BM. A related stochastic control problem

Mean-field SDE driven by a fractional BM. A related stochastic control problem Rainer Buckdahn, Université de Bretagne Occidentale, Brest Durham Symposium on Stochastic Analysis, July 1th to July 2th, 217

Outline Introduction Preliminaries Mean-field SDE driven by a fbm The mean-field control problem

Introduction Our objective: Study of the control problem with the dynamics driven by a fbm B H with Hurst parameter H (1/2, 1) X u t = x + t σ(p X u s )db H s + t b(p (X u s,u s), X u s, u s )ds, where x R, u U(, T ) - adapted control process with values in a convex open set U R m, and with the cost functional: J(u) = E [ f ( P (X u t,u t), Xt u ) ( ) ], u t dt + g X u T, P X u ; T Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 3/25

characterisation of an optimal control u U(, T ) s.t. J takes its minimum over U(, T ) at u. Short state-of-art: Several recent works on Pontryagin s maximum principle for mean-field control problems driven by a BM: + Carmona, Delarue (215),..., + Buckdahn, Li, Ma (Pontryagin s principle for a mean-field control problem with partial observation, 216); Several recent works on Peng s maximum principle (using spike control) for mean-field control problems driven by a BM: + Buckdahn, Djehiche, Li (211), + Buckdahn, Li, Ma (216); The fbm is not a semimartingale which makes the analysis much more subtle; much less works on maximum principle for control problems driven by a fbm: + Biagini, Hu, Oksendal, Sulem (22), Hu, Zhou (linear control system, Riccati equation is a linear BSDE driven by a BM and a fbm, 25), Han, Hu, Song Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 4/25

(213) (Pontryagin s maximum principle with regularity assumptions -as Malliavin differentiability- on the optimal control process; adjoint equation is a linear BSDE driven by a BM and a fbm); + Buckdahn, Jing: Peng s maximum principle for control problem driven by a BM and a fbm (214). With Shuai Jing we try to avoid regularity conditions on the optimal control. Preliminaries (Ω, F, P ) complete probability space; 1.Fractional BM. A minimalist overview: fbm with Hurst parameter H (, 1): centered Gaussian process B H with cov(b H t, B H s ) = 1 2 (t2h + s 2H t s 2H ), s, t. Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 5/25

For H (1/2, 1), BM W s.t. B H t = t K H (t, s)dw s, t [, T ], with kernel t K H (t, s) = c H s 1/2 H (u s) H 3/2 u H 1/2 du, t > s, s with constant c H = [H(2H 1)/β(2 2H, H 1/2)] 1/2, where β(α, γ) = Γ(α + γ)/(γ(α)γ(γ)) (Beta function), Γ(α) = x α 1 e x dx (Gamma function). Stochastic integral w.r.t. fbm: Let f L 2 ([, T ] Ω); f Dom(δ H ), if δ H (f) L 2 (Ω, F, P ) s.t., for all G D H 1,2, If fi [s,t] Dom(δ H ), Fractional calculus: E [Gδ H (f)] = t s E [ f(t)d H t G ] dt. f(r)db H r := δ H (fi [s,t] ). Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 6/25

Malliavin derivatives. Dt H instead of a BM; D H s G = - classical Malliavin derivative but now w.r.t. B H H(2H 1) s r 2H 2 D H r Gdr = (K H K HD. H G)(s), G S, { where K H is an operator on H := ( t K H(t, s) ˆf(s)ds) t [,T ], ˆf } L 2 ([, T ]) defined by (K H ψ)(s) = c H Γ(H 1/2)s 1/2 H I H 1/2 + (u H 1/2 ψ(u))(s) x with I+(f)(x) α = 1 f(u) Γ(α) (x u) 1 α du, for almost all x [, T ], and K H is its adjoint operator on H. Important: for all ψ H, ψ(t)db H (t)= (K Hψ)(t)dW (t), ψ(t)dw (t)= (K H 1 ψ)(t)db H (t). Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 7/25

Therefore, F := F W = F BH. L 2 -norm of stochastic integral: For all f Dom(δ H ), E [ f(t)dbt H 2] = E [ KHf(t) 2 dt ] + 2E [ s D H s f(r)d H r f(s)drds ]. Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 8/25

Mean-field SDE driven by a fbm Given ξ L 2 (Ω, F, P ), Θ L 2 ([, T ] Ω; R m ), we consider the SDE: (1) X t = ξ + t ( ) t γs X s + σ(s, P (Xs,Θ s) db H s + b(s, P (Xs,Θ s), X s )ds, where σ : [, T ] P 2 (R R m ) R and b : Ω [, T ] P 2 (R R m ) R R satisfy the following conditions: (H1) For any s [, T ], x, x R, η, η L 2 (Ω, F, P ) and Θ L 2 (Ω, F, P ; R m ), there exists a constant C > such that σ(s, P (η,θ) ) C, b(s, P (η,θ), x) C(1 + W 2 (P (η,θ), P (,Θ) ) + x ), σ(s, P (η,θ) ) σ(s, P (η,θ)) W 2 (P (η,θ), P (η,θ)), b(s, P (η,θ), x) b(s, P (η,θ), x ) C(W 2 (P (η,θ), P (η,θ)) + x x ). Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 9/25

The SDE will be solved using Girsanov transformation: For γ L ([, T ]) H, T t (ω) = ω + A t (ω) = ω t t K H(γI [,t] )(s)ds, K H(γI [,t] )(s)ds, t [, T ], ω Ω, Clearly, A t T t (ω) = T t A t (ω) = ω. Moreover, for any F S, from the Girsanov theorem E[F ] = E[F (T t )ε 1 t (T t )] = E[F (A t )ε t ], where ε t { t = exp γ s dbs H 1 t ( K 2 H (γi [,t] ) ) } 2 (s)ds { t = exp KH(γI [,t] )(s)dw s 1 t ( K 2 H (γi [,t] ) ) } 2 (s)ds, Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 1/25

and[ E sup ε p t t [,T ] ] < + and E sup t [,T ] [ sup ε p t (T t ) t [,T ] ] < +, for all p R. Let L 2, ([, T ]; R) be the Banach space of F-adapted processes {ϕ(t), t [, T ]} such that E [ ϕ(t) 2 ε 1 ] t < +. Theorem. The above SDE (1) has a unique solution X L 2, ([, T ]; R). Remark. Proof is based on the following statement with rather technical proof: Proposition. 1) X L 2, ([, T ]; R) is a solution of our SDE iff it solves equ. (2) Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 11/25

= ξ + X t (T t )ε 1 t (T t ) t σ(s, P (Xs,Θ s))ε 1 s (T s )dbs H + t b(s, T s, P (Xs,Θ s), X s (T s ))ε 1 (T s )ds. 2) For all deterministic Θ L ([, T ]), ( Θ s ε 1 s (T s ) ) s [,T ] Dom(δ H). 3) The above SDE (2) has a unique solution X L 2, ([, T ]; R) Proof. Main tools: + The method of Girsanov transformation; + The formula for E [ f(t)dbh t 2] (involving f and its Malliavin derivative Df); + Picard s iteration with the distance function (E[ X t X t 2 ε 1 t ]) 1/2. s Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 12/25

The mean-field control problem Recall: B H fbm with Hurst parameter H (1/2, 1); control state space U R m nonempty, convex, bounded; U([, T ]) := L F ([, T ]; U). Given x R, u U([, T ]) we consider the control problem with dynamics (3) X u t = x + t with the cost functional (4) J(u) = E [ σ(p X u s )db H s + t b(p (X u s,u s), X u s, u s )ds, f ( P (X u t,u t), Xt u ) ( ) ], u t dt + g X u T, P X u. T Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 13/25

Assumptions: (H2) σ : P 2 (R) R, b : P 2 (R U) R U R, g : R P 2 (R) R and f : P 2 (R U) R U U are bounded and Lipschitz (in all variables); (H3) σ, b, f and g are C 1 in (x, µ, u) (with µ P 2 (R) and P 2 (R U), respectively), and all first order derivatives are bounded and Lipschitz. This means, e.g., that, for some C R, for any (µ, y), (µ, y ) P 2 (R) R, µ σ(µ, y) µ σ(µ, y ) C(W 2 (µ, µ ) + y y ). Under the above assumptions we have seen the existence and the uniqueness for every u U([, T ]). Assume that there is a u U([, T ]) minimising J : U([, T ]) R; X := X u. Objective: Characterisation of u by convex perturbation: For u U([, T ]), ε, put u ε := u + ε(u u ) U([, T ]), X ε := X uε. Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 14/25

Lemma The following SDE obtained by formal differentiation of (3) for u = u ε w.r.t. ε at ε =, for t [, T ], Y t = [ ( t Ẽ µ σ P X s, X ) ] s Ỹs dbs H + [ ( t Ẽ µ b P (X s,u s ), Xs, u s, X ) ) ] s, ũ s, (Ỹs, ũ s ũ s ds + t xb ( ) P (X s,u s ), Xs, u s Ys ds + t ub ( P (X s,u s ), Xs, us) (us u s)ds, has a unique solution Y L 2 F ([, [ T ]). Moreover, lim sup E ε Y t Xε t Xt 2] =. ε t [,T ] Above, ( X, Ỹ, ũ, ũ ) is an independent copy of (X, Y, u, u ) defined on a probability space ( Ω, F, P ). The expectation Ẽ[ ] under P only concerns ( X, Ỹ, ũ, ũ ) but not (X, Y, u, u ). Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 15/25

As (X, u ) is optimal, J(u ε ) J(u ), ε [, 1]. Thus, d dε J(uε ) 1 := lim ε= <ε ε (J(uε ) J(u )), and d dε J(uε ) = E [ x g(xt, P X )Y ] [ T T + E [Ẽ µ g(xt, P X, X ]] T T )ỸT ε= +E [ x f(p (X t,u t ), Xt, u t )Y t dt ] +E [ [ Ẽ ( µ f) 1 (P (X t,u t ), Xt, u t, X ] t, ũ t )Ỹt dt ] +E [ [ Ẽ ( µ f) 2 (P (X t,u t ), Xt, u t, X ] t, ũ t )(ũ t ũ t ) dt ] +E [ u f(p (X t,u t ), Xt, u t )(u t u t )dt ]. (5) Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 16/25

We introduce the adjoint BSDE with its F-adpated solution (P, β): P t = P T α s ds β s dw s, t [, T ], t t P T = x g(xt, P X [ ) + Ẽ T µ g( X ] T, P X T, X T ), where α L 2 F ([, T ]) will be specified later. Applying Itô s formula (with BH ) taking the expectation and rearranging terms, we get E [ ] Y T P T = E [ {Ẽ[ ( Y s Ps ( µ b) 1 P(X s,u s ), X s, ũ s, Xs )] + Ps x b ( P (X s,u s ), Xs, u ) s +α s + µ σ ( P X s, Xs ) [ ]}] E D H s P s ds + E [( Ẽ [ ( Ps ( µ b) 2 P(X s,u s ), X s, ũ s, u )] s +P s u b ( P (X s,u s ), X s, u s) ) (u s u s) ] ds. (6) Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 17/25

Combining (5) with (6) we obtain E [ {Ẽ[ ( Y s Ps ( µ b) 1 P(X s,u s ), X s, ũ s, Xs )] +Ps x b ( P (X s,u s ), Xs, u ) s + + µ σ ( P X s, Xs ) [ ] E D H s P s + x f(p (X s,u s ), Xs, u s) + α s +Ẽ[ ( µ f) 1 (P (X s,u s ), X s, ũ s, Xs, u s) ]}] ds E [ (u s u s) { Ẽ [ ( µ f) 2 (P (X s,u s ), X s, ũ s, X s, u s) ] (7) + u f(p (X s,u s ), X s, u s) +Ẽ[ Ps ( µ b) 2 ( P(X s,u s ), X s, ũ s, X s, u s)] +Ps u b ( P (X s,u s ), X s, u s)}] ds. Putting the first integral equal to zero, we choose α s = Ẽ[ ( Ps ( µ b) 1 P(X s,u s ), X )] s, ũ s, Xs Ps x b ( ) P (X s,u s ), Xs, u s µ σ ( ) [ ] P X s, Xs E D H s P s x f(p (X s,u s ), Xs, u s) Ẽ[ ( µ f) 1 (P (X s,u s ), X s, ũ s, Xs, u s) ]. Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 18/25

This gives the following form of the BSDE for P t = P T t α s ds t β s dw s : { P t = P T + Ẽ [ ( Ps ( µ b) 1 P(X s,u s ), X s, ũ s, Xs, u )] s t +P s x b ( P (X s,u s ), Xs, u ) s + µ σ ( P X s, Xs ) [ ] E D H s P s + x f(p (X s,u s ), Xs, u s) +Ẽ[ ( µ f) 1 (P (X s,u s ), X s, ũ s, X s, u s) ]} ds t (8) β s dw s, which is a mean-field BSDE driven by the standard Brownian motion W. Let us assume that the above BSDE has an F-adapted solution (P, β) (Discussion on existence and unique for the BSDE later). Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 19/25

Then (7) becomes E [ (u s u s) { Ẽ [ ( µ f) 2 (P (X s,u s ), X s, ũ s, X s, u s) ] (9) + u f(p (X s,u s ), X s, u s) +Ẽ[ Ps ( µ b) 2 ( P(X s,u s ), X s, ũ s, X s, u s)] +Ps u b ( P (X s,u s ), X s, u s)}] ds. As U is open and u U([, T ]) arbitrary, we have = Ẽ[ ( µ f) 2 (P (X t,u t ), X t, ũ t, Xt, u t ) ] + u f(p (X t,u t ), Xt, u t ) +Ẽ[ ( Pt ( µ b) 2 P(X t,u t ), X )] t, ũ t, Xt, u t + Pt u b ( ) P (X t,u t ), Xt, u t, dp -a.s., dt-a.e. Hence, we have following necessary conditions of Pontryagin-type. Theorem. If (X, u ) is an optimal pair of our control problem, then (X, u ) satisfies the following system: Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 2/25

X t = x + P T = P t = t σ(p X s )db H s + t b(p (X s,u s ), X s, u s)ds, (1) x g(xt, P X ) + Ẽ[ T µ g( X T, P X, X T T ) ], { P T β s dw s + Ẽ [ ( Ps ( µ b) 1 P(X s,u s ), X s, ũ s, Xs, u )] s t t +P s x b ( P (X s,u s ), Xs, u s) + µ σ ( P X, ) [ ] X s s E D H s P s + x f(p (X s,u s ), X s, u s) + Ẽ[ ( µ f) 1 (P (X s,u s ), X s, ũ s, X s, u s) ]} ds, = Ẽ[ ( Pt ( µ b) 2 P(X t,u t ), X t, ũ t, Xt, u )] t + Pt u b ( P (X t,u t ), Xt, u ) t +Ẽ[ ( µ f) 2 (P (X t,u t ), X t, ũ t, Xt, u t ) ] + u f(p (X t,u t ), Xt, u t ), dp -a.s., dt-a.e. After this necessary condition we can give also a sufficient one: Theorem. In addition to our standard assumptions, assume that g = g(x, µ) is Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 21/25

convex in (x, µ) and the Hamiltonian H(µ, x, u, y, z) := f(µ, x, u) + b(µ, x, u)y + σ(µ)z be jointly convex in (µ, x, u). Let (u, X ) satisfy the above system. Then (u, X ) is optimal: J(u ) = inf u U([,T ]) J(u). Remark. The convexity assumption on the Hamiltonian implies practically the linearity of b and σ. Another assumption in which b and σ don t need to be linear: + g : R P 2 (R) R jointly convex in (x, µ), with x g, µ g ; + b(µ, x, u) : P 2 (R U) R U R is jointly convex in (µ, x, u) with ( µ b) 1 (µ, x, u, y), strictly convex in (µ, u); + f(µ, x, u) : P 2 (R U) R U R is jointly convex in (µ, x, u), and strictly convex in (µ, u), with ( µ f) 1 (η, x, u, y), x f(η, x, u, y) ; + σ(µ) σ R. Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 22/25

Recall (Carmona, Delarue, 215): g convex (strictly convex), if there exists exists λ (resp., > ) such that for all x, x R, ξ, ξ L 2 (Ω, F, P ), g(x, P ξ ) g(x, P ξ ) x g(x, P ξ )(x x) + E [ µ g(x, P ξ, ξ)(ξ ξ) ] + λ ( x x 2 + E[ ξ ξ 2 ] ). Theorem. If in addition to our standard assumptions the above assumption is satisfied and (u, X ) solves the above system, then (u, X ) is optimal: J(u ) = inf u U([,T ]) J(u). Concerning the solvability of the coupled forward-backward system (1) under the conditions of the preceding Theorem, we proceed as follows: For any given (P, ξ) L 2 (F t ) L 2 (F t ), we suppose that there is some η L 2 (F t ; U) such that: Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 23/25

= Ẽ[ ( µ f) 2 ( P(ξ,η), ξ, η, ξ, η )] + u f(p (ξ,η), ξ, η) +Ẽ[ P ( µ b) 2 ( P(ξ,η), ξ, η, ξ, η )] + P ( u b)(p (ξ,η), ξ, η). Lemma. The mapping (P, ξ) η is unique and η = η(p, ξ) is Lipschitz in (P, ξ) under L 2 -norm. Theorem. Under the assumptions of the preceding theorem (i.e., in particular σ(µ) = σ R) and the preceding assumption, then, if the time horizon T > is sufficiently small, there exists a unique solution (X, (P, β), u = η(x, P )) of the coupled FBSDE (1). Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 24/25

THANK YOU VERY MUCH! Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 25/25