Stochastic control of mean-field SPDEs with jumps
aa r X i v : . [ m a t h . O C ] A p r Stochastic control of mean-field SPDEs with jumps
Roxana Dumitrescu ∗ Bernt Øksendal † Agn`es Sulem ‡ April 12, 2017
Abstract
We study the problem of optimal control for mean-field stochastic partial differen-tial equations (stochastic evolution equations) driven by a Brownian motion and anindependent Poisson random measure, in the case of partial information control. Oneimportant novelty of our problem is represented by the introduction of general mean-field operators, acting on both the controlled state process and the control process.We first formulate a sufficient and a necessary maximum principle for this type of con-trol. We then prove existence and uniqueness of the solution of such general forwardand backward mean-field stochastic partial differential equations. We finally apply ourresults to find the explicit optimal control for an optimal harvesting problem.
Keywords:
Mean-field stochastic partial differential equation (MFSPDE); optimal con-trol; mean-field backward stochastic partial differential equation (MFBSPDE); stochasticmaximum principles.
As a motivation for the problem studied in this paper, we consider the following optimalharvesting problem: Suppose we model the density Y ( t, x ) of a fish population in a lake ∗ Department of Mathematics, King’s College London, United Kingdom, email: [email protected] † Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway, email: [email protected] ‡ INRIA Paris, Equipe-projet MathRisk, 3 rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France, email: [email protected] at time t and at point x ∈ D by an equation of the form: dY ( t, x ) = E [ Y ( t, x )] b ( t, x ) dt + 12 d X i =1 ∂ ∂ x i Y ( t, x ) dt + Y ( t, x ) σ ( t, x ) dW t + Y ( t, x ) Z R ∗ θ ( t, x, e ) ˜ N ( dt, de ) .Y (0 , x ) = y ( x ) , x ∈ D, (1.1)where D is a bounded domain in R d and y ( x ) , b ( t, x ) , σ ( t, x ) , θ ( t, x, e ) are given boundeddeterministic functions. Here W t is a Brownian motion and ˜ N ( dt, de ) = N ( dt, de ) − ν ( de ) dt isan independent compensated Poisson random measure, respectively, on a filtered probabilityspace (Ω , F , F = {F t } , P ).We may heuristically regard (1.1) as a limit as n → ∞ of a large population interactingsystem of the form dy j,n ( t, x ) = " n n X l =1 y l,n ( t, x ) b ( t, x ) dt + 12 d X i =1 ∂ ∂ x i y j,n ( t, x ) dt + y j,n ( t, x ) σ ( t, x ) dW t + y j,n ( t, x ) Z R ∗ θ ( t, x, e ) ˜ N ( dt, de ) , j = 1 , , ..., ny j,n ( t, x )(0 , x ) = y ( x ) , (1.2)where we have divided the whole lake into a grid of size n and y j,n ( t, x ) represents the densityin box j of the grid. Now suppose we introduce a harvesting-rate process u ( t, x ). The densityof the corresponding population Y ( t, x ) = Y u ( t, x ) is thus modeled by a controlled mean-fieldstochastic partial differential equation with jumps of the form: dY ( t, x ) = E [ Y ( t, x )] b ( t, x ) dt + 12 d X i =1 ∂ ∂ x i Y ( t, x ) dt + Y ( t, x ) σ ( t, x ) dW t + Y ( t, x ) Z R ∗ θ ( t, x, e ) ˜ N ( dt, de ) − Y ( t, x ) u ( t, x ) dt. (1.3)The performance functional is assumed to be of the form J ( u ) = E (cid:20)Z T Z D log( Y ( t, x ) u ( t, x )) dxdt + Z D α ( x ) Y ( T, x ) dx (cid:21) . (1.4)This may be regarded as the expected total logarithmic utility of the harvest up to time T plus the value of the remaining population at time T .The problem is thus to find u ∗ such that J ( u ∗ ) = sup u ∈A J ( u ) , (1.5)where A represents the set of admissible controls. This process u ∗ ( t, x ) is called an optimalharvesting rate. 2his is an example of an optimal control problem of a mean-field stochastic reaction-diffusion equation. In the next sections, we will give necessary and sufficient conditionsfor optimality of a control in the case of partial information , as well as results of existenceand uniqueness for forward and backward mean-field stochastic partial differential equationswith a general mean-field operator . Finally, we apply our results in order to solve the optimalharvesting problem presented above.To the best of our knowledge, the only paper that deals with optimal control of mean-field SPDEs is [15]. Our paper extends [15] in four ways: (i) we consider a more generalmean-field operator; (ii) we introduce an additional general mean-field operator which actson the control process; (iii) we add jumps; (iv) we study the optimal control problem in thecase of partial information .The paper is organized as follows: in Section 2 we show the sufficient and necessarymaximum principles and apply the results to the optimal harvesting example. In Section3, we investigate the existence and the uniqueness of the solution of mean-field SPDEswith jumps and general mean-field operator. In Section 4, we prove the existence and theuniqueness of mean-field backward SPDEs with jumps and general mean-field operator. Let (Ω , F , IF = {F t } ≤ t ≤ T , P ) be a filtered probability space. Let W be a one-dimensionalBrownian motion. Let E := R ∗ and B ( E ) be its Borel filtration. Suppose that it is equippedwith a σ -finite positive measure ν , satisfying R E | e | ν ( de ) < ∞ and let N ( dt, de ) be a in-dependent Poisson random measure with compensator ν ( de ) dt . We denote by ˜ N ( dt, de ) itscompensated process, defined as ˜ N ( dt, de ) = N ( dt, de ) − ν ( de ) dt . For simplicity, we consider d = 1 . We introduce the following notation: • L ( P ):= the set of random variables X such that E [ | X | ] < ∞ . • L ( R ):= the set of measurable functions k : ( R , B ( R )) → ( R , B ( R )) with R R k ( x ) dx < ∞ . • H := the set of real-valued predictable processes Z ( t, x ) with E [ R T R D Z ( t, x ) dxdt ] < ∞ , where D a bounded domain in R . • L ν := the set of measurable functions l : ( E , B ( E )) → ( R , B ( R )) such that k l k L ν := R E l ( e ) ν ( de ) < ∞ . The set L ν is a Hilbert space equipped with the scalar product < l, l ′ > ν := R E l ( e ) l ′ ( e ) ν ( de ) for all l, l ′ ∈ L ν × L ν .3 H ν : the set of predictable real-valued processes k ( t, x, · ) with E [ R T R D k k ( t, x, · ) k L ν ] < ∞ .Assume that we are given a subfiltration E t ⊆ F t ; t ∈ [0 , T ] , representing the information available to the controller at time t . For example, we couldhave E t = F ( t − δ ) + ( δ > delayed information flow compared to F t .Consider a controlled mean-field stochastic partial differential equation Y ( t, x ) = Y u ( t, x )at ( t, x ) of the following form dY ( t, x ) = (cid:2) LY ( t, x ) + b ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) (cid:3) dt + σ ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) dW t + Z E θ ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) , e ) ˜ N ( dt, de ); ( t, x ) ∈ (0 , T ) × D. (2.6)with boundary conditions Y (0 , x ) = ξ ( x ); x ∈ D (2.7) Y ( t, x ) = η ( t, x ); ( t, x ) ∈ (0 , T ) × ∂D. (2.8)We interpret Y as a weak (variational) solution to (2.6), in the sense that for φ ∈ C ∞ ( D ) ,< Y t , φ > L ( D ) = < y , φ > L ( D ) + Z t < Y s , L ∗ φ > ds + Z t < b ( s, Y s ) , φ > L ( D ) ds + Z t < σ ( s, Y s ) , φ > L ( D ) dW s + Z t Z E < θ ( s, Y s , e ) , φ > L ( D ) ˜ N ( ds, de ) , (2.9)where L ∗ corresponds to the adjoint operator of L and < · , · > represents the duality productbetween W , ( D ) and W , ( D ) ∗ , with W , ( D ) the Sobolev space of order 1. Existence andthe uniqueness of the solution are proved in Section 4.Under this framework the Itˆo formula can be applied to such SPDEs. See e.g. Pardoux[10], Pr´evot and Rockner [12].Here, dY ( t, x ) = d t Y ( t, x ) is the differential with respect to t and L is a bounded lin-ear differential operator acting on x . The process u ( t, x, ω ) is our control process, takingvalues in a set A ⊂ R . The functions b : [0 , T ] × Ω × D × R × A × R R ; ( t, ω, x, y, ¯ y, u, ¯ u ) b ( t, ω, x, y, ¯ y, u, ¯ u ), σ : [0 , T ] × Ω × D × R × A × R R ; ( t, ω, x, y, ¯ y, u, ¯ u ) σ ( t, ω, x, y, ¯ y, u, ¯ u ), θ : [0 , T ] × Ω × D × R × A × R × E R ; ( t, ω, x, y, ¯ y, u, ¯ u, e ) θ ( t, ω, x, y, ¯ y, u, ¯ u, e ) aregiven F t -predictable processes for each x, y, ¯ y, ¯ u ∈ R , u ∈ A , e ∈ E . We assume that b, σ, θ are C b and have linear growth with respect to y, ¯ y, u, ¯ u . We denote by A E a given family of admissible controls, contained in the set of E t -predictable stochastic processes u ( t, x ) ∈ A E [ R T R D u ( t, x ) dxdt ] < ∞ and such that (2.6)-(2.7)-(2.8) has a unique c`adl`agsolution Y ( t, x ).In the above equation, F , G : L ( P ) R are Fr´echet differentiable operators. Oneimportant example is represented by the expectation operator E [ · ].Let f : [0 , T ] × D × R × A × R R and g : D × R R be a given profit rate functionand bequest rate function, respectively. Moreover, we suppose that E (cid:20)Z T (cid:18)Z D | f ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) | dx (cid:19) dt + Z D | g ( x, Y ( T, x ) , F( Y ( t, x ) ) | dx (cid:21) < ∞ , where f ( t, x, y, ¯ y, u, ¯ u ) , g ( x, y, ¯ y ) are of class C b with respect to ( y, ¯ y, u, ¯ u ) and continuousw.r.t to t . E denotes the expectation with respect to P .For each u ∈ A E , we define the performance functional J ( u ) by J ( u ) = E (cid:20)Z T (cid:18)Z D f ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) dx (cid:19) dt + Z D g ( x, Y ( T, x ) , F( Y ( t, x ) ) | dx (cid:21) . (2.10)We aim to maximize J ( u ) over all u ∈ A E and our problem is the following:Find u ∗ ∈ A E such that sup u ∈A J ( u ) = J ( u ∗ ) . (2.11)Such a process u ∗ is called an optimal control (if it exists), and the number J = J ( u ∗ ) is the value of this problem. In this section, we prove necessary and sufficient maximum principles for optimal controlwith partial information in the case of a process described by a mean-field stochastic partial differential equation (in short MFSPDE) driven by a Brownian motion ( W ) and a Poissonrandom measure ˜ N . The drift and the diffusion coefficients as well as the performancefunctional depend not only on the state and the control but also on the distribution of thestate process, and also on the one of the control.Define the Hamiltonian H : [0 , T ] × D × R × A × R × L ν R as follows: H ( t, x, y, ¯ y, u, ¯ u, p, q, γ ) = f ( t, x, y, ¯ y, u, ¯ u ) + b ( t, x, y, ¯ y, u, ¯ u ) p + σ ( t, x, y, ¯ y, u, ¯ u ) q + Z E θ ( t, x, y, ¯ y, u, ¯ u, e ) γ ( e ) ν ( e ) . (2.12)5n our case, since the state process and the cost functional are of mean-field type, it turnsout that the adjoint equation will be a mean-field backward SPDE , denoted in the sequelMFBSPDEs.We now introduce the adjoint operator of the operator L , denoted by L ∗ , which satisfies( L ∗ φ, ψ ) = ( φ, Lψ ) , for all φ, ψ ∈ C ∞ ( R ) , (2.13)where < φ , φ > L ( R ) := ( φ , φ ) = Z R φ ( x ) φ ( x ) dx is the inner product in L ( R ) . For u ∈ A E , we consider the following mean field backward stochastic partial differen-tial equation (the adjoint equation) in the three unknown processes p ( t, x ) ∈ R , q ( t, x ) ∈ R , γ ( t, x, · ) ∈ L ν ; called the adjoint processes : dp ( t, x ) = − (cid:20) L ∗ p ( t, x ) + ∂H∂y ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) , p ( t, x ) , q ( t, x ) , γ ( t, x, · )) (cid:21) dt − E (cid:20) ∂H∂ ¯ y ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) , p ( t, x ) , q ( t, x ) , γ ( t, x, · )) (cid:21) ∇ F( Y ( t, x ) ) dt + q ( t, x ) dW t + Z E γ ( t, x, e ) ˜ N ( dt, de ); ( t, x ) ∈ (0 , T ) × D. (2.14) p ( T, x ) = ∂g∂y ( x, Y ( T, x ) , F( Y ( T, x ) ) ) + E (cid:20) ∂g∂ ¯ y ( x, Y ( T, x ) , F( Y ( T, x ) ) (cid:21) ∇ F( Y ( T, x ) ) ; x ∈ D (2.15) p ( t, x ) = 0; ( t, x ) ∈ (0 , T ) × ∂D. (2.16)Note that (2.14) is equivalent to dp ( t, x ) = − (cid:20) L ∗ p ( t, x ) + ∂f∂y ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) )+ ∂b∂y ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) p ( t, x ) (cid:21) dt − (cid:20) ∂σ∂y ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) q ( t, x )+ Z E ∂θ∂y ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) , e ) γ ( t, x, e ) ν ( de ) (cid:21) dt − E (cid:20) ∂f∂ ¯ y ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) )6 ∂b∂ ¯ y ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) p ( t, x ) (cid:21) ∇ F( Y ( t, x ) ) dt − E (cid:20) ∂σ∂ ¯ y ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) q ( t, x ) (cid:21) ∇ F( Y ( t, x ) ) dt − E (cid:20)Z E ∂θ∂ ¯ y ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) , e ) γ ( t, x, e ) ν ( de ) (cid:21) ∇ F( Y ( t, x ) ) dt + q ( t, x ) dW t + Z E γ ( t, x, e ) ˜ N ( dt, de ) , x ∈ D. (2.17)We now show the sufficient maximum principle. Theorem 2.1 (Sufficient Maximum Principle for mean-field SPDEs with jumps)
Let ˆ u ∈ A E with corresponding solution ˆ Y ( t, x ) and suppose that ˆ p ( t, x ) , ˆ q ( t, x ) and ˆ γ ( t, x, · ) is a solution of the adjoint MFBSPDE (2.14) - (2.15) - (2.16) . Assume the following hold:(i) The maps Y g ( x, Y, F ( Y )) and ( Y, u ) H ( Y, u ) := H ( t, x, Y, F ( Y ) , u, G ( u ) , ˆ p ( t, x ) , ˆ q ( t, x ) , ˆ γ ( t, x, · )) (2.18) are concave functions with respect to Y and ( Y, u ) , respectively, for all ( t, x ) ∈ [0 , T ] × ¯ D .(ii) (The maximum condition) E h H ( t, x, ˆ Y ( t, x ) , F ( ˆ Y ( t, x )) , ˆ u ( t, x ) , G (ˆ u ( t, x )) , ˆ p ( t, x ) , ˆ q ( t, x ) , ˆ γ ( t, x, · )) |E t i = ess sup v ∈A E E h H ( t, x, ˆ Y ( t, x ) , F ( ˆ Y ( t, x )) , v ( t, x ) , G ( v ( t, x )) , ˆ p ( t, x ) , ˆ q ( t, x ) , ˆ γ ( t, x, · )) |E t i a.s . (2.19) for all t ∈ [0 , T ] and x ∈ ¯ D .Then ˆ u ( t ) is an optimal control for the random jump field control problem (2.11) . Proof. Define a sequence of stopping times τ n ; n = 1 , , .... as follows: τ n := inf { t >
0; max {k ˆ p ( t ) k L ( D ) , k ˆ q ( t ) k L ( D ) , k ˆ γ ( t ) k L ( D × E ) , k σ ( t ) − ˆ σ ( t ) k L ( D ) , k θ ( t ) − ˆ θ ( t ) k L ( D × E ) , k Y ( t ) − ˆ Y ( t ) k L ( D ) ≥ n } ∧ T. Then τ n → T as n → ∞ and E (cid:20)Z τ n (cid:18)Z D ˆ p ( t, x )( σ ( t, x ) − ˆ σ ( t, x )) dx (cid:19) dW t + Z τ n Z E (cid:18)Z D ( θ ( t, x, e ) − ˆ θ ( t, x, e )) dx (cid:19) ˜ N ( dt, de ) (cid:21) = E (cid:20)Z τ n (cid:18)Z D ( Y ( t, x ) − ˆ Y ( t, x ))ˆ q ( t, x ) dx (cid:19) dW t + Z τ n Z E (cid:18)Z D ( Y ( t, x ) − ˆ Y ( t, x ))ˆ γ ( t, x, e ) dx (cid:19) ˜ N ( dt, de ) (cid:21) = 0 for all n. u ∈ A E and let Y ( t, x ) = Y u ( t, x ) be the associated solution of (2.6). Define: ( ˆ f := f ( t, x, ˆ Y ( t, x ) , F ( ˆ Y ( t, x )) , ˆ u ( t, x ) , G (ˆ u ( t, x ))); f := f ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G ( u ( t, x )));ˆ g := g ( x, ˆ Y ( T, x ) , F ( ˆ Y ( T, x ))); g := g ( x, Y ( T, x ) , F( Y ( T, x ) ) ); (2.20)and ˆ b := b ( t, x, ˆ Y ( t, x ) , F ( ˆ Y ( t, x )) , ˆ u ( t, x ) , G (ˆ u ( t, x ))); b := b ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) );ˆ σ := σ ( t, x, ˆ Y ( t, x ) , F ( ˆ Y ( t, x )) , ˆ u ( t, x ) , G (ˆ u ( t, x ))); σ := σ ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) );ˆ θ := θ ( t, x, ˆ Y ( t, x ) , F ( ˆ Y ( t, x )) , ˆ u ( t, x ) , G (ˆ u ( t, x )) , e ); θ := θ ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) , e ) . We also set ( ˆ H := H ( t, x, ˆ Y ( t, x ) , F ( ˆ Y ( t, x )) , ˆ u ( t, x ) , G (ˆ u ( t, x )) , ˆ p ( t, x ) , ˆ q ( t, x ) , ˆ γ ( t, x, · )); H := H ( t, x, Y ( t, x ) , F( Y ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) , ˆ p ( t, x ) , ˆ q ( t, x ) , ˆ γ ( t, x, · )) . Using the above definitions and the definition of the performance functional J , we get that: J ( u ) − J (ˆ u ) = J + J , (2.21)where J := E [ R T R D ( f − ˆ f ) dxdt ] and J := E [ R D ( g − ˆ g ) dx ] . Now, let us notice the following relations: ( ˆ f = ˆ H − ˆ b ˆ p ( t, x ) − ˆ σ ˆ q ( t, x ) − R E ˆ θ ˆ γ ( t, x, e ) ν ( de );ˆ f = H − b ˆ p ( t, x ) − σ ˆ q ( t, x ) − R E θ ˆ γ ( t, x, e ) ν ( de ) , which imply J = E (cid:20)Z T Z D (cid:18) H − ˆ H − ( b − ˆ b ) · ˆ p − ( σ − ˆ σ ) · ˆ q − Z E ( θ − ˆ θ ) · ˆ γν ( de ) (cid:19)(cid:21) . (2.22)Fix x ∈ D . Since the map Y g ( x, Y, F ( Y )) is concave for each x ∈ ¯ D , we obtain: g − ˆ g ≤ ∂g∂y ( x, ˆ Y ( T, x ) , F ( ˆ Y ( T, x ))) ˜ Y ( T, x )+ ∂g∂ ¯ y ( x, ˆ Y ( T, x ) , F ( ˆ Y ( T, x ))) < ∇ F( ˆ Y ( T, x ) ) , ˜ Y ( T, x ) > L ( P ) , where ˜ Y ( t, x ) = Y ( t, x ) − ˆ Y ( t, x ) . We thus obtain, by taking the expectation and applying the Itˆo formula for jump-diffusion8rocesses, J ≤ E (cid:20)Z D (cid:18) ∂g∂y ( x, ˆ Y ( T, x ) , F ( ˆ Y ( T, x ))) ˜ Y ( T, x )+ ∂g∂ ¯ y ( x, ˆ Y ( T, x ) , F ( ˆ Y ( T, x ))) < ∇ F ( ˆ Y ( T, x )) , ˜ Y ( T, x ) > L ( P ) (cid:19) dx (cid:21) = E (cid:20)Z D < ˆ p ( T, x ) , ˜ Y ( T, x ) > dx (cid:21) = E (cid:20)Z D (cid:18) ˆ p (0 , x ) · ˜ Y (0 , x ) + Z T (cid:16) < ˜ Y ( t, x ) , d ˆ p ( t, x ) > +ˆ p ( t, x ) d ˜ Y ( t, x ) + ( σ − ˆ σ )ˆ q ( t, x ) (cid:17) dt (cid:19) dx (cid:21) + E (cid:20)Z D (cid:18)Z T Z E ( θ − ˆ θ )ˆ γ ( t, x, e ) N ( dt, de ) (cid:19) dx (cid:21) = E (cid:20)Z D Z T ˆ p ( t, x ) (cid:16) L ˜ Y ( t, x ) + ( b − ˆ b ) (cid:17) + ˜ Y ( t, x ) (cid:18) − L ∗ ˆ p ( t, x ) − ∂H∂y − E (cid:20) ∂H∂ ¯ y (cid:21) < ∇ F ( ˆ Y ( t, x )) , ˜ Y ( t, x ) > L ( P ) (cid:19) dtdx (cid:21) + E (cid:20)Z D Z T (cid:18) ( σ − ˆ σ )ˆ q ( t, x ) + Z E ( θ − ˆ θ )ˆ γ ( t, x, e ) ν ( de ) (cid:19) dtdx (cid:21) , (2.23)where ∂H∂y := ∂H∂y ( t, x, ˆ Y ( t, x ) , F( ˆ Y ( t, x ) ) , ˆ u ( t, x ) , G( ˆ u ( t, x ) ) , ˆ p ( t, x ) , ˆ q ( t, x ) , ˆ γ ( t, x, · ))and ∂H∂ ¯ y := ∂H∂ ¯ y ( t, x, ˆ Y ( t, x ) , F( ˆ Y ( t, x ) ) , ˆ u ( t, x ) , G( ˆ u ( t, x ) ) , ˆ p ( t, x ) , ˆ q ( t, x ) , ˆ γ ( t, x, · )) . From (2.21), (2.22) and (2.23), we derive J ( u ) − J (ˆ u ) ≤ E (cid:20)Z T (cid:18)Z D ˆ p ( t, x ) L e Y ( t, x ) − e Y ( t, x ) L ∗ ˆ p ( t, x ) dx (cid:19) dt (cid:21) + E (cid:20)Z D (cid:18)Z T (cid:18) H − ˆ H − ∂H∂y · ˜ Y ( t, x ) − E (cid:20) ∂H∂ ¯ y (cid:21) < ∇ F( ˆ Y ( t, x ) ) , ˜ Y ( t, x ) > L ( P ) (cid:19) dt (cid:19) dx (cid:21) . Since ˜ Y ( t, x ) = ˆ p ( t, x ) = 0 for all ( t, x ) ∈ [0 , T ] × ∂D , we obtain by an easy extension of(2.13) using Green’s formula that Z D ˜ Y ( t, x ) L ∗ ˆ p ( t, x ) dx = Z D ˆ p ( t, x ) L ˜ Y ( t, x ) , for all t ∈ (0 , T ). We therefore get J ( u ) − J (ˆ u ) ≤ E (cid:20)Z D (cid:18)Z T (cid:18) H − ˆ H − ∂H∂y · ˜ Y ( t, x ) + E (cid:20) ∂H∂ ¯ y (cid:21) < ∇ F( ˆ Y ( t, x )) , ˜ Y ( t, x ) > L ( P ) (cid:19) dt (cid:19) dx (cid:21) .
9y the concavity assumption (2.18) we have H − ˆ H ≤ ∂H∂y ( ˆ Y , F( ˆ Y ) , ˆ u, G( ˆ u ) )( Y − ˆ Y ) + ∂H∂ ¯ y ( ˆ Y , F( ˆ Y ) , ˆ u, G( ˆ u ) ) < ∇ F( ˆ Y ) , ( Y − ˆ Y ) > L ( P ) + ∂H∂u ( ˆ Y , F( ˆ Y ) , ˆ u, G( ˆ u ))( u − ˆ u ) + ∂H∂ ¯ u ( ˆ Y , F( ˆ Y ) , ˆ u, G( ˆ u )) < ∇ G( ˆ u ) , ( u − ˆ u ) > L ( P ) . Combining the two above relations we get: J ( u ) − J (ˆ u ) ≤ E (cid:20)Z D Z T (cid:18) ∂H∂u ( ˆ Y , F( ˆ Y ) , ˆ u, G( ˆ u ))( u − ˆ u )+ ∂H∂ ¯ u ( ˆ Y , F( ˆ Y ) , ˆ u, G( ˆ u )) < ∇ G( ˆ u ) , ( u − ˆ u ) > L ( P ) (cid:19) dtdx (cid:21) . (2.24)By the maximum condition (2.19) , we obtain: E (cid:20) ∂H∂u ( ˆ Y , F( ˆ Y ) , ˆ u, G( ˆ u )) |E t (cid:21) ( u − ˆ u ) + E (cid:20) ∂H∂ ¯ u ( ˆ Y , F( ˆ Y ) , ˆ u, G( ˆ u )) |E t (cid:21) < ∇ G( ˆ u ) , u − ˆ u > L ( P ) ≤ , (2.25)for all ( t, x ) ∈ [0 , T ] × D . From (2.24) and (2.25) we conclude that J ( u ) ≤ J (ˆ u ) . By arbitrariness of u , we conclude that ˆ u is optimal. (cid:3) As in many applications the concavity condition may not hold, we prove a version of themaximum principle which does not need this assumption. Instead, we assume the following: (A1)
For all s ∈ [0 , T ) and all bounded E s -measurable random variables θ ( ω, x ) thecontrol β defined by β t ( ω, x ) = θ ( ω, x ) χ ( s,T ] ( t ); t ∈ [0 , T ] , x ∈ D is in A E . (A2) For all u, β ∈ U where β is bounded there exists δ > u ( t ) + yβ ( t ); t ∈ [0 , T ]belongs to A E for all y ∈ ( − δ, δ ).Let us give an auxiliary lemma. 10 emma 2.2 Let u ∈ A E and v ∈ A E . The derivative process Y ( t, x ) := lim z + Y u + zβ ( t, x ) − Y u ( t, x ) z (2.26) exists and belongs to L ( dx × dt × dP ) . We then have that Y satisfies the following mean-fieldSPDE: d Y ( t, x ) = L Y ( t, x ) + (cid:18) ∂b∂y ( t, x, Y u ( t, x ) , F( Y u ( t, x ) ) , G( u ( t, x ) ) ) Y ( t, x )+ ∂b∂ ¯ y ( t, x, Y u ( t, x ) , F( Y u ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) < ∇ F( Y u ( t, x ) ) , Y ( t, x ) > L ( P ) + ∂b∂u ( t, x, Y u ( t, x ) , F( Y u ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) β ( t, x )+ ∂b∂ ¯ u ( t, x, Y u ( t, x ) , F( Y u ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) < ∇ G( u ( t, x ) ) , β ( t, x ) > L ( P ) (cid:19) dt + (cid:18) ∂σ∂y ( t, x, Y u ( t, x ) , F( Y u ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) Y ( t, x )+ ∂σ∂ ¯ y ( t, x, Y u ( t, x ) , F( Y u ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) < ∇ F( Y u ( t, x ) ) , Y ( t, x ) > L ( P ) + ∂σ∂u ( t, x, Y u ( t, x ) , F( Y u ( t, x ) ) , u ( t, x ) , G( Y u ( t, x ) ) ) β ( t, x )+ ∂σ∂ ¯ u ( t, x, Y u ( t, x ) , F( Y u ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) ) < ∇ G( u ( t, x ) ) , β ( t, x ) > L ( P ) (cid:19) dW t + Z E (cid:18) ∂θ∂y ( t, x, Y u ( t, x ) , F( Y u ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) , e ) Y ( t, x )+ ∂θ∂ ¯ y ( t, x, Y u ( t, x ) , F( Y u ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) , e ) < ∇ F( Y u ( t, x ) ) , Y ( t, x ) > L ( P ) + ∂θ∂u ( t, x, Y u ( t, x ) , F( Y u ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) , e ) β ( t, x )+ ∂θ∂ ¯ u ( t, x, Y u ( t, x ) , F( Y u ( t, x ) ) , u ( t, x ) , G( u ( t, x ) ) , e ) β ( t, x )) < ∇ G( u ( t, x ) ) , β ( t, x ) > L ( P ) ) ˜ N ( dt, de ) , Y ( t, x ) = 0 , ( t, x ) ∈ (0 , T ) × ∂D ; Y (0 , x ) = 0 , x ∈ D. Proof. The result follows by applying the mean theorem. We omit the details. (cid:3)
We now provide the necessary-type maximum principle for our optimal control problemfor mean-field SPDEs.
Theorem 2.3 (Necessary-type maximum principle for mean-field SPDEs with jumps)
Let ˆ u ∈ A E with corresponding solutions (2.6) - (2.7) - (2.8) and (2.14) - (2.15) - (2.16) . Assume hat Assumptions (A1)-(A2) hold. Then the following are equivalent: (i) ddy J (ˆ u + yβ ) | y =0 = 0 for all bounded β ∈ A E . (ii) E h ∇ ˆ H ( t, x ) |E t i = 0 , for all ( t, x ) ∈ [0 , T ] × D a.s. , where ∇ ˆ H ( t, x ) := ∂H∂u ( t, x, ˆ u ( t, x ) , ˆ Y ( t, x ) , F( ˆ Y ( t, x ) ) , G( ˆ u ( t, x ) ) , ˆ p ( t, x ) , ˆ q ( t, x ) , ˆ γ ( t, x, · ))+ E (cid:20) ∂H∂ ¯ u ( t, x, ˆ u ( t, x ) , ˆ Y ( t, x ) , F( ˆ Y ( t, x ) ) , G( ˆ u ( t, x ) ) , ˆ p ( t, x ) , ˆ q ( t, x ) , ˆ γ ( t, x, · )) (cid:21) ∇ G( ˆ u ( t, x ) ) , for all ( t, x ) ∈ [0 , T ] × D . Proof. The assumptions on the coefficients together with the mean theorem and relation(2.26) yield to:lim y → y ( J (ˆ u + yβ ) − J (ˆ u )) = E (cid:20)Z T Z D ( ∂f∂y ( t, x, ˆ Y ( t, x ) , F( ˆ Y ( t, x ) ) , ˆ u ( t, x ) , G( ˆ u ( t, x ) ) ) Y ( t, x )+ ∂f∂y ( t, x, ˆ Y ( t, x ) , F( ˆ Y ( t, x ) ) , ˆ u ( t, x ) , G( ˆ u ( t, x ) ) ) < ∇ F( ˆ Y ( t, x ) ) , Y ( t, x ) > L ( P ) + ∂f∂u ( t, x, ˆ Y ( t, x ) , F( ˆ Y ( t, x ) ) , ˆ u ( t, x ) , G( ˆ u ( t, x ) ) ) β ( t, x )) dxdt ]+ ∂f∂ ¯ u ( t, x, ˆ Y ( t, x ) , F( ˆ Y ( t, x ) ) , ˆ u ( t, x ) , G( ˆ u ( t, x ) ) ) dxdt ] < ∇ G( ˆ u ( t, x ) ) , β ( t, x ) > L ( P ) (cid:21) + E (cid:20)Z D ( ∂g∂y ( T, x, ˆ Y ( T, x ) , F( ˆ Y ( T, x ) ) ) Y ( T, x )+ ∂g∂y ( T, x, ˆ Y ( T, x ) , F( ˆ Y ( T, x ) ) , ˆ u ( T, x )) < ∇ F( ˆ Y ( T, x ) ) , Y ( T, x ) > L ( P ) dx ) (cid:21) . (2.27)The definition of the Hamiltonian H implies: ∂f∂y ( t, x, ˆ Y ( t, x ) , F( ˆ Y ( t, x ) ) , ˆ u ( t, x ) , G( ˆ u ( t, x ) ) ) == ∂ ˆ H∂y ( t, x ) − ∂ ˆ b∂y ( t, x )ˆ p ( t, x ) − ∂ ˆ σ∂y ( t, x )ˆ q ( t, x ) − Z E ∂ ˆ θ∂y ( t, x, e )ˆ γ ( t, x, e ) ν ( de ) (2.28) ∂f∂y ( t, x, ˆ Y ( t, x ) , F( ˆ Y ( t, x ) ) , ˆ u ( t, x ) , G( ˆ u ( t, x ) ) ) == ∂ ˆ H∂y ( t, x ) − ∂ ˆ b∂y ( t, x )ˆ p ( t, x ) − ∂ ˆ σ∂y ( t, x )ˆ q ( t, x ) − Z E ∂ ˆ θ∂y ( t, x, e )ˆ γ ( t, x, e ) ν ( de ) (2.29)12 f∂u ( t, x, ˆ Y ( t, x ) , F( ˆ Y ( t, x ) ) , ˆ u ( t, x ) , G( ˆ u ( t, x ) ) )= ∂ ˆ H∂u ( t, x ) − ∂ ˆ b∂u ( t, x )ˆ p ( t, x ) − ∂ ˆ σ∂u ( t, x )ˆ q ( t, x ) − Z E ∂ ˆ θ∂u ( t, x, e )ˆ γ ( t, x, e ) ν ( de ) (2.30) ∂f∂ ¯ u ( t, x, ˆ Y ( t, x ) , F( ˆ Y ( t, x ) ) , ˆ u ( t, x ) , G( ˆ u ( t, x ) ) )= ∂ ˆ H∂u ( t, x ) − ∂ ˆ b∂u ( t, x )ˆ p ( t, x ) − ∂ ˆ σ∂u ( t, x )ˆ q ( t, x ) − Z E ∂ ˆ θ∂u ( t, x, e )ˆ γ ( t, x, e ) ν ( de ) (2.31)Using (2.27), (2.28), (2.29), (2.30), (2.31) we derive:lim y → y ( J (ˆ u + yβ ) − J (ˆ u ))= E "Z T Z D ∂ ˆ H∂y ( t, x ) − ∂ ˆ b∂y ( t, x )ˆ p ( t, x ) − ∂ ˆ σ∂y ( t, x )ˆ q ( t, x ) − Z E ∂ ˆ θ∂y ( t, x, e )ˆ γ ( t, x, e ) ν ( de ) ! Y ( t, x ) dxdt + E [ Z T Z D ( ∂ ˆ H∂y ( t, x ) − ∂ ˆ b∂y ( t, x )ˆ p ( t, x ) − ∂ ˆ σ∂y ( t, x )ˆ q ( t, x ) − Z E ∂ ˆ θ∂y ( t, x, e )ˆ γ ( t, x, e ) ν ( de ))+ < ∇ F( ˆ Y ( t, x ) ) , Y ( t, x ) > L ( P ) dxdt ]+ E "Z T Z D ∂ ˆ H∂u ( t, x ) − ∂ ˆ b∂u ( t, x )ˆ p ( t, x ) − ∂ ˆ σ∂u ( t, x )ˆ q ( t, x ) − Z E ∂ ˆ θ∂u ( t, x, e )ˆ γ ( t, x, e ) ν ( de ) ! β ( t, x ) dxdt + E [ Z T Z D ( ∂ ˆ H∂u ( t, x ) − ∂ ˆ b∂u ( t, x )ˆ p ( t, x ) − ∂ ˆ σ∂u ( t, x )ˆ q ( t, x ) − Z E ∂ ˆ θ∂u ( t, x, e )ˆ γ ( t, x, e ) ν ( de ))+ < ∇ G( ˆ u ( t, x ) ) , β ( t, x ) > dxdt ] + E [ Z D < ˆ p ( T, x ) , Y ( T, x ) > dx ] . Applying Itˆo formula to < ˆ p ( T, x ) , Y ( T, x ) > and using the dynamics of the adjoint equa-tions, we finally getlim y → y ( J (ˆ u + yβ ) − J (ˆ u )) = E (cid:20)Z T Z D E h < ∇ u ˆ H ( t, x ) , β ( t, x ) > |E t i dxdt (cid:21) , (2.32)where < ∇ u ˆ H ( t, x ) , β ( t, x ) > = ∂H∂u ( t, x ) β ( t, x ) + ∂H∂ ¯ u ( t, x ) < ∇ G( ˆ u ) , β ( t, x ) > L ( P ) . We conclude that lim y → y ( J (ˆ u + yβ ) − J (ˆ u )) = 0if and only if E (cid:20)Z T Z D E [ < ∇ u ˆ H ( t, x ) , β ( t, x ) > |E t ] dxdt (cid:21) = 0 . In particular this holds for all β ∈ A E which takes the form β ( t, x ) = θ ( ω, x ) χ [ s,T ] ( t ); t ∈ [0 , T ] , s ∈ [0 , T ) , where θ ( ω, x ) is a bounded E s -measurable random variable. We thusget that this is again equivalent to E (cid:20)Z Ts Z D E h < ∇ u ˆ H ( t, x ) , θ > |E t i dxdt (cid:21) = 0 . We now differentiate with respect to s and derive that E (cid:20)Z D E h < ∇ u ˆ H ( s, x ) , θ> |E s i dx (cid:21) = 0 . Since this holds for all bounded E s -measurable random variable θ , we can easily concludethat lim y → y ( J (ˆ u + yβ ) − J (ˆ u )) = 0is equivalent to E " ∂ ˆ H∂u ( t, x ) |E t + E " ∂ ˆ H∂ ¯ u ( t, x ) ∇ G( ˆ u ( t, x ) ) = 0 a.s., for all ( t, x ) ∈ [0 , T ] × D. (cid:3) We now return to the problem of optimal harvesting from a population in a lake D statedin the motivating example. Thus we suppose the density Y ( t, x ) of the population at time t ∈ [0 , T ] and at point x ∈ D is given by the stochastic reaction-diffusion equation (1.1),and the performance criterion is assumed to be as in (1.4). For simplicity, we choose d = 1and E t = F t . In this case the Hamiltonian gets the following form H ( t, x, y, ¯ y, u, ¯ u, p, q, γ ) = log( yu ) + [ b ( t, x )¯ y − yu ] p + σ ( t, x ) yq + Z R ∗ θ ( t, x, e ) yγ ( e ) ν ( de ) , and the adjoint BSDE becomes dp ( t, x ) = [ − ∂ ∂ x p ( t, x ) + 1 Y ( t, x ) + σ ( t, x ) q ( t, x ) + Z R ∗ θ ( t, x, e ) γ ( t, x, e ) ν ( de ) − u ( t, x ) p ( t, x ) − E [ b ( t, x ) p ( t, x )]] dt + q ( t, x ) dW t + Z R ∗ γ ( t, x, e ) ˜ N ( dt, de ) p ( T, x ) = α ( x ) , x ∈ D,p ( t, x ) = 0 , ( t, x ) ∈ (0 , T ) × ∂D. We now apply the necessary maximum principle which implies the fact that if u is an optimalcontrol then it satisfies the first order condition u ( t, x ) = 1 Y ( t, x ) p ( t, x ) . We summarize our results as follows: 14 heorem 2.4
Assume that the conditions of Theorem 2.3 hold. Suppose a harvesting rateprocess u ( t, x ) is optimal for the optimization problem (1.5) . Then u ( t, x ) = 1 Y ( t, x ) p ( t, x ) , (2.33) where p ( t, x ) solves the MFBSPDE dp ( t, x ) = [ − ∂ ∂ x p ( t, x ) + 1 Y ( t, x ) + σ ( t, x ) q ( t, x ) + Z R ∗ θ ( t, x, e ) γ ( t, x, e ) ν ( de ) − E [ b ( t, x ) p ( t, x )] − u ( t, x ) p ( t, x )] dt + q ( t, x ) dW t + Z R ∗ γ ( t, x, e ) ˜ N ( dt, de ) p ( T, x ) = α ( x ) , x ∈ D.p ( t, x ) = 0 , ( t, x ) ∈ (0 , T ) × ∂D. We address here the problem of existence and uniqueness of the forward mean-field SPDE(2.6) with general mean-field operator, introduced in Section 2. In order to do this, we firstintroduce the general framework. Let
V, H be two separable Hilbert spaces such that V iscontinously, densely imbedded in H . Identifying H with its dual we have V ⊂ H ≅ H ∗ ⊂ V ∗ , where we have denoted by V ∗ the topological dual of V . Let L be a bounded linear operatorfrom V to V ∗ satisfying the following coercivity hypothesis: There exist constants χ > ζ ≥ < − Lu, u > + χ | u | H ≥ ζ || u || V for all u ∈ V, (3.34)where < Lu, u > = Lu ( u ) denotes the action of Lu ∈ V ∗ on u ∈ V and | · | H (resp. || · || V )the norm associated to the Hilbert space H (resp. V ).Let us introduce the notation adopted in this section. • L ν ( H ) is the set of measurable functions k : ( E , B ( E )) ( H, B ( H )) such that || k || L ν ( H ) := (cid:0)R E | k ( e ) | H ν ( de ) (cid:1) < ∞ ; • L (Ω , H ) is the set of measurable functions k : (Ω , F ) ( H, B ( H )) such that E [ | k | H ] < ∞ ; • L (Ω , L ν ( H )) is the set of measurable functions k : (Ω , F ) ( L ν ( H ) , B ( L ν ( H ))) suchthat E [ || k || L ν ( H ) ] < ∞ L (Ω × [0 , T ] , H ) (resp. L (Ω × [0 , T ] , V )) is the set of F t -adapted H -valued (resp. V -valued) processes Φ : Ω × [0 , T ] H (resp.V) such that k Φ k L (Ω × [0 ,T ] ,H ) := E [ R T | Φ( t ) | H dt ] < ∞ (resp. k Φ k L (Ω × [0 ,T ] ,V ) := E [ R T k Φ( t ) k V dt ] < ∞ ). • L (Ω × [0 , T ] × E , H ) is the set of all the P × B ( E )-measurable H -valued maps θ :Ω × [0 , T ] × E H satisfying k θ k L (Ω × [0 ,T ] × E ,H ) := E [ R T R E | Φ( t, e ) | H ν ( de ) dt ] < ∞ . • S (Ω × [0 , T ] , H ) denotes the set of F t -adapted H -valued cadlag processes Φ : Ω × [0 , T ] H such that k Φ k S (Ω × [0 ,T ] ,H ) := E (cid:2) sup ≤ t ≤ T | Φ( t ) | H (cid:3) < ∞ . The mean field SPDE under study is: dY t = [ LY t + b ( t, Y t , F( Y t ) ) ] dt + σ ( t, Y t , F( Y t ) ) dW t + Z E θ ( t, Y t , F( Y t ) , e ) ˜ N ( dt, de ); ( t, x ) ∈ (0 , T ) × D. We recall that this equation should be understood in the weak sense.Before giving the main result of this section, we make the following assumption on thecoefficients b, σ, θ and the operator F which appear in the above mean-field SPDE. Assumption 3.1
The maps b : Ω × [0 , T ] × H × H H , σ : Ω × [0 , T ] × H × H H are P × B ( H ) × B ( H ) / B ( H ) -measurable. The map θ : Ω × [0 , T ] × H × H × E H is P × B ( E ) × B ( H ) × B ( H ) / B ( H ) -measurable. There exist a constant C < ∞ such that | b ( t, y , ¯ y ) − b ( t, y , ¯ y ) | H + | σ ( t, y , ¯ y ) − σ ( t, y , ¯ y ) | H + Z E | θ ( t, y , ¯ y , e ) − θ ( t, y , ¯ y , e ) | ν ( de ) ≤ C ( | y − y | H + | ¯ y − ¯ y | H ) a.s. for all ( ω, t ) ∈ Ω × [0 , T ] . We also assume that there exists
C < ∞ such that: | b ( t, y, ¯ y ) | H + | σ ( t, y, ¯ y ) | H + Z E | θ ( t, y, ¯ y, e ) | H ν ( de ) ≤ C (1 + | y | H + | ¯ y | H ) , ∀ ( ω, t ) ∈ Ω × [0 , T ] , y, ¯ y ∈ R . Finally, we assume that the operator F : L (Ω; H ) H is Fr´echet differentiable. Theorem 3.1
Under Assumption 3.1, there exists a unique H-valued progressively measur-able process Y t , t ≥ satisfying the mean-field SPDE:(i) Y ∈ L (Ω × [0 , T ] , V ) ∩ S (Ω × [0 , T ] , H ); (ii) Y t = h + R t [ LY s + b ( s, Y s , F ( Y s ))] ds + R t σ ( s, Y s , F ( Y s )) dW s + R t R E θ ( s − , Y s − , F ( Y s − ) , e ) ˜ N ( dt, de ); (iii) Y = h ∈ H . Proof.
I. Existence of the solution
Let Y t := h, t ≥ . For n ≥
0, we define Y n +1 ∈ L ([0 , T ]; V ) ∩ S ([0 , T ]; H ) to be theunique solution to the following equation: dY n +1 t = LY n +1 t dt + b ( t, Y n +1 t , F ( Y nt )) dt + σ ( t, Y n +1 t , F ( Y nt )) dW t + Z E θ ( t − , Y n +1 t − , F ( Y nt − ) , e ) ˜ N ( dt, de ) . (3.35)16he solution Y n +1 of this equation follows by Proposition 3.1 in [11]. Let us now showthat the sequence { Y n , n ≥ } is a Cauchy sequence in the spaces L (Ω × [0 , T ] , V ) and S (Ω × [0 , T ] , H ). By applying Itˆo formula, we get | Y n +1 t − Y nt | H = 2 Z t < Y n +1 s − Y ns , L ( Y n +1 s − Y ns ) > ds + 2 Z t < Y n +1 s − Y ns , b ( s, Y ns , F ( Y ns )) − b ( s, Y n − s , F ( Y n − s )) > H ds + 2 Z t < Y n +1 s − Y ns , σ ( s, Y n +1 s , F ( Y ns )) − σ ( s, Y ns , F ( Y n − s )) > H dW s + Z t | σ ( s, Y ns , F ( Y ns )) − σ ( s, Y n − s , F ( Y n − s )) | H ds + Z t Z E [ | θ ( s, Y ns − , F ( Y ns − ) , e ) − θ ( s, Y n − s − , F ( Y n − s − ) , e ) | H ] ˜ N ( ds, de )+ 2 Z t Z E < Y n +1 s − − Y ns − , θ ( s, Y ns − , F ( Y ns − ) , e ) − θ ( s, Y n − s − , F ( Y n − s − ) , e ) > H ˜ N ( ds, de )+ Z t Z E [ | θ ( s, Y ns − , F ( Y ns − ) , e ) − θ ( s, Y n − s − , F ( Y n − s − ) , e ) | H ν ( de ) ds. Using Burkholder-Davis-Gundy and Cauchy-Schwarz inequalities and the coercivity assump-tion (3.34) on the operator L , we obtain that E (cid:20) sup ≤ s ≤ t | Y n +1 s − Y ns | H (cid:21) ≤ − χ E [ Z t | Y n +1 s − Y ns | V ds ] + C E [ Z t [ | Y n +1 s − Y ns | H ds ]+ 12 E [ sup ≤ s ≤ t | Y n +1 s − Y ns | H ds ] + C E [ Z t [ | b ( s, Y ns , F ( Y ns )) − b ( s, Y n − s , F ( Y n − s )) | H ds ]+ C E [ Z t [ | σ ( s, Y ns , F ( Y ns )) − σ ( s, Y n − s , F ( Y n − s )) | H ds ]+ C E [ Z t Z E [ | θ ( s, Y ns , F ( Y ns ) , e ) − θ ( s, Y n − s , F ( Y n − s ) , e ) | H ν ( de ) ds ] . (3.36)By the Lipschitz properties of b, σ and θ , we deduce E (cid:20) sup ≤ s ≤ t | Y n +1 s − Y ns | H (cid:21) ≤ C E [ Z t [ | Y n +1 s − Y ns | H ds ] + C E [ Z t [ | Y ns − Y n − s | H ds ]+ C E [ Z t | F ( Y ns ) − F ( Y n − s ) | H ds ] . (3.37)We use the mean theorem and obtain the existence for each n ∈ N , t ∈ [0 , T ] of a randomvariables ˜ Y n ( t ) ∈ L (Ω , H ) such that | F ( Y nt ) − F ( Y n − t ) | H ≤ k∇ F ( ˜ Y n ( t )) kk Y nt − Y n − t k L (Ω; H ) . (3.38)The two above relations (3.37) and (3.38) lead to: E (cid:20) sup ≤ s ≤ t | Y n +1 s − Y ns | H (cid:21) ≤ C E [ Z t [ | Y n +1 s − Y ns | H ds ] + C E [ Z t | Y ns − Y n − s | H ds ] . (3.39)17et us now define a nt = E (cid:20) sup ≤ s ≤ t | Y ns − Y n − s | H ds (cid:21) A nt = Z t a ns ds. Using (3.39), we obtain: a n +1 t ≤ CA n +1 t + CA nt . (3.40)We multiply the above inequality by e − Ct and derive d ( A n +1 t e − Ct ) dt ≤ Ce − Ct A nt , which allows us to conclude that A n +1 t ≤ Ce Ct Z t e − Cs A ns ds ≤ Ce Ct tA nt . This inequality together with (3.40) gives a n +1 t ≤ C e Ct tA nt + CA nt ≤ C T Z t A ns ds, where C T is a given constant. By iteration for all n , we finally obtain E [ sup ≤ s ≤ T | Y n +1 s − Y ns | H ] ≤ C ( C T T ) n n ! . This implies that we can find Y ∈ S (Ω × [0 , T ]; H ) such thatlim n →∞ E [ sup ≤ s ≤ t | Y ns − Y s | H ds ] = 0 . By (3.36), we remark that Y n also converges to Y in L (Ω × [0 , T ] , V ) . Passing to the limitin (3.35) , we obtain that Y satisfies this equation. II. Uniqueness of the solution
Let Y and Y be two solutions in S (Ω × [0 , T ] , H ) ∩ L (Ω × [0 , T ] , V ) . Applying Itoformula, we have | Y t − Y t | H = − Z t < Y s − Y s , L ( Y s − Y s ) > ds + 2 Z t < Y s − Y s , b ( s, Y s , F( Y s ) ) − b ( s, Y s , F( Y s ) ) > H ds + 2 Z t < Y s − Y s , σ ( s, Y s , F( Y s ) ) − σ ( s, Y s , F( Y s ) ) > H dW s + Z t | σ ( s, Y s , F( Y s ) ) − σ ( s, Y s , F( Y s ) ) | H ds + Z t Z E (cid:2) | θ ( s, Y s − , F( Y s − ) , e ) − θ ( s, Y s − , F( Y s − ) , e ) | H +2 < Y s − − Y s − , θ ( s, Y s − , F( Y s − ) , e ) − θ ( s, Y s − , F( Y s − ) , e ) > (cid:3) ˜ N ( ds, de )+ Z t Z E | θ ( s, Y s − , F( Y s − ) , e ) − θ ( s, Y s − , F( Y s − ) , e ) | H dsν ( de ) . L , the Lipschitz property of b, σ, θ andthe boundness of the Fr´echet derivative of the operator F , we finally obtain: E [ | Y t − Y t | H ] ≤ − α E [ Z t | Y s − Y s | V ds ] + C E [ Z t | Y s − Y s | H ds ]+ 12 E [ sup ≤ s ≤ t | Y t − Y t | H ] + C E [ Z t | b ( s, Y s , F( Y s ) ) − b ( s, Y s , F( Y s ) ) | H ds ]+ C E [ Z t | σ ( s, Y s , F( Y s ) ) − σ ( s, Y s , F( Y s ) ) | H ds ]+ C E [ Z t Z E | θ ( s, Y s , F( Y s ) ) − θ ( s, Y s , F( Y s ) ) | H ν ( de ) ds ] ≤ C E [ Z t | Y s − Y s | H ds ] . We thus deduce that Y t = Y t . In this section we give an existence and uniqueness result for mean-field backward SPDEswith jumps. The analysis will be carried out in a general case, where there exists a generalmean-field operator acting on each composant of the solution.We consider the same framework as in the previous section. Let A be a bounded linearoperator from V to V ∗ satisfying the following coercivity hypothesis: There exist constants α > λ ≥ < Au, u > + λ | u | H ≥ α || u || V for all u ∈ V, where < Au, u > = Au ( u ) denotes the action of Au ∈ V ∗ on u ∈ V . Assumption 4.2
Let f : [0 , T ] × Ω × H × H × H × H × L ν ( H ) × L ν ( H ) → H be a P × B ( H ) × B ( H ) × B ( H ) × B ( H ) × B ( L ν ( H )) × B ( L ν ( H )) / B ( H ) measurable. There existsa constant C < ∞ such that | f ( t, ω, y , ˜ y , z , ˜ z , q , ˜ q ) − f ( t, ω, y , ˜ y , z , ˜ z , q , ˜ q ) | H ≤ C ( | y − y | H + | ˜ y − ˜ y | H + | z − z | H + | ˜ z − ˜ z | H + | q − q | L ν ( H ) + | ˜ q − ˜ q | L ν ( H ) ) for all t, y , ˜ y , z , ˜ z , q , ˜ q , y , ˜ y , z , ˜ z , q , ˜ q . We also assume the integrability condition E [ Z T | f ( t, , , , , , | H dt ] < ∞ . (4.41)We now give our main result of existence and uniqueness.19 heorem 4.1 Assume Assumption 4.2 holds. Let ξ ∈ L (Ω; H ) . Let H : L (Ω; H ) H , J : L (Ω; H ) H and K : L (Ω , L ν ( H )) L ν ( H ) be Fr´echet differentiable operators.There exists a unique H × H × L ν ( H ) -valued progressively measurable process ( Y t , Z t , U t ) such that (i) E [ | Y t | H ] < ∞ , E [ Z T | Z t | H ] < ∞ , E [ Z T | U t | L ν ( H ) dt ] < ∞ . (ii) ξ = Y t + Z Tt AY s ds + Z Tt f ( s, Y s , H ( Y s ) , Z s , J ( Z s ) , U s , K ( U s )) ds + Z Tt Z s dW s + Z Tt Z E U s ( e ) ˜ N ( ds, de ) , for all ≤ t ≤ T .The equation ( ii ) should be understood in the dual space V ∗ . Proof.
I. Existence of the solution
Set Y t = 0; Z t = 0; U t = 0. We denote by ( Y nt , Z nt , U nt ) the unique solution of the mean-fieldbackward stochastic equation: ( dY nt = AY nt dt + f ( t, Y nt , H ( Y n − t ) , Z nt , J ( Z n − t ) , U nt , K ( U n − t )) dt + Z nt dW t + R E U nt ( e ) ˜ N ( dt, de ) Y nT = ξ. The existence and the uniqueness of a solution ( Y nt , Z nt , U nt ) of such an equation has beenproved in [7]. By applying Itˆo’s formula, we get0 = | Y n +1 T − Y nT | H = | Y n +1 t − Y nt | H + 2 Z Tt < A ( Y n +1 s − Y ns ) , Y n +1 s − Y ns > ds + 2 Z Tt < f ( s, Y n +1 s , H ( Y ns ) , Z n +1 s , J ( Z ns ) , U n +1 s , K ( U ns )) − f ( s, Y ns , H ( Y n − s ) , Z ns , J ( Z n − s ) , U ns , K ( U n − s )) , Y n +1 s − Y ns > H ds + Z Tt Z E (cid:2) | Y n +1 s − − Y ns − + U n +1 s − U ns | H − | Y n +1 s − − Y ns − | H (cid:3) ˜ N ( ds, de ) + Z Tt Z E [ | U n +1 s ( e ) − U ns ( e ) | H ] ν ( de ) ds + 2 Z Tt < Y n +1 s − Y ns , d ( Z n +1 s − Z ns ) > H + Z Tt | Z n +1 s − Z ns | H ds, where Z nt := R t Z ns dW s .We thus get, by taking the expectation and using the coercivity assumption on the20perator A E [ | Y n +1 t − Y nt | H ] + E [ Z Tt | Z n +1 s − Z ns | H ds ] + E [ Z Tt Z E | U n +1 s − U ns | H ν ( de ) ds ] = (4.42) − E [ < A ( Y n +1 s − Y ns ) , Y n +1 s − Y ns > ds ] − E [ Z Tt < f ( s, Y n +1 s , H ( Y ns ) , Z n +1 s , J ( Z ns ) , U n +1 s , K ( U ns )) − f ( s, Y ns , H ( Y n − s ) , Z ns , J ( Z n − s ) , U ns , K ( U n − s )) , Y n +1 s − Y ns > ds ] ≤≤ λ E [ Z Tt | Y n +1 s − Y ns | H ds ] − α E [ Z Tt | Y n +1 s − Y ns | V ds ] −− E [ Z Tt < f ( s, Y n +1 s , H ( Y ns ) , Z n +1 s , J ( Z ns ) , U n +1 s , K ( U ns )) − f ( s, Y ns , H ( Y n − s ) , Z ns , J ( Z n − s ) , U ns , K ( U n − s )) , Y n +1 s − Y ns > H ds ] . By using the Cauchy Schwarz inequality and the Lipschitz property of the generator f , foreach ( t, ω ) ∈ [0 , T ] × Ω, we obtain: < f ( s, Y n +1 s , H ( Y n − s ) , Z n +1 s , J ( Z n − s ) , U n +1 s , K ( U n − s )) − f ( s, Y ns , H ( Y n − s ) , Z ns , J ( Z n − s ) , U ns , K ( U n − s )) , Y n +1 s − Y ns > H ≤ | f ( s, Y n +1 s , H ( Y n − s ) , Z n +1 s , J ( Z n − s ) , U n +1 s , K ( U n − s )) − f ( s, Y ns , H ( Y n − s ) , Z ns , J ( Z n − s ) , U ns , K ( U n − s )) | H · | Y n +1 s − Y ns | H ≤ C (cid:0) |H ( Y ns ) − H ( Y n − s ) | H + |J ( Z ns ) − J ( Z n − s ) | H + |K ( U ns ) − K ( U n − s ) | L ν ( H ) (cid:1) | Y n +1 s − Y ns | H + C (cid:0) | Y n +1 s − Y ns | H + | Z n +1 s − Z ns | H + | U n +1 s − U ns | L ν ( H ) (cid:1) | Y n +1 s − Y ns | H . (4.43)We now appeal to the mean theorem in Hilbert spaces and obtain the existence foreach t ∈ [0 , T ] of some random variables ˜ Y n ( t ) ∈ L (Ω , H ), ˜ Z n ( t ) ∈ L (Ω , H ) , ˜ U n ( t ) ∈ L (Ω , L ν ( H )) such that |H ( Y nt ) − H ( Y n − t ) | H ≤ k∇H ( ˜ Y n ( t )) kk Y nt − Y n − t k L (Ω ,H ) |J ( Z nt ) − J ( Z n − t ) | H ≤ k∇J ( ˜ Z n ( t )) kk Z nt − Z n − t k L (Ω ,H ) |K ( U nt ) − K ( U n − t ) | H ≤ k∇K ( ˜ U n ( t )) kk U nt − U n − t k L (Ω , L ν ( H )) . (4.44)Using (4.42), (4.43), (4.44) together with the boundness of the Fr´echet derivatives of theoperators H , J , K and the inequality 2 ab ≤ εa + ε b , we obtain: E [ | Y n +1 t − Y nt | H ] + E [ Z Tt | Z n +1 s − Z ns | H ds ] + E [ Z Tt Z E | U n +1 s ( e ) − U ns ( e ) | H ν ( de ) ds ] ≤≤ λ E [ Z Tt | Y n +1 s − Y ns | H ds ] − α E [ Z Tt | Y n +1 s − Y ns | V ds ] − + Cε E [ Z Tt (cid:16) | Y ns − Y n − s | H + | Z ns − Z n − s | H + | U ns − U n − s | L ν ( H ) (cid:17) ds ] + 1 ε E [ Z Tt | Y n +1 s − Y ns | H ds ]+ Cβ E [ Z Tt (cid:16) | Y n +1 s − Y ns | H + | Z n +1 s − Z ns | H + | U n +1 s − U ns | L ν ( H ) (cid:17) ds ] + 1 β E [ Z Tt | Y n +1 s − Y ns | H ds ] , C is a constant dependind on the Lipschitz constant of f and the bounding constantsof the Fr´echet derivative operators of H , J , K .Let us choose ε ≤ C and β ≤ C . We set γ := λ + Cβ + ε + β + and then multiply theprevious inequality by e γt . We thus get − ddt (cid:18) e γt E [ Z Tt | Y n +1 s − Y ns | H ds ] (cid:19) + 12 e γt E [ Z Tt | Z n +1 s − Z ns | H ds ]+ 12 E [ Z Tt | Y n +1 s − Y ns | H ds ] e γt + 12 e γt E [ Z Tt | U n +1 s − U ns | L ν ( H ) ds ] + αe γt E [ Z Tt | Y n +1 s − Y ns | V ds ] ≤ E [ Z Tt | Y ns − Y n − s | H ] e γt + 14 E [ Z Tt | Z ns − Z n − s | H ] e γt + 14 E [ Z Tt | U ns − U n − s | L ν ( H ) ds ] e γt . (4.45)We now integrate between 0 and T and obtain: E [ Z T | Y n +1 s − Y ns | H ds ] + 12 Z T E [ Z Tt | Y n +1 s − Y ns | H ds ] e γt dt + 12 Z T E [ Z Tt | Z n +1 s − Z ns | H ds ] e γt dt + 12 Z T e γt E [ Z Tt | U n +1 s − U ns | L ν ( H ) ds ] + Z T α E [ Z Tt | Y n +1 s − Y ns | V ds ] e γt dt ≤ Z T E [ Z Tt | Y ns − Y n − s | H ds ] e γt dt + 14 Z T E [ Z Tt | Z ns − Z n − s | H ds ] e γt dt + 14 Z T E [ Z Tt | U ns − U n − s | L ν ( H ) ds ] e γt dt. (4.46)From the above inequality it follows that Z T E [ Z Tt | Y ns − Y n − s | H ] e γt dt + Z T E [ Z Tt | Z ns − Z n − s | H ] e γt dt + Z T E [ Z Tt | U ns − U n − s | L ν ( H ) ds ] e γt dt ≤ n C. From (4.46) one can deduce E [ Z T | Y n +1 s − Y ns | H ds ] ≤ n C. We now appeal to (4.45) and derive12 E [ Z T | Y n +1 s − Y ns | H ds ] + 12 Z T E [ Z T | Z n +1 s − Z ns | H ds ] + 12 Z T E [ Z T | U ns − U n − s | L ν ( H ) ds ] ≤ γ n C + 14 E [ Z T | Y ns − Y n − s | H ds ] + 14 E [ Z T | Z ns − Z n − s | H ds ] + 14 E [ Z T | U n +1 s − U ns | L ν ( H ) ds ] , which implies that E [ Z T | Y n +1 s − Y ns | H ds ]+ Z T E [ Z T | Z n +1 s − Z ns | H ds ]+ Z T E [ Z T | U ns − U n − s | L ν ( H ) ds ] ≤ n − Cγn. E [ Z T | Y n +1 s − Y ns | V ds ] ≤ ( 12 ) n − ( n + 1) Cγ.
Hence, we can conclude that the sequence ( Y n , Z n , U n ), n ≥ L (Ω × [0 , T ] , V ) × L (Ω × [0 , T ] , H ) × L (Ω × [0 , T ] , L ( ν )), and thus convergesin the corresponding spaces to ( Y, Z, U ). The limit (
Y, Z, U ) satisfies: Y t + Z Tt AY s ds + Z Tt f ( s, Y s , H ( Y s ) , Z s , J ( Z s ) , U s , K ( U s )) ds + Z Tt Z s dW s + Z Tt Z E U s ˜ N ( ds, de ) = ξ a.s. II. Uniqueness of the solution
The proof of the uniqueness of the solution is classical, but we give it for convenience of thereader. Suppose ( Y t , Z t , U t ) and ( ˜ Y t , ˜ Z t , ˜ U t ) are two solutions. By applying Itˆo formula, weobtain E [ | Y t − ˜ Y t | H ] + E [ Z Tt | Z s − ˜ Z s | H ds ] + E [ Z Tt | U s − ˜ U s | L ν ( H ) ] ds ] = − E [ < A ( Y s − ˜ Y s ) , Y s − ˜ Y s > ds ] − E [ Z Tt < f ( s, Y s , H ( Y s ) , Z s , J ( Z s ) , U s , K ( U s )) − f ( s, ˜ Y s , H ( ˜ Y s ) , ˜ Z s , J ( ˜ Z s ) , ˜ U s , K ( ˜ U s )) , ˜ Y s − ˜ Y s > H ds ] ≤ λ E [ Z Tt | Y s − ˜ Y s | H ds ] − α E [ Z Tt | Y s − ˜ Y s | V ds ] + K E [ Z Tt | Y s − ˜ Y s | H ds ]+ 12 E [ Z Tt | Z s − ˜ Z s | H ds ] + 12 E [ Z Tt | U s − ˜ U s | L ν ( H ) ds ] . We thus derive that E [ | Y t − ˜ Y t | H ] ≤ ( λ + K ) E [ Z Tt | Y s − ˜ Y s | H ]Hence, by Gronwall lemma, we get Y t = ˜ Y t . This also implies that Z t = ˜ Z t and U t = ˜ U t . A Some results on Banach theory
We recall here some basic concepts and results from Banach space theory. Let V be an opensubset of a Banach space X with norm k · k and let F : V R . (i) We say that F has a directional derivative (or Gˆateaux derivative ) at x ∈ X in thedirection y ∈ X if D y F ( x ) := lim ε → ε ( F ( x + εy ) − F ( x ))23xists. (ii) We say that F is Fr´echet differentiable at x ∈ V if there exists a linear map L : X 7→ R such that lim h → h ∈X k h k | F ( x + h ) − F ( x ) − L ( h ) | = 0 . In this case we call L the gradient (or Fr´echet derivative) of F at x and we write L = ∇ F. (iii) If F is Fr´echet differentiable, then F has a directional derivative in all directions y ∈ X and D y F ( x ) = ∇ x F ( y ) =: < ∇ x F, y > .
In particular, if X = L ( P ) the Fr´echet derivative of F at X ∈ L ( P ), denoted by ∇ F ( X ),is a bounded linear functional on L ( P ), which we can identify by Riesz theorem with arandom variable in L ( P ). For example, if F ( X ) = E [ φ ( X )]; X ∈ L ( P ) , where φ is a real C -function such that φ ( X ) ∈ L ( P ) and ∂φ∂x ( X ) ∈ L ( P ), then ∇ F ( X ) = ∂φ∂x ( X ) and ∇ F ( X )( Y ) = (cid:28) ∂φ∂x ( X ) , Y (cid:29) L ( P ) = E (cid:20) ∂φ∂x ( X ) Y (cid:21) , for Y ∈ L ( P ) . References [1] A. Bensoussan, Maximum principle and dynamic programming approaches of the op-timal control of partially observed diffusions,
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Lecture notes in Mathematics
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