Appl Math Optim DOI 10.1007/s00245-017-9425-1

Stochastic Control of Memory Mean-Field Processes Nacira Agram1 · Bernt Øksendal1

© Springer Science+Business Media, LLC 2017

Abstract By a memory mean-field process we mean the solution X (·) of a stochastic mean-field equation involving not just the current state X (t) and its law L(X (t)) at time t, but also the state values X (s) and its law L(X (s)) at some previous times s < t. Our purpose is to study stochastic control problems of memory mean-field processes. We consider the space M of measures on R with the norm ||·||M introduced by Agram and Øksendal (Model uncertainty stochastic mean-field control. arXiv:1611.01385v5, [2]), and prove the existence and uniqueness of solutions of memory mean-field stochastic functional differential equations. We prove two stochastic maximum principles, one sufficient (a verification theorem) and one necessary, both under partial information. The corresponding equations for the adjoint variables are a pair of (time-advanced backward stochastic differential equations (absdes), one of them with values in the space of bounded linear functionals on path segment spaces. As an application of our methods, we solve a memory mean–variance problem as well as a linear–quadratic problem of a memory process. Keywords Mean-field stochastic differential equation · Law process · Memory · Path segment spaces · Random probability measures · Stochastic maximum principle · Operator-valued absde · Mean–variance problem

Dedicated to the memory of Salah-Eldin Mohammed.

B

Bernt Øksendal [email protected] Nacira Agram [email protected]

1

Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway

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Appl Math Optim

Mathematics Subject Classification 60H05 · 60H20 · 60J75 · 93E20 · 91G80 · 91B70

1 Introduction In this work we are studying a general class of controlled memory mean-field stochastic functional differential equations (mf-sfdes) of the form ⎧ d X (t) = b(t, X (t), X t , M(t), Mt , u(t), u t )dt + σ (t, X (t), X t , M(t), Mt , u(t), u t )d B(t) ⎪ ⎪  ⎪ ⎨ (dt, dζ ); t ∈ [0, T ], + R0 γ (t, X (t), X t , M(t), Mt , u(t), u t , ζ ) N ⎪ X (t) = ξ(t); t ∈ [−δ, 0], ⎪ ⎪ ⎩ u(t) = u 0 (t); t ∈ [−δ, 0],

(1.1) on a filtered probability space (, F, P) satisfying the usual conditions, i.e., the filtration F = (Ft )t≥0 is right-continuous and increasing, and each Ft , t ≥ 0, contains all P-null sets in F. Here M(t) := L(X (t)) is the law of X (t) at time t, δ ≥ 0 is a given (constant) memory span and X t := {X (t + s)}s∈[−δ,0] ,

(1.2)

is the path segment of the state process X (·), while Mt := {M(t + s)}s∈[−δ,0] ,

(1.3)

is the path segment of the law process M(·) = L(X (·)). These processes represent the memory terms of the Eq. (1.1). The terms B(t) and N˜ (dt, dζ ) in the mf-sfde (1.1) denote a one-dimensional Brownian motion and an independent compensated Poisson random measure, respectively, such that N˜ (dt, dζ ) = N (dt, dζ ) − ν(dζ )dt, where N (dt, dζ ) is an independent Poisson random measure and ν(dζ ) is the Lévy measure of N . For the sake of simplicity, we only consider the one-dimensional case, i.e., X (t) ∈ R, B(t) ∈ R and N (t, ζ ) ∈ R, for all t, ζ. Following Agram and Øksendal [2], we now introduce the following Hilbert spaces: Definition 1.1 • M is the Hilbert space of random measures μ on R equipped with the norm μ2M := E



2 −y 2 | μ(y)| ˆ e dy , R

where μˆ is the Fourier transform of the measure μ, i.e., μ(y) ˆ :=

123



Re

ixy

dμ(x);

y ∈ R.

Appl Math Optim

• Mδ is the Hilbert space of all path segments μ = {μ(s)}s∈[−δ,0] of processes μ(·) with μ(s) ∈ M for each s ∈ [−δ, 0], equipped with the norm μMδ :=

0

−δ μ(s)M ds.

(1.4)

• M0 and Mδ0 denote the set of deterministic elements of M and Mδ , respectively. For simplicity of notation, in some contexts we regard M as a subset of Mδ and M0 as a subset of Mδ . The structure of this space M equipped with the norm obtained by the Fourier transform is an alternative to the Wasserstein metric space P2 equipped with the Wasserstein distance W2 . Moreover, the Hilbert space M deals with any random measure on R, however the Wasserstein space P2 deals with Borel probability measures on R with finite second moments. Using the Hilbert space structure for this type of problems has been proposed by P. L. Lions, to simplify the technicalities of the Wasserstein metric space where he considers the Hilbert space of square integrable random variables. Our Hilbert space, however is now. In the following, we denote by C := C([−δ, 0]; R) the Banach space of all paths of càdlàg F-adapted processes X t := {X (t + s)}s∈[−δ,0] : [−δ, 0] → R, equipped with the norm

 X C := E

sup |X (s)| .

(1.5)

s∈[−δ, 0]

To simplify the writing, we introduce some notations and the same notations E and E  differ but they are clear from the context. The coefficients b(t, x, x, m, m, u, u) = b(t, x, x, m, m, u, u, ω) : E → R, σ (t, x, x, m, m, u, u) = σ (t, x, x, m, m, u, u, ω) : E → R, γ (t, x, x, m, m, u, u, ζ ) = γ (t, x, x, m, m, u, u, ζ, ω) : E  → R, where E := [0, T ] × R × C × M0 × Mδ0 × R × C ×  and E  := [0, T ] × R × C × M0 × Mδ0 × R × C × R0 ×  and R0 = R − {0}. We remark that the functionals b, σ and γ on the mf-sfde depend on more than the solution X (t) and its law L(X (t)), both the segment X t and the law of this segment L(X t ) and this is a new-type of mf-sfdes with memory. Let us give some examples: let X (t) satisfies the following mean-field delayed sfde ⎧ X(t), E[X(t)], u(t))dt + σ (t, X(t), E[X(t)], u(t))d B(t) ⎪ ⎨ d X (t) = b(t,  + R0 γ (t, X(t), E[X(t)], u(t), ζ ) N˜ (dt, dζ ); t ∈ [0, T ], (1.6) ⎪ ⎩ X (t) = ξ(t); t ∈ [−δ, 0], 0 where we denote by the bold X(t) = −δ X (t + s)μ(ds) for some bounded Borelmeasure μ. As noted in Agram and Røse [3] and Banos et al. [5], we have the following:

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Appl Math Optim

• If this measure μ is a Dirac-measure concentrated at 0, i.e., X(t) = X (t) then Eq. (1.6) is a classical mean-field stochastic differential equation, we refer for example to Anderson and Djehiche [4] and Hu et al. [14] for stochastic control of such a systems. • It could also be the Dirac measure concentrated at −δ then X(t) = X (t − δ) and in that case the state equation is called a mean-field sde with discrete delay, see for instance Meng and Shen [17] and for delayed systems without a mean-field term, we refer to Chen and Wu [10], Dahl et al. [11] and Øksendal et al. [20]. • If we choose now μ(ds) = g(s)ds for any function g ∈ L 1 ([−δ, 0]) thus X(t) = 0 −δ g(s)X (t + s)ds and the state is a mean-field distributed delay. It is worth mentioning the papers by Lions [16], Cardaliaguet [7], Carmona and Delarue [8,9], Buckdahn et al. [6] and Agram [1] for more details about systems driven by mean-field equations and stochastic control problems for such a system. These papers, however, use the Wasserstein metric space of probability measures and not our Hilbert space of measures. The paper is organized as follows: in Sect. 2, we give some mathematical background and define some concepts and spaces which will be used in the paper. In Sect. 3, we prove existence and uniqueness of memory McKean–Vlasov equations. Section 4 contains the main results of this paper, including a sufficient and a necessary maximum principle for the optimal control of stochastic memory mean-field equations. In Sect. 5, we illustrate our results by solving a mean–variance and a linear–quadratic (LQ) problems of a memory processes.

2 Generalities In this section, we recall some concepts which will be used on the sequel. (a) We first discuss the differentiability of functions defined on a Banach space. Let X , Y be two Banach spaces with norms  · X ,  · Y , respectively, and let F: X → Y. • We say that F has a directional derivative (or Gâteaux derivative) at v ∈ X in the direction w ∈ X if Dw F(v) := lim

ε→0

1 (F(v + εw) − F(v)), ε

exists. • We say that F is Fréchet differentiable at v ∈ X if there exists a continuous linear map A: X → Y such that 1 F(v + h) − F(v) − A(h)Y = 0, h→0 hX lim

h∈X

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Appl Math Optim

where A(h) = A, h is the action of the linear operator A on h. In this case we call A the gradient (or Fréchet derivative) of F at v and we write A = ∇v F. • If F is Fréchet differentiable at v with Fréchet derivative ∇v F, then F has a directional derivative in all directions w ∈ X and Dw F(v) = ∇v F(w) = ∇v F, w . In particular, note that if F is a linear operator, then ∇v F = F for all v. (b) Throughout this work, we will use the following spaces: • S 2 is the set of R-valued F-adapted càdlàg processes (X (t))t∈[−δ,T ] such that

X S 2 := E



2

sup

|X (t)|

sup

|X (t)|

t∈[−δ, T ]

< ∞,

2

(alternatively (X (t))t∈[0,T +δ] with

X S 2 = E



2

t∈[0, T +δ]

2

< ∞,

depending on the context). • L2 is the set of R-valued F-adapted processes (Q(t))t∈[0,T ] such that Q2L2 := E

 T 0

|Q(t)|2 dt < ∞.

• U ad is a set of all stochastic processes u required to have values in a convex subset U of R and adapted to a given subfiltration G = {Gt }t≥0 , where Gt ⊆ Ft for all t ≥ 0. We call U ad the set of admissible control processes u(·). • L 2 (Ft ) is the set of R-valued square integrable Ft -measurable random variables. • L2ν is the set of R-valued F-adapted processes Z : R0 → R such that ||Z ||2L2 := E



ν

2 |Z (t, ζ )| ν(dζ )dt < ∞. R0

• R is the set of measurable functions r : R0 → R. • K is the set of bounded linear functionals K : M0 → R equipped with the operator norm ||K ||K :=

sup

m∈M0 , ||m||M0 ≤1

|K (m)|.

(2.1)

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Appl Math Optim 2 is the set of F-adapted stochastic processes p: [0, T + δ] ×  → K such • SK that



|| p||2SK := E

sup

t∈[0, T +δ]

|| p(t)||2K < ∞.

(2.2)

• L2K is the set of F-adapted stochastic processes q: [0, T + δ] ×  → K such that  T +δ (2.3) ||q||2L2 := E 0 ||q(t)||2K dt < ∞. K

• L2ν,K is the set of F-adapted stochastic processes r : [0, T + δ] × R0 ×  → K such that  T +δ  2 (2.4) ||r ||2L2 := E 0 R0 ||r (t, ζ )||K ν(dζ )dt < ∞. ν,K

3 Solvability of Memory Mean-Field sfde For a given constant δ > 0, we consider a memory mf-sfdes of the following form: ⎧ X (t), X t , M(t), Mt )dt + σ (t, X (t), X t , M(t), Mt )d B(t) ⎪ ⎨ d X (t) = b(t,  (dt, dζ ); t ∈ [0, T ], + R0 γ (t, X (t), X t , M(t), Mt , ζ ) N ⎪ ⎩ X (t) = ξ(t); t ∈ [−δ, 0]. (3.1) Here E := [0, T ]×R×C×M0 ×Mδ0 ×, E  := [0, T ]×R×C×M0 ×Mδ0 ×R0 × and the coefficients = b(t, x, x, m, m, ω) : E → R, b(t, x, x, m, m) = σ (t, x, x, m, m, ω) : E → R, σ (t, x, x, m, m) γ (t, x, x, m, m, ζ ) = γ (t, x, x, m, m, ζ, ω) : E  → R, are supposed to be Ft -measurable and the initial value function ξ is F0 -measurable. For more information about sfdes, we refer to the seminal work of S. E. A. Mohammed [18] and a recent paper by Banos et al. [5]. In order to prove an existence and uniqueness result for the mf-sfde (3.1), we first need the following lemma: Lemma 3.1 (i) Let X (1) and X (2) be two random variables in L 2 (P). Then  

 2 2  √ (1) (2) (1) (2) . −L X ≤ πE X − X L X M0

(ii) Let {X (1) (t)}t≥0 , {X (2) (t)}t≥0 be two processes such that E

123



T 0

X (i)2 (s)ds < ∞ for all T with i = 1, 2.

Appl Math Optim

Then 

 (1) (2) 2 − L Xt L X t

Mδ0

≤

√

πE

 0 −δ

X

(1)

(s) − X

(2)

(s)

2

 ds .

Proof By definition of the norms and standard properties of the complex exponential function, we have 

 2 L X (1) − L X (2) M0

 2   (1)   X  X (2) (y) e−y 2 dy = R L (y) − L



 2    2  = R  R ei x y d L X (1) (x) − R ei x y d L X (2) (x) e−y dy     (1) (2) 2 2 = R E ei y X − ei y X  e−y dy 

  

 2    2  = R E cos y X (1) − cos y X (2) + iE sin y X (1) − sin y X (2)  e−y dy    

 2 

 2   2 (1) (2) (1) (2) = R E cos y X e−y dy + E sin y X − cos y X − sin y X  

 

 2 

 2   2     e−y dy ≤ R E cos y X (1) − cos y X (2)  + E sin y X (1) − sin y X (2)      2  2   2     e−y dy ≤ R E y X (1) − X (2)  + E y X (1) − X (2)   2   (1) 2 −y 2 (2)  y e dyE − X X  R    2 √ , ≤ π E X (1) − X (2) ≤2



and similarly, we get that 

 (1) (2) 2 − L Xt L X t

 2 0 (1) (2) X (s) − X (s) ds L −δ Mδ0 M0  2  0 √ ≤ π E −δ X (1) (s) − X (2) (s) ds . ≤

  We are now able to state the theorem of existence and uniqueness of a solution of Eq. (3.1). As before we put E := [0, T ] × R × C × M0 × Mδ0 ×  and E  := [0, T ] × R × C × M0 × Mδ0 × R0 × . Then we have Theorem 3.2 Assume that ξ(t) ∈ C, b, σ : E → R and γ : E  → R are progressively measurable and satisfy the following uniform Lipschitz condition dt P(dω)-a.e.:

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Appl Math Optim

There is some constant L ∈ R such that |b(t, x, x, m, m, ω) − b(t, x  , x  , m  , m  , ω)|2 + |σ (t, x, x, m, m, ω) − σ (t, x  , x  , m  , m  , ω)|2  + |γ (t, x, x, m, m, ζ, ω) − γ (t, x  , x  , m  , m  , ζ, ω)|2 ν(dζ ) R0 

≤ L |x − x  |2 + ||x − x  ||2C + ||m − m  ||2M0 + ||m − m  ||2Mδ , for a.a. t, ω, 0

(3.2) and |b (t, 0, 0, μ0 , μ0 , ω)|2 + |σ (t, 0, 0, μ0 , μ0 , ω)|2  |γ (t, 0, 0, μ0 , μ0 , ζ, ω)|2 ν(dζ ) ≤ L for a.a. t, ω, + R0

(3.3)

where μ0 is the Dirac measure with mass at zero. Then there is a unique solution X ∈ S 2 of the mf-sfde (3.1). Proof For X ∈ S 2 [−δ, T ] and for t0 ∈ (0, T ], we introduce the norm

||X ||2t0

:= E

 sup |X (t)|

t∈[−δ, t0 ]

2

.

The space Ht0 equipped with this norm is a Banach space. Define the mapping : Ht0 → Ht0 by (x) = X where X ∈ S 2 is defined by ⎧ x(t), xt , m(t), m t )dt + σ (t, x(t), xt , m(t), m t )d B(t) ⎪ ⎨ d X (t) = b(t,  (dt, dζ ); t ∈ [0, T ], + R0 γ (t, x(t), xt , m(t), m t , ζ ) N ⎪ ⎩ X (t) = ξ(t); t ∈ [−δ, 0]. We want prove that is contracting in Ht0 under the norm || · ||t0 for small enough t0 . For two arbitrary elements (x 1 , x 2 ) and (X 1 , X 2 ), we denote their difference by X = X 1 − X 2 , respectively. In the following C < ∞ will denote a  x = x 1 − x 2 and  constant which is big enough for all the inequalities to hold. Applying the Itô formula to  X 2 (t), we get t

      X (s) b s, x 1 (s), xs1 , m 1 (s), m 1s − b s, x 2 (s), xs2 , m 2 (s), m 2s ds t      +2 0 X (s) σ s, x 1 (s), xs1 , m 1 (s), m 1s − σ s, x 2 (s), xs2 , m 2 (s), m 2s d B(s) t       (ds, dζ ) +2 0 X (s) R0 γ s, x 1 (s), xs1 , m 1 (s), m 1s , ζ − γ s, x 2 (s), xs2 , m 2 (s), m 2s , ζ N t      1 1 1 1 2 2 2 2 2 ds + 0 σ s, x (s), xs , m (s), m s − σ s, x (s), xs , m (s), m s  t     2 1 1 1 1 2 2 2 + 0 R0 γ s, x (s), xs , m (s), m s , ζ − γ s, x (s), xs , m (s), m 2s , ζ ν(dζ )ds.

 X 2 (t) = 2

0

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Appl Math Optim

By the Lipschitz assumption (3.2) combined with standard majorization of the square of a sum (resp. integral) via the sum (resp. integral) of the square (up to a constant), we get t

| X (s)| t0 ds 0      t   t (ds, dζ ) + tC (2) +  0 X (s) σ (s)d B(s) +  0 R0  X (s) γ (s, ζ ) N t0 ,

 X 2 (t) ≤ C

where 

t0 := || x ||S 2 + || x||C + || m ||M0 + ||m|| Mδ0 , (2)

 2 δ.

t0 := || x ||2S 2 + || x||2C + || m ||2M0 + ||m|| M0 By the Burkholder–Davis–Gundy inequalities,     1   t0 2 2  t  2  ≤ Ct0 ||  E sup  0  (s) σ (s)ds X ||t0 t0 , X (s) σ (s)d B(s) ≤ CE X 0 t≤t0

(3.4) and     1   t0 2 2  t 2 (ds, dζ ) ≤ CE  (s) γ (s)ν(dζ )ds X (s) γ (s) N X E sup  0  0 t≤t0

≤ Ct0 ||  X ||t0 t0 .

(3.5)

Combining the above and using that 

(2) ||  X ||t0 t0 ≤ C ||X ||2t0 + t0 , we obtain  

 2 2   || X ||t0 := E sup X (t) ≤ Ct0 ||  X ||2t0 + (2) . t0 t≤t0

By definition of the norms, we have

(2) x ||2t0 . t0 ≤ C||

(3.6)

Thus we see that if t0 > 0 is small enough we obtain ||  X (t)||2t0 ≤

1 || x (s)||2t0 , 2

(3.7)

and hence is a contraction on Ht0 . Therefore the equation has a solution up to t0 . By the same argument we see that the solution is unique. Now we repeat the argument

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Appl Math Optim

above, but starting at t0 instead of starting at 0. Then we get a unique solution up to   2t0 . Iterating this, we obtain a unique solution up to T for any T < ∞.

4 Optimal Control of Memory mf-sfde Consider again the controlled memory mf-sfde (1.1) ⎧ X (t), X t , M(t), Mt )dt + σ (t, X (t), X t , M(t), Mt )d B(t) ⎪ ⎨ d X (t) = b(t,  (dt, dζ ); t ∈ [0, T ], + R0 γ (t, X (t), X t , M(t), Mt , ζ ) N ⎪ ⎩ X (t) = ξ(t); t ∈ [−δ, 0]. (4.1) The coefficients b, σ and γ are supposed to satisfy the assumptions of Theorem 3.2, uniformly w.r.t. u ∈ U, then we have the existence and the uniqueness of the solution X (t) ∈ S 2 of the controlled mf-sfde (4.1). Moreover, b, σ and γ have Fréchet derivatives w.r.t. x, m, m and are continuously differentiable in the variables x and u. The performance functional is assumed to be of the form J (u) = E

 T 0

 (t, X (t), X t , M(t), Mt , u(t), u t ) dt + h(X (T ), M(T ) ; u ∈ U. (4.2)

With E := [0, T ] × R × C × M0 × Mδ0 × U × C × , E  := R × M0 ×  we assume that the functions (t, x, x, ¯ m, m, ¯ u, u) ¯ = (t, x, x, ¯ m, m, ¯ u, u, ¯ ω) : E → R, h(x, m) = h(x, m, ω) : E  → R, admit Fréchet derivatives w.r.t. x, m, m and are continuously differentiable w.r.t. x and u. We allow the integrand in the performance functional (4.2) to depend on the path process X t and also its law process L(X t ) =: Mt , and we allow the terminal value to depend on the state X (T ) and its law M(T ). Consider the following optimal control problem. It may regarded as a partial information control problem (since u is required to be G-adapted) but only in the limited sense, since G does not depend on the observation. Problem 4.1 Find u ∗ ∈ U such that J (u ∗ ) = sup J (u). u∈U

(4.3)

To study this problem we first introduce its associated Hamiltonian, as follows: Definition 4.2 The Hamiltonian H : [0, T + δ] × R × C × M0 × Mδ0 × U × C × R × R × R × K ×  → R,

123

Appl Math Optim

associated to this memory mean-field stochastic control problem (4.3) is defined by

 H t, x, x, m, m, u, u, p 0 , q 0 , r 0 (·), p 1

 = H t, x, x, m, m, u, u, p 0 , q 0 , r 0 (·), p 1 , ω = (t, x, x, m, m, u, u) + p 0 b(t, x, x, m, m, u, u) + q 0 σ (t, x, x, m, m, u, u)    + R0 r 0 (t, ζ )γ (t, ζ )ν(dζ ) + p 1 , m  ; t ∈ [0, T ], and H (t, x, x, m, m, u, u, p 0 , q 0 , r 0 (·), p 1 ) = 0; t > T. Here m  =

(4.4) d dt

m(t).

The Hamiltonian H is assumed to be continuously differentiable w.r.t. x, u and to admit Fréchet derivatives w.r.t. x, m and m. In the following we let L 20 denote the set of measurable stochastic processes Y (t) on R such that Y (t) = 0 for t < 0 and for t > T and T 0

Y 2 (t)dt < ∞ a.s.

(4.5)

The map Y →

T 0

∇x H (t), Yt  dt; Y ∈ L 20 ,

is a bounded linear functional on L 20 . Therefore, by the Riesz representation theorem there exists a unique process (t) ∈ L 20 such that T 0

(t)Y (t)dt =

T 0

∇x H (t), Yt  dt,

(4.6)

for all Y ∈ L 20 . Here ∇x H (t), Yt  denotes the action of the operator ∇x H (t) to the segment Yt = {Y (t + s)}s∈[−δ,0] , where H (t) is a shorthand notation for

 H t, X (t), X t , M(t), Mt , u(t), u t , p 0 (t), q 0 (t), r 0 (t, ·), p 1 (t), ω . As a suggestive notation (see below) for  we will in the following write ∇x H t := (t).

(4.7)

Lemma 4.3 Consider the case when H (t, x, x, p, q) = f (t, x) + F(x) p + σ q. Then   (t) := ∇x F, p t ,

(4.8)

123

Appl Math Optim

satisfies (4.6), where p t := { p(t + r )}r ∈[0,δ] = { p(t − s)}s∈[−δ,0] . Proof Choose Y ∈ L 20 and consider T    ∇x¯ F, p t Y (t)dt = 0 ∇x¯ F, { p(t + r )}r ∈[0,δ] Y (t)dt T   = 0 ∇x¯ F, {Y (t) p(t + r )}r ∈[0,δ] dt     T +r = ∇x¯ F, r Y (u − r ) p(u)du r ∈[0,δ]     T = ∇x¯ F, 0 Y (u − r ) p(u)du

T  0

= =

r ∈[0,δ]

T 0

∇x¯ F, Yu  p(u)du

0

∇x¯ H (u), Yu  du.

T

  Example 4.4 (i) For example, if F(x) ¯ =

0

−δ a(s)x(s)ds,

(4.9)

when x¯ = {x(s)}s∈[−δ,0] , then 

δ    0 ∇x¯ F, p t = F, p t = −δ a(s) p(t − s)ds = 0 a(−r ) p(t + r )dr.

(4.10)

(ii) Similarly, if G(x) ¯ = x(−δ) when x¯ = {x(s)}s∈[−δ,0] ,

(4.11)

then 

 ∇x¯ G, p t = p(t + δ).

(4.12)

For u ∈ U ad with corresponding solution X = X u , define p = ( p 0 , p 1 ), q = q 1 ) and r = (r 0 , r 1 ) by the following two adjoint equations:

(q 0 ,

• The advanced backward sfde (absfde) in the unknown ( p 0 , q 0 , r 0 ) ∈ S 2 ×L2 ×L2ν is given by ⎧ 0 d p (t) ⎪ ⎪ ⎪ ⎪ ⎨ p 0 (t) ⎪ q 0 (t) ⎪ ⎪ ⎪ ⎩ 0 r (t, ·)

=− =

!

∂H t ∂ x (t) + E(∇x H |Ft )

∂h ∂ x (X (T ),

dt + q 0 (t)d B(t) +

 R0

(dt, dζ ); t ∈ [0, T ], r 0 (t, ζ ) N

M(T )); t ≥ T,

= 0; t > T, = 0; t > T,

(4.13) where ∇m¯ H t is defined in the similar way as ∇x¯ H t above,

123

Appl Math Optim

• the operator-valued mf-absfde (ov-mf-absfde) in the unknown ( p 1 , q 1 , r 1 ) 2 × L2 × L2 ∈ SK K ν,K is given by ⎧ 1 dp (t) ⎪ ⎪ ⎪ ⎪ ⎨ p 1 (t) ⎪ q 1 (t) ⎪ ⎪ ⎪ ⎩ 1 r (t, ·)

= −[∇m H (t) + E(∇m H t |Ft )]dt + q 1 (t)d B(t) +

 R0

(dt, dζ ); t ∈ [0, T ], r 1 (t, ζ ) N

= ∇m h(X (T ), M(T )); t ≥ T, = 0; t > T, = 0; t > T.

(4.14) absdes have been studied by Peng and Yang [21] in the Brownian setting and for the jump case, we refer to Øksendal et al. [20], Øksendal and Sulem [19]. It was also extended to the context of enlargement progressive of filtration by Jeanblanc et al. [15]. When Agram and Røse [3] used the maximum principle to study optimal control of mean-field delayed sfde (1.6), they obtained a mean-field absfde. The question of existence and uniqueness of the solutions of the equations above will not be studied here. 4.1 A Sufficient Maximum Principle We are now able to derive the sufficient version of the maximum principle. Theorem 4.5 (Sufficient maximum principle) Let  u ∈ U ad with corresponding solu2 × L2 × L2 2 0 0 0 2 2 2  p , q , r ) ∈ S × L × Lν and ( p1 ,  q 1,  r 1 ) ∈ SK tions X ∈ S , ( K ν,K of the forward and backward stochastic differential equations (3.1), (4.13) and (4.14), respectively. For arbitrary u ∈ U, put 

 t , u(t), u t ,  M p 0 (t),  q 0 (t),  r 0 (t, ·),  p 1 (t) , H (t) := H t,  X (t),  X t , M(t), (4.15) 

 t ,  (t) := H t,  M u (t),  ut ,  p 0 (t),  q 0 (t),  r 0 (t, ·),  p 1 (t) . H X (t),  X t , M(t), (4.16) Suppose that • (Concavity) The functions   (x, x, m, m, u, u) → H t, x, x, m, m, u, u,  p0 ,  q 0,  r 0 (·),  p1 , (x, m)

→ h(x, m), are concave P-a.s. for each t ∈ [0, T ]. • (Maximum condition) ! (t)|Gt = sup E [H (t)|Gt ] , E H

(4.17)

u∈U ad

123

Appl Math Optim

P-a.s. for each t ∈ [0, T ]. Then  u is an optimal control for the problem (4.2). Proof We want to derive that J (u) ≤ J ( u ) for all u ∈ U ad . Application of definition ad (4.2) gives for fixed u ∈ U that J (u) − J ( u ) = I1 + I2 ,

(4.18)

where I1 = E

 T 0

{(t) −  (t)}dt ,

 ))], X (T ), M(T I2 = E[h(X (T ), M(T )) − h(  with    t , u(t), u t , (t) :=  t,  X (t),  X t , M(t), M     t ,  (t) :=  t,  X (t),  X t , M(t), M u (t),  ut ,

(4.19) (4.20)

and similarly with b(t),  b(t), etc. later. Applying the definition of the Hamiltonian (4.4), we get   T (t) −  I1 = E 0 H (t) − H p 0 (t) b(t) −  q 0 (t) σ (t)   0   (t) dt , − R0 r (t, ζ )γ˜ (t, ζ )ν(dζ ) −  p 1 (t) M

(4.21)

where  b(t) = b(t) −  b(t), etc., and     (t) = d M(t) = d (M(t) − M(t)). M dt dt Using concavity of h and the definition of the terminal values of the absfde (4.13) and (4.14), we get I2 ≤ E



∂ h    ∂ x (T ) X (T ) + ∇m h(T ) M(T )



  ) . X (T ) +  p 1 (T ) M(T =E  p 0 (T ) 

 we have Applying the Itô formula to  p0  X and  p 1 M,  T T 0 ! T 0 E  p 0 (T )  X (T ) = E 0  p (t)d  X (t) + 0  q (t) σ (t)dt X (t)d  p 0 (t) + 0  T 0 r (t, ζ ) γ (t, ζ )ν(dζ )dt + 0 R0

123

(4.22)

Appl Math Optim

 =E

T

 T ∂ H



  "t |Ft  X (t)dt E ∇x H 0 T 0 T 0 + 0 q (t) σ (t)dt + 0 R0 r (t, ζ ) γ (t, ζ )ν(dζ )dt , 0

 p 0 (t) b(t)dt −

0 ∂x

(t)  X (t)dt −

T

(4.23) and   ! T 1 1   + T M(t)d  E  p 1 (T ) M(t) =E 0  p (t)d M(t) (t)  p 0  T T   T 1  "t |Ft M(t)dt  (t) − ∇m H (t) M(t)dt   H p (t) M E ∇ =E 0  − , m 0 0

(4.24) (dt, dζ ) integrals have mean zero with the where we have used that the d B(t) and N necessary integrability condition. On substituting (4.21), (4.23) and (4.24) into (4.18), we obtain     T T "t  (t) − T ∂ H (t)  J (u) − J ( u ) ≤ E 0 H (t) − H H X (t)dt − ∇ X (t) dt x 0 ∂x 0 T  T  "t M(t)dt (t), M(t)   − 0 ∇m H dt − 0 ∇m H . Since  X (t) = 0 for all t ∈ [−δ, 0] and for all t > T we see that  X ∈ L 20 and therefore by (4.6), we have T 0

t  ∇x¯ H X (t)dt =

T  0

 (t),  ∇x¯ H X t dt.

(4.25)

Similar considerations can be done to get T 0

t M(t)dt  ∇m H =

T  0

 (t), M t dt. ∇m H

(4.26)

By the assumption that H is concave and that the process u is Gt -adapted, we therefore get  

 (t),  ∇u H u t dt 

    T (t),  = E 0 E ∂∂uH (t) u (t) + ∇u H u t |Gt dt     !   T (t)|Gt ,  = E 0 E ∂∂uH (t)|Gt  u (t) + E ∇u H u t dt ≤ 0.

J (u) − J ( u) ≤ E

T 0

 ∂H u (t) + ∂u (t)



(t)|Gt ] has a maximum at  For the last inequality to hold, we use that E[ H u (t).

 

4.2 A Necessary Maximum Principle We now proceed to study the necessary maximum principle. Let us then impose the following set of assumptions.

123

Appl Math Optim

(i) On the coefficient functionals: • The functions b, σ and γ admit bounded partial derivatives w.r.t. x, x, m, m, u, u. (ii) On the performance functional: • The function  and the terminal value h admit bounded partial derivatives w.r.t. x, x, m, m, u, u and w.r.t. x, m, respectively. (ii) On the set of admissible processes: • Whenever u ∈ U ad and π ∈ U ad is bounded, there exists  > 0 such that u + λπ ∈ U ad , for each λ ∈ [−, ]. • For each t0 ∈ [0, T ] and all bounded Gt0 -measurable random variables α, the process π(t) = α1(t0 ,T ] (t), belongs to U ad . In general, if K u (t) is a process depending on u, we define the operator D on K by D K u (t) := D π K u (t) =

d u+λπ K (t)|λ=0 , dλ

(4.27)

whenever the derivative exists. Define the derivative process Z (t) by Z (t) := D X (t) :=

u+λπ d |λ=0 . dλ X

Using matrix notation, note that Z (t) satisfies the equation ⎧ d Z (t) = (∇b(t))T (Z (t), Z t , D M(t), D Mt , π(t), πt )dt ⎪ ⎪ ⎪ ⎪ ⎨ + (∇σ (t))T (Z (t), Z t , D M(t), D Mt , π(t), πt )B(t)  (dt, dζ ); t ∈ [0, T ], ⎪ + R0 (∇γ (t, ζ ))T (Z (t), Z t , D M(t), D Mt , π(t), πt , ζ ) N ⎪ ⎪ ⎪ ⎩ Z (t) = 0; t ∈ [−δ, 0],

(4.28)  T ∂b , ∇u b , (·)T denotes matrix transposed where (∇b)T = ∂∂bx , ∇x b, ∇m b, ∇m b, ∂u and we mean by ∇x b(t)Z t (∇m b(t)D Mt ) the action of the operator ∇x b(t) (∇m b(t)) to the segment Z t = {Z (t +s)}s∈[−δ,0] (D Mt = {D M(t +s)}s∈[−δ,0] ), i.e., ∇x b(t), Z t  (∇m b(t), D Mt ) and similar considerations for σ and γ . Theorem 4.6 (Necessary maximum principle) Let  u ∈ U ad with corresponding solu2 × L2 × L2 2 0 0 0 2 2 2  p , q , r ) ∈ S × L × Lν and ( p1 ,  q 1,  r 1 ) ∈ SK tions X ∈ S and ( K ν,K of the forward and backward stochastic differential equations (3.1) and (4.13)–(4.14), respectively, with the corresponding derivative process  Z ∈ S 2 given by (4.28). Then the following, (i) and (ii), are equivalent:

123

Appl Math Optim

(i) For all π ∈ U ad and for all u ∈ U ad d dλ J (u

+ λπ )|λ=0 = 0.

(ii) E

∂H ∂u

 (t) + ∇u Ht  Gt ] = 0 for all t ∈ [0, T ).

Proof Before starting the proof, let us first clarify some notation: note that  ∇m

p11 (t),

   d d 1 m = p1 (t), (·) , dt dt

and hence 

       d d ∇m p11 (t), m , D M(t) = p11 (t), D M(t) = p11 (t), D M  (t) dt dt = p11 (t)D M  (t).

Also, note that d D M(t) = D M  (t)dt.

(4.29)

Assume that (i) holds. Then 0= =

d dλ J (u + λπ )|λ=0   T E 0 (∇(t))T (Z (t),

 Z t , D M(t), D Mt , π(t), πt ) dt ! + ∂∂hx (T )Z (T ) + ∇m h(T )D M(T ) .

Hence, by the definition of H (4.4) and the terminal values of the absfde p 0 (T ) and p 1 (T ), we have 0= =

d dλ J (u + λπ )|λ=0   T E 0 (∇ H (t))T

(Z (t), Z t , D M(t), D Mt , π(t), πt )

− p (t)(∇b(t)) (Z (t), Z t , D M(t), D Mt , π(t), πt ) 0

T

− q 0 (t)(∇σ (t))T (Z (t), Z t , D M(t), D Mt , π(t), πt )   − R0 r 0 (t, ζ )(∇γ (t, ζ ))T (Z (t), Z t , D M(t), D Mt , π(t), πt ) ν(dζ ) dt T − 0 p 1 (t)D M  (t)dt + p 0 (T )Z (T ) + p 1 (T )D M(T ) .

123

Appl Math Optim

Applying Itô formula to both p 0 Z and p 1 D M, we get  T ! ! T E p 0 (T )Z (T ) = E 0 p 0 (t)d Z (t) + 0 Z (t)d p 0 (t) + p 0 , Z T  T = E 0 p 0 (t)(∇b(t))T (Z (t), Z t , D M(t), D Mt , π(t), πt ) dt T # $ − 0 ∂∂Hx (t) + ∇x H t Z (t)dt T + 0 q 0 (t)(∇σ (t))T (Z (t), Z t , D M(t), D Mt , π(t), πt ) dt T + 0 R0 r 0 (t, ζ )(∇γ (t, ζ ))T (Z (t), Z t , D M(t), D Mt , π(t), πt ) ν(dζ )dt ,

and   T T E p 1 (T )D M(T ) = E 0 p 1 (t)D M  (t)dt + 0 D M(t)dp 1 (t)  T # $ T = E 0 p11 (t)D M  (t)dt − 0 ∇m H (t) + ∇m H t D M(t)dt . Proceeding as in (4.25)–(4.26), we obtain T T 0

0

∇x H t Z (t)dt =

∇m H D M(t)dt = t

T 0

∇x H (t),

0

∇m H (t), D Mt  dt.

T

Z t  dt,

Combining the above, we get 0=E

  ∇ (t)π(t) + H (t), π t dt . u ∂u

  T ∂H 0

(4.30)

Now choose π(t) = α1(t0 ,T ] (t), where α = α(ω) is bounded and Gt0 -measurable and t0 ∈ [0, T ). Then πt = α{1(t0 ,T ] (t + s)}s∈[−δ,0] and (4.30) gives 0=E



T ∂H t0 ∂u (t)αdt

+

T t0

# $ ∇u H (t), α 1(t0 ,T ] (t + s) s∈[−δ,0] dt .

Differentiating with respect to t0 , we obtain E

∂H ∂u

 ! (t0 ) + ∇u Ht0 α = 0.

Since this holds for all such α, we conclude that E

∂H ∂u

 ! (t0 ) + ∇u Ht0 |Gt0 = 0, which is (ii).

This argument can be reversed, to prove that (ii) ⇒ (i). We omit the details.

5 Applications We illustrate our results by studying some examples.

123

 

Appl Math Optim

5.1 Mean–Variance Portfolio with Memory We apply the results obtained in the previous sections to solve the memory mean– variance problem by proceeding as it has been done in Framstad et al. [12], Anderson and Djehiche [4] and Røse [22]. Consider the state equation X π (t) = X (t) on the form ⎧  ⎨ d X (t) = X (t − δ)π(t) b (t)dt + σ (t)d B(t) +  γ (t, ζ ) N (dt, dζ ) ; t ∈ [0, T ], 0 0 R0 0 ⎩ X (t) = ξ(t); t ∈ [−δ, 0],

(5.1) for some bounded deterministic function ξ(t); t ∈ [−δ, 0]. We assume that the admissible processes are càdlàg processes in L 2 (, [0, T ]), that are adapted to the filtration Ft and such that a unique solution exists. The coefficients b0 , σ0 and γ0 > −1 are supposed to be bounded F-adapted processes with |b0 (t)| > 0 and σ02 (t) +



2 R0 γ0 (t,

ζ )ν(dζ ) > 0 a.s. for all t.

We want to find an admissible portfolio π(t) which maximizes   1 2 J (π ) = E − (X (T ) − a) , 2

(5.2)

over the set of admissible processes U ad and for a given constant a ∈ R. The Hamiltonian for this problem is given by



  H t, x, π, p 0 , q 0 , r 0 (·) = π G(x) b0 p 0 + σ0 q 0 + R0 γ0 (ζ )r 0 (ζ )ν(dζ ) , (5.3) where G(x) ¯ = x(−δ) when x¯ = {x(s)}s∈[−δ,0] .

(5.4)

See Example 4.4 (i). Hence by Lemma 4.3 the triple ( p 0 , q 0 , r 0 ) ∈ S 2 × L2 × L2ν is the adjoint process which satisfies ⎧ dp 0 (t) = −E[π(t + δ)(b0 (t + δ) p 0 (t + δ) + σ0 (t + δ)q 0 (t + δ) ⎪ ⎪ ⎪  ⎪ ⎪ + R0 γ0 (t + δ, ζ )r 0 (t + δ, ζ )ν(dζ ))|Ft ]dt + q 0 (t)d B(t) ⎪ ⎨  (dt, dζ ); t ∈ [0, T ], + R0 r 0 (t, ζ ) N ⎪ ⎪ ⎪ p 0 (t) = −(X (T ) − a); t ≥ T, ⎪ ⎪ ⎪ ⎩ 0 q (t) = r 0 (·) = 0; t > T.

(5.5)

Existence and uniqueness of equations of type (5.5) have been studied by Øksendal et al. [20].

123

Appl Math Optim

Suppose that  π is an optimal control. Then by the necessary maximum principle, we get for each t that

 t, " Xt ,  π (t),  p 0 (t),  q 0 (t),  r 0 (t, ·) (5.6)

  = X (t − δ) b0 (t) p 0 (t) + σ0 (t) q 0 (t) + R0 γ0 (t, ζ ) r 0 (t, ζ )ν(dζ ) .

0=

 ∂H ∂π

So we search for a candidate  π satisfying 0 = b0 (t) p 0 (t) + σ0 (t) q 0 (t) +



R0 γ0 (t,

ζ ) r 0 (t, ζ )ν(dζ ), for all t.

This gives the following adjoint equation: ⎧  0 (dt, dζ ); t ∈ [0, T ], q 0 (t)d B(t) + R0 r (t, ζ ) N p 0 (t) =  ⎪ ⎨d 0  p (t) = −(X (T ) − a); t ≥ T, ⎪ ⎩ 0  q (t) =  r 0 (·) = 0; t > T.

(5.7)

(5.8)

We start by guessing that  p 0 has the form X (t) + ψ(t),  p 0 (t) = ϕ(t) 

(5.9)

for some deterministic functions ϕ, ψ ∈ C 1 [0, T ] with ϕ(T ) = −1, ψ(T ) = a.

(5.10)

Using the Itô formula to find the integral representation of  p 0 and comparing with the adjoint equation (5.8), we find that the following three equations need to be satisfied: X (t) + ψ  (t) + ϕ(t)  X (t − δ) π (t)b0 (t), 0 = ϕ  (t) 

(5.11)

 q 0 (t) = ϕ(t)  X (t − δ) π (t)σ0 (t),

(5.12)

 r 0 (t, ζ ) = ϕ(t)  X (t − δ) π (t)γ0 (t, ζ ).

(5.13)

Assuming that  X (t) = 0P × dt-a.e. and ϕ(t) = 0 for each t, we find from Eq. (5.11) that  π needs to satisfy  π (t) = −

ϕ  (t)  X (t) + ψ  (t) . ϕ(t)  X (t − δ)b0 (t)

Now inserting the expressions for the adjoint processes (5.11)–(5.13) into (5.7), the following equation need to be satisfied:    X (t) + ψ(t)] + ϕ(t)  X (t − δ) π (t) σ02 (t) + R0 γ02 (t, ζ )ν(dζ ) . 0 = b0 (t)[ϕ(t) 

123

Appl Math Optim

This means that the control  π also needs to satisfy  π (t) = − 

 b0 (t)[ϕ(t) X(t)+ψ(t)] . X(t−δ) σ02 (t)+ R γ02 (t, ζ )ν(dζ ) ϕ(t)  0

(5.14)

By comparing the two expressions for  π , we find that b02 (t)[ϕ(t)  X (t) + ψ(t)]

  = σ02 (t) + R0 γ02 (t, ζ )ν(dζ ) [ϕ  (t)  X (t) + ψ  (t)].

(5.15)

Now define (t) :=

b2 (t)  0 . σ02 (t)+ R γ02 (t, ζ )ν(dζ )

(5.16)

0

Then from Eq. (5.15), we need to have ϕ  (t) − (t)ϕ(t) = 0, ψ  (t) − (t)ψ(t) = 0. Together with the terminal values (5.10), these equations have the solution

  T ϕ(t) = − exp − t (s)ds ,

  T ψ(t) = a exp − t (s)ds . Then from Eq. (5.14) we can compute

 π (t) =

  ψ(t) b0 (t)  X (t)− ϕ(t)





σ02 (t)+ R γ02 (t, ζ )ν(dζ ))  X (t−δ) 0

= (t)

ψ(t)  X (t)− ϕ(t)



b0 (t)  X (t−δ)

=

(t) ( X (t) − a). b0 (t)  X (t−δ)

Now, with our choice of  π , the corresponding state equation is the solution of %

d X (t) =

(t)  b0 (t) ( X (t) − a)



b0 (t)dt + σ0 (t)d B(t) +

 X (t) = x0 (t); t ∈ [−δ, 0].



R0 γ0 (t,

(dt, dζ ) ; t ∈ [0, T ], ζ )N

(5.17)

Put Y (t) =  X (t) − a, then  dY (t) = Y (t) (t)b0 (t)dt +

(t) b0 (t) σ0 (t)d B(t) +

 R0

(t) b0 (t) γ0 (t,

  ζ ) N (dt, dζ ) . (5.18)

123

Appl Math Optim

The linear equation (5.18) has the following explicit solution  Y (t) = Y (0) exp

t



t

(s)b0 (s)ds +

0

0

(s) b0 (s) σ0 (s)d B(s) +

 t 0

R0

(s) b0 (s) γ0 (s,

 (ds, dζ ) . ζ )N

So if Y (0) > 0 then Y (t) > 0 for all t. We have proved the following: Theorem 5.1 (Optimal mean–variance portfolio) Suppose that ξ(t) > a for all t ∈ [−δ, 0]. Then  X (t − δ) > 0 for all t ≥ 0 and the solution  π ∈ U ad of the mean– variance portfolio problem (5.2) is given in feedback form as  π (t) =

(t) ( X (t) − a), b0 (t)  X(t−δ)

where  X (t) and (t) are given by Eqs. (5.17) and (5.16), respectively. 5.2 A Linear–Quadratic (LQ) Problem with Memory We now consider a LQ control problem for a controlled system X (t) = X u (t) driven by a distributed delay, of the form %

d X (t) =

 δ

 (dt, dζ ); t ∈ [0, T ], a(s)X (t + s)ds + u(t) dt + α0 d B(t) + R0 β0 (ζ ) N 0

X (t) = ξ(t); t ∈ [−δ, 0],

(5.19) where ξ(·) and a(·) are given bounded deterministic functions, α0 is a given constant, β0 is a given function from R0 into R with 

R0 β0 (ζ )ν(dζ )

< ∞,

2 and u ∈ U ad is our control process. We want to minimize the expected  value of X (T ) T 2 with a minimal average use of energy, measured by the integral E 0 u (t)dt , i.e., the performance functional is of the quadratic type

 T J (u) = − 21 E X 2 (T ) + 0 u 2 (t)dt . Our goal is to find  u ∈ U ad , such that J ( u ) = sup J (u).

(5.20)

u∈U ad

The Hamiltonian is that case takes the form 

 H t, x, u, p 0 , q 0 = − 21 u 2 + (F(x) + u) p 0 + β0 q 0 + R0 r 0 (ζ )β0 (ζ )ν(dζ ),

123

Appl Math Optim

where F(x) ¯ =

0

−δ a(s)x(s)ds

when x¯ = {x(s)}s∈[−δ,0] .

(5.21)

By Lemma 4.3 and Example 4.4 (i) we see that the adjoint absde for ( p 0 , q 0 , r 0 ) is the following linear absde ⎧  ⎨ d p 0 (t) = −E  δ (a(−r ) p(t + r )dr |F + q 0 (t)d B(t) +  r 0 (t, ζ ) N (dt, dζ ); t ∈ [0, T ], t 0 R0 ⎩ p 0 (T ) = −X (T ); t ≥ T.

(5.22) q 0 (t), r 0 (t, ζ )) is maximal when The function u → H (t,  X (t − δ),  u (t),  p 0 (t),  u(t) =  u (t) =  p 0 (t),

(5.23)

where  p 0 (t) is a part of the solution of the linear absde ⎧   0 δ ⎨ d (dt, dζ ); t ∈ [0, T ], p (t + r )dr |Ft +  r (t, ζ ) N p 0 (t) = −E 0 (a(−r ) q 0 (t)d B(t) + R0 ⎩ X (T ); t ≥ T. p 0 (T ) = − 

(5.24) We have proved: Theorem 5.2 The optimal control uˆ of the LQ memory problem (5.20) is given by (5.23), where the quadruplet ( Xˆ (t) = X uˆ (t), pˆ 0 (t), qˆ 0 (t), r 0 (t, ζ )) solves the coupled system (5.19) and (5.24) of forward–backward stochastic differential equations with distributed delay. Remark 5.3 We may regard this coupled system (5.19) and (5.24) as the corresponding Riccati equation to our LQ memory problem. See, e.g., [13, p. 1747]. Acknowledgements This research was carried out with support of the Norwegian Research Council, within the Research Project Challenges in Stochastic Control, Information and Applications (STOCONINF), Project Number 250768/F20.

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