## Stochastic Control of Memory Mean-Field Processes

### Nacira A

GRAM^{1,2}

### and Bernt Ø

KSENDAL^{1,2}

### 4 September 2017

### Dedicated to the memory of Salah-Eldin Mohammed

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 spaceMof measures onRwith the norm|| · ||M introduced by Agram and Øksendal in [?], 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, 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.

MSC(2010): 60H05, 60H20, 60J75, 93E20, 91G80,91B70.

Keywords: Mean-field stochastic differential equation; law process; memory; path seg- ment spaces; random probability measures; stochastic maximum principle; operator-valued advanced backward stochastic differential equation; mean-variance problem.

1Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N–0316 Oslo, Norway.

Email: naciraa@math.uio.no, oksendal@math.uio.no

2This 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.

### 1 Introduction

In this work we are studying a general class of controlled memory mean-field stochastic functional differential equations (mf-sfde) of the form

dX(t) = b(t, X(t), X_{t}, M(t), M_{t}, u(t), u_{t})dt+σ(t, X(t), X_{t}, M(t), M_{t}, u(t), u_{t})dB(t)
+R

R0γ(t, X(t), X_{t}, M(t), M_{t}, u(t), u_{t}, ζ)Ne(dt, dζ);t∈[0, T],
X(t) =ξ(t);t∈[−δ,0],

u(t) =u_{0}(t);t∈[−δ,0],

(1.1) {mfsfde}

on a filtered probability space (Ω,F,P) satisfying the usual conditions, i.e. the filtration
F= (F_{t})t≥0 is right-continuous and increasing, and each F_{t},t ≥0, contains allP-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

M_{t}:={M(t+s)}s∈[−δ,0] (1.3)

is the path segment of the law process M(·) = L(X(·)). The process u(t) is our control
process, and u_{t}:={u(t+s)}s∈[−δ,0] is its memory path segment. The path processes X_{t}, M_{t}
andu_{t}represent the memory terms of the equation (??). The termsB(t) and ˜N(dt, dζ) in the
mf-sfde (??) 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 L´evy 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 [?], we now introduce the following Hilbert spaces:

Definition 1.1

• M is the Hilbert space of random measures µon R equipped with the norm
kµk^{2}_{M} := E[R

R|µ(y)|ˆ ^{2}e^{−y}^{2}dy],
where µˆ is the Fourier transform of the measure µ, i.e.

ˆ

µ(y) := R

Re^{ixy}dµ(x); y ∈R.

• M^{δ} 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

kµk_{M}δ :=R0

−δkµ(s)k_{M}ds. (1.4)

• M_{0} and M^{δ}_{0} denote the set of deterministic elements of M and M^{δ}, respectively.

For simplicity of notation, in some contexts we regard M as a subset of M^{δ} and M_{0}
as a subset of M^{δ}.

The structure of this spaceMequipped with the norm obtained by the Fourier transform,
is an alternative to the Wasserstein metric spaceP2 equipped with the Wasserstein distance
W_{2}. Moreover, the Hilbert space M deals with any random measure on R, however the
Wasserstein spaceP_{2} deals with Borel probability measures onRwith 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 ¯x :=

{x(s)}_{s∈[−δ,0]}, equipped with the norm

||¯x||C :=E[ sup

s∈[−δ,0]

|x(s)|]. (1.5)

To simplify the writing, we introduce some notations and the same notationsE andE^{0} 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^{0} →R,

whereE := [0, T]×R× C × M_{0}× M^{δ}_{0}×R× C ×Ω and E^{0} := [0, T]×R× C × M_{0}× M^{δ}_{0}×
R× C ×R0×Ω and R0 =R− {0}.

We remark that the functionals b, σ and γ on the mf-sfde depend on more than the so-
lution 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 mean-field stochastic functional differential equations with memory.

Let us give some examples: Let X(t) satisfies the following mean-field delayed sfde

dX(t) =b(t,X(t),E[X(t)], u(t))dt+σ(t,X(t),E[X(t)], u(t))dB(t) +R

R0γ(t,X(t),E[X(t)], u(t), ζ) ˜N(dt, dζ);t∈[0, T], X(t) =ξ(t);t∈[−δ,0],

(1.6) {mfd}

where we denote by the bold X(t) = R0

−δX(t+s)µ(ds) for some bounded Borel-measure µ.

As noted in Agram and Røse [?] and Banos et al [?], we have the following:

• If this measureµis a Dirac-measure concentrated at 0 i.e. X(t) = X(t) then equation (??) is a classical mean-field stochastic differential equation, we refer for example to Anderson and Djehiche in [?] and Hu el al in [?] 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 [?] and for delayed systems without a mean-field term, we refer to Chen and Wu [?], Dahl et al [?] and Øksendal et al [?].

• If we choose now µ(ds) = g(s)ds for any function g ∈ L^{1}([−δ,0]) thus X(t) =
R0

−δg(s)X(t+s)ds and the state is a mean-field distributed delay.

It is worth mentioning the papers by Lions [?], Cardaliaguet [?], Carmona and Delarue [?], [?], Buckdahn et al [?] and Agram [?] 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 section 2, we give some mathematical background and define some concepts and spaces which will be used in the paper. In section 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 section 5, we illustrate our results by solving a mean-variance and a linear-quadratic 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.

LetX,Y be two Banach spaces with normsk · kX,k · kY, respectively, and let F :X → Y.

• We say that F has a directional derivative (or Gˆateaux derivative) at v ∈ X in the direction w∈ X if

DwF(v) := lim

ε→0

1

ε(F(v+εw)−F(v)) exists.

• We say thatF is Fr´echet differentiable atv ∈ X if there exists a continuous linear map A:X → Y such that

h→0lim

h∈X

1 khkX

kF(v+h)−F(v)−A(h)kY = 0,

whereA(h) = hA, hiis the action of the linear operator Aonh. In this case we call A the gradient (or Fr´echet derivative) of F atv and we write

A=∇_{v}F.

• IfF is Fr´echet differentiable atv with Fr´echet derivative∇_{v}F, thenF has a directional
derivative in all directionsw∈ X and

D_{w}F(v) =∇_{v}F(w) = h∇_{v}F, wi.

In particular, note that if F is a linear operator, then∇vF =F for all v.

b) Throughout this work, we will use the following spaces:

• S^{2} is the set of R-valued F-adapted c`adl`ag processes (X(t))t∈[−δ,T] such that
kXk^{2}_{S}2 :=E[ sup

t∈[−δ,T]

|X(t)|^{2}] < ∞,
(alternatively (X(t))t∈[0,T+δ] with

kXk^{2}_{S}2 =E[ sup

t∈[0,T+δ]

|X(t)|^{2}] < ∞,
depending on the context.)

• L^{2} is the set of R-valued F-adapted processes (Q(t))t∈[0,T] such that
kQk^{2}

L^{2} :=E[RT

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 callU^{ad} the set of admissible control processes u(·).

• L^{2}(F_{t}) is the set of R-valued square integrable F_{t}-measurable random variables.

• L^{2}ν is the set of R-valued F-adapted processes Z :R0 →R such that

||Z||^{2}

L^{2}ν :=E[R

R0|Z(t, ζ)|^{2}ν(dζ)dt] < ∞.

• R is the set of measurable functions r:R^{0} →R.

• Ca([0, T],M0) denotes the set of absolutely continuous functions m : [0, T]→ M0.

• K is the set of bounded linear functionals K : M_{0} → R equipped with the operator
norm

||K||_{K} := sup

m∈M0,||m||M0≤1

|K(m)|. (2.1)

• S^{2}

K is the set of F-adapted stochastic processes p: [0, T +δ]×Ω7→K such that

||p||^{2}_{S}

K :=E[ sup

t∈[0,T+δ]

||p(t)||^{2}

K]<∞. (2.2)

• L^{2}_{K} is the set of F-adapted stochastic processes q : [0, T +δ]×Ω7→K such that

||q||^{2}

L^{2}_{K} :=E[RT+δ

0 ||q(t)||^{2}

Kdt]<∞. (2.3)

• L^{2}ν,K is the set of F-adapted stochastic processesr : [0, T +δ]×R0×Ω7→Ksuch that

||r||^{2}

L^{2}_{ν,}_{K} :=E[RT+δ
0

R

R0||r(t, ζ)||^{2}_{K}ν(dζ)dt]<∞. (2.4)

### 3 Solvability of memory mean-field sfde

For a given constantδ >0, we consider a memory mean-field stochastic functional differential equations (mf-sfde) of the following form:

dX(t) = b(t, X(t), X_{t}, M(t), M_{t})dt+σ(t, X(t), X_{t}, M(t), M_{t})dB(t)
+R

R0γ(t, X(t), X_{t}, M(t), M_{t}, ζ)Ne(dt, dζ);t∈[0, T],
X(t) =ξ(t);t∈[−δ,0].

(3.1) {sfde}

HereE := [0, T]×R× C × M_{0}× M^{δ}_{0} ×Ω,E^{0} := [0, T]×R× C × M_{0}× M^{δ}_{0}×R0×Ω and
the coefficients

b(t, x, x, m, m) =b(t, x, x, m, m, ω) :E →R,
σ(t, x, x, m, m) =σ(t, x, x, m, m, ω) :E →R,
γ(t, x, x, m, m, ζ) = γ(t, x, x, m, m, ζ, ω) :E^{0} →R,

are supposed to beF_{t}-measurable and the initial value function ξ is F_{0}-measurable.

For more information about stochastic functional differential equations, we refer to the seminal work of S.E.A. Mohammed [?] and a recent paper by Banos et al [?].

In order to prove an existence and uniqueness result for the mf-sfde (??), 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
L(X^{(1)})− L(X^{(2)})

2

M_{0} ≤ √

πE[(X^{(1)}−X^{(2)})^{2}].

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

0 X^{(i)2}(s)ds]<∞ for all T with i= 1,2.

Then

||L(X_{t}^{(1)})− L(X_{t}^{(2)})||^{2}_{M}δ
0

≤ √ πE[R0

−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

L(X^{(1)})− L(X^{(2)})

2 M0

=R

R|L(Xb ^{(1)})(y)−L(Xb ^{(2)})(y)|^{2}e^{−y}^{2}dy

=R

R

R

Re^{ixy}dL(X^{(1)})(x)−R

Re^{ixy}dL(X^{(2)})(x)

2e^{−y}^{2}dy

=R

R|E[e^{iyX}^{(1)} −e^{iyX}^{(2)}]|^{2}e^{−y}^{2}dy

=R

R|E[cos(yX^{(1)})−cos(yX^{(2)})] +iE[sin(yX^{(1)})−sin(yX^{(2)})]|^{2}e^{−y}^{2}dy

=R

R(E[cos(yX^{(1)})−cos(yX^{(2)})]^{2}+E[sin(yX^{(1)})−sin(yX^{(2)})]^{2})e^{−y}^{2}dy

≤R

R(E[|cos(yX^{(1)})−cos(yX^{(2)})|]^{2}+E[|sin(yX^{(1)})−sin(yX^{(2)})|]^{2})e^{−y}^{2}dy

≤R

R(E[|y(X^{(1)}−X^{(2)})|]^{2}+E[|y(X^{(1)}−X^{(2)})|]^{2})e^{−y}^{2}dy

≤2R

Ry^{2}e^{−y}^{2}dyE[|X^{(1)}−X^{(2)}|]^{2}

≤√

πE[(X^{(1)}−X^{(2)})^{2}],
and similarly, we get that

||L(X_{t}^{(1)})− L(X_{t}^{(2)})||^{2}_{M}δ
0

≤ R0

−δ

L(X^{(1)}(s)−X^{(2)}(s))

2 M0ds

≤ √ πE[R0

−δ(X^{(1)}(s)−X^{(2)}(s))^{2}ds].

We also need the following result, which is Lemma 2.3 in [?]:

Lemma 3.2 Suppose that X(t) is an Itˆo-L´evy process of the form (dX(t) =α(t)dt+β(t)dB(t) +R

R0γ(t, ζ) ˜N(dt, dζ); t∈[0, T],

X(0) =x∈R, (3.2) {eq2.1}

where α, β and γ are predictable processes.

Then the map t 7→M(t) : [0, T]→ M0 is absolutely continuous.

It follows thatt 7→M(t) is differentiable for a.a.t. We will in the following use the notation
M^{0}(t) = dM(t)

dt . (3.3)

We are now able to state the theorem of existence and uniqueness of a solution of equation
(??). As before we putE := [0, T]×R× C × M_{0}× M^{δ}_{0}×Ω and E^{0} := [0, T]×R× C × M_{0}×
M^{δ}_{0}×R0×Ω. Then we have

{existence}
Theorem 3.3 Assume that ξ(t) ∈ C, b, σ : E → R and γ : E^{0} → R are progressively

measurable and satisfy the following uniform Lipschitz condition dtP(dω)-a.e.:

There is some constant L∈R such that

|b(t, x, x, m, m, ω)−b(t, x^{0}, x^{0}, m^{0}, m^{0}, ω)|^{2}+|σ(t, x, x, m, m, ω)−σ(t, x^{0}, x^{0}, m^{0}, m^{0}, ω)|^{2}
+R

R0|γ(t, x, x, m, m, ζ, ω)−γ(t, x^{0}, x^{0}, m^{0}, m^{0}, ζ, ω)|^{2}ν(dζ)

≤L(|x−x^{0}|^{2}+||x−x^{0}||^{2}_{C}+||m−m^{0}||^{2}_{M}

0 +||m−m^{0}||^{2}_{M}δ
0

), for a.a. t, ω,

(3.4) {Lip}

and

|b(t,0,0, µ_{0}, µ_{0}, ω)|^{2}+|σ(t,0,0, µ_{0}, µ_{0}, ω)|^{2}
+R

R0|γ(t,0,0, µ_{0}, µ_{0}, ζ, ω)|^{2}ν(dζ)≤L for a.a. t, ω, (3.5) {Bou}

where µ_{0} is the Dirac measure with mass at zero. Then there is a unique solution X ∈ S^{2} of
the mf-sfde (??).

Proof. For X ∈ S^{2}[−δ, T] and for t_{0} ∈(0, T], we introduce the norm

||X||^{2}_{t}

0 :=E[ sup

t∈[−δ,t_{0}]

|X(t)|^{2}].

The spaceH^{t}0 equipped with this norm is a Banach space. Define the mapping Φ : H^{t}0 →
Ht0 by Φ(x) = X whereX ∈ S^{2} is defined by

dX(t) =b(t, x(t), x_{t}, m(t), m_{t})dt+σ(t, x(t), x_{t}, m(t), m_{t})dB(t)
+R

R0γ(t, x(t), x_{t}, m(t), m_{t}, ζ)Ne(dt, dζ);t∈[0, T],
X(t) =ξ(t);t∈[−δ,0].

We want prove that Φ is contracting in Ht0 under the norm || · ||_{t}_{0} for small enough t_{0}. For
two arbitrary elements (x^{1}, x^{2}) and (X^{1}, X^{2}), we denote their difference by xe=x^{1}−x^{2} and
Xe = X^{1} −X^{2} respectively. In the following C < ∞ will denote a constant which is big
enough for all the inequalities to hold.

Applying the Itˆo formula toXe^{2}(t), we get

Xe^{2}(t) = 2Rt

0X(s)(b(s, xe ^{1}(s), x^{1}_{s}, m^{1}(s), m^{1}_{s})−b(s, x^{2}(s), x^{2}_{s}, m^{2}(s), m^{2}_{s}))ds
+ 2Rt

0X(s)(σ(s, xe ^{1}(s), x^{1}_{s}, m^{1}(s), m^{1}_{s})−σ(s, x^{2}(s), x^{2}_{s}, m^{2}(s), m^{2}_{s}))dB(s)
+ 2Rt

0X(s)e R

R0(γ(s, x^{1}(s), x^{1}_{s}, m^{1}(s), m^{1}_{s}, ζ)−γ(s, x^{2}(s), x^{2}_{s}, m^{2}(s), m^{2}_{s}, ζ))Ne(ds, dζ)
+Rt

0(σ(s, x^{1}(s), x^{1}_{s}, m^{1}(s), m^{1}_{s})−σ(s, x^{2}(s), x^{2}_{s}, m^{2}(s), m^{2}_{s}))^{2}ds
+Rt

0

R

R0(γ(s, x^{1}(s), x^{1}_{s}, m^{1}(s), m^{1}_{s}, ζ)−γ(s, x^{2}(s), x^{2}_{s}, m^{2}(s), m^{2}_{s}, ζ))^{2}ν(dζ)ds.

By the Lipschitz assumption (??) 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

Xe^{2}(t)≤CRt

0|X(s)|∆e _{t}_{0}ds
+|Rt

0X(s)e σ(s)dB(s)|e +|Rt 0

R

R0X(s)e eγ(s, ζ)Ne(ds, dζ)|+tC∆^{(2)}_{t}_{0} ,
where

∆_{t}_{0} :=||x||e S^{2}+||ex||C+||m||e M0 +||m||e _{M}^{δ}

0

∆^{(2)}_{t}_{0} :=||x||e ^{2}_{S}2+||ex||^{2}_{C}+||m||e ^{2}_{M}_{0} +||m||e ^{2}_{M}δ
0

.

By the Burkholder-Davis-Gundy inequalities, E[sup

t≤t_{0}

|Rt

0X(s)e σ(s)dB(s)|]e ≤CE[(Rt0

0 Xe^{2}(s)eσ^{2}(s)ds)^{1}^{2}]≤Ct0||X||e t0∆t0, (3.6)
and

E[sup

t≤t0

|Rt

0X(s)e eγ(s)Ne(ds, dζ)|]≤CE[(Rt0

0 Xe^{2}(s)eγ^{2}(s)ν(dζ)ds)^{1}^{2}]≤Ct_{0}||X||e _{t}_{0}∆_{t}_{0}. (3.7)
Combining the above and using that

||X||e _{t}_{0}∆_{t}_{0} ≤C(||X||^{2}_{t}_{0} + ∆^{(2)}_{t}_{0} ),
we obtain

||X||e ^{2}_{t}_{0} :=E[sup

t≤t0

Xe^{2}(t)]≤Ct_{0}(||X||e ^{2}_{t}_{0} + ∆^{(2)}_{t}_{0} ).

By definition of the norms, we have

∆^{(2)}_{t}_{0} ≤C||x||e ^{2}_{t}_{0}. (3.8)
Thus we see that if t_{0} >0 is small enough we obtain

||X(t)||e ^{2}_{t}_{0} ≤ 1

2||ex(s)||^{2}_{t}_{0}, (3.9)

and hence Φ is a contraction onH^{t}0. Therefore the equation has a solution up tot_{0}. By the
same argument we see that the solution is unique. Now we repeat the argument above, but
starting at t_{0} instead of starting at 0. Then we get a unique solution up to 2t_{0}. 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 (??)

dX(t) = b(t, X(t), X_{t}, M(t), M_{t}, u(t), u_{t})dt+σ(t, X(t), X_{t}, M(t), M_{t}, u(t), u_{t})dB(t)
+R

R0γ(t, X(t), X_{t}, M(t), M_{t}, u(t), u_{t}, ζ)Ne(dt, dζ);t∈[0, T],
X(t) =ξ(t);t∈[−δ,0].

(4.1) {exmfsfde}

The coefficients b, σ and γ are supposed to satisfy the assumptions of Theorem ??, uni-
formly w.r.t. u ∈ U^{ad}, then we have the existence and the uniqueness of the solution
X(t)∈ S^{2} of the controlled mf-sfde (??).

Moreover, b, σ and γ have Fr´echet derivatives w.r.t. x,m, m and are continuously differ- entiable in the variables x and u.

The performance functional is assumed to be of the form J(u) = E[RT

0 `(t, X(t), X_{t}, M(t), M_{t}, u(t), u_{t})dt+h(X(T), M(T)]; u∈ U. (4.2) {perf}
With E := [0, T]×R× C × M_{0} × M^{δ}_{0}× U^{ad} × C ×Ω, E^{0} :=R× M_{0}×Ω 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^{0} →R,

admit Fr´echet 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 (??) to depend on the path process
X_{t} and also its law processL(X_{t}) =: M_{t}, 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 (sinceu is required to beG-adapted) but only in the limited sense, since G does not depend on the observation.

Problem 4.1 Find u^{∗} ∈ U^{ad} such that

J(u^{∗}) = sup

u∈U^{ad}

J(u). (4.3) {eq4.3}

To study this problem we first introduce its associated Hamiltonian, as follows:

Definition 4.2 The Hamiltonian

H : [0, T +δ]×R× C × M_{0}× M^{δ}_{0}× U^{ad}× C ×R×R× R ×K×Ω→R
associated to this memory mean-field stochastic control problem (??) 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)

+R

R0r^{0}(t, ζ)γ(t, ζ)ν(dζ) +hp^{1}, m^{0}i; t ∈[0, T],

(4.4) {haml}

and H(t, x, x, m, m, u, u, p^{0}, q^{0}, r^{0}(·), p^{1}) = 0;t > T.

The HamiltonianH is assumed to be continuously differentiable w.r.t. x, uand to admit Fr´echet derivatives w.r.t. x, m and m.

In the following we let L^{2}_{0} denote the set of measurable stochastic processes Y(t) on R
such that Y(t) = 0 for t <0 and fort > T and

RT

0 Y^{2}(t)dt <∞ a.s. (4.5)

The map

Y 7→RT

0 <∇_{x}H(t), Y_{t}> dt; Y ∈L^{2}_{0}

is a bounded linear functional on L^{2}_{0}. Therefore, by the Riesz representation theorem there
exists a unique process Γ_{x}_{¯}(t)∈L^{2}_{0} such that

RT

0 Γ_{x}_{¯}(t)Y(t)dt =RT

0 <∇_{x}H(t), Y_{t} > dt, (4.6) {eq4.6}

for all Y ∈ L^{2}_{0}. Here < ∇_{x}H(t), Y_{t} > denotes the action of the operator ∇_{x}H(t) to the
segment Y_{t}={Y(t+s)}s∈[−δ,0], where H(t) is a shorthand notation for

H(t, X(t), X_{t}, M(t), M_{t}, u(t), u_{t}, p^{0}(t), q^{0}(t), r^{0}(t,·), p^{1}(t), ω).

As a suggestive notation (see below) for Γ_{¯}_{x} we will in the following write

∇_{x}H^{t}:= Γ_{x}_{¯}(t). (4.7)

Lemma 4.3 Consider the case when

H(t, x, x, p, q) =f(t, x) +F(x)p+σq, Then

Γ_{x}_{¯}(t) :=<∇_{x}F, p^{t} > (4.8) {eq4.8}
satisfies (??), where p^{t}:={p(t+r)}r∈[0,δ]={p(t−s)}s∈[−δ,0].

Proof. We must verify that if we define Γ_{x}_{¯}(t) by (??), then (??) holds. To this end,
choose Y ∈L^{2}_{0} and consider

RT

0 Γ¯x(t)Y(t)dt=RT

0 <∇x¯F, p^{t}> Y(t)dt =RT

0 <∇x¯F,{p(t+r)}r∈[0,δ] > Y(t)dt

=RT

0 <∇_{x}_{¯}F,{Y(t)p(t+r)}r∈[0,δ]> dt

=<∇_{x}_{¯}F,{RT+r

r Y(u−r)p(u)du}_{r∈[0,δ]} >

=<∇_{x}_{¯}F,{RT

0 Y(u−r)p(u)du}r∈[0,δ]

=RT

0 <∇_{x}_{¯}F, Y_{u} > p(u)du

=RT

0 <∇x¯H(u), Yu > du.

Example 4.4 (i) For example, if

F(¯x) = R0

−δa(s)x(s)ds (4.9)

when x¯={x(s)}_{s∈[−δ,0]}, then

<∇_{¯}_{x}F, p^{t}>=< F, p^{t}>=R0

−δa(s)p(t−s)ds=Rδ

0a(−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^{0}, q^{1}) and
r= (r^{0}, r^{1}) by the following two adjoint equations:

• The advanced backward stochastic functional differential equation (absfde) in the un-
known (p^{0}, q^{0}, r^{0})∈ S^{2}×L^{2}×L^{2}ν is given by

dp^{0}(t) =−[^{∂H}_{∂x}(t) +E(∇_{x}H^{t}|F_{t})]dt+q^{0}(t)dB(t) +R

R0r^{0}(t, ζ)Ne(dt, dζ);t∈[0, T],
p^{0}(t) = ^{∂h}_{∂x}(X(T), M(T));t≥T,

q^{0}(t) = 0;t > T,
r^{0}(t,·) = 0;t > T.

(4.13) {p0}

• The operator-valued mean-field advanced backward stochastic functional differential
equation (ov-mf-absfde) in the unknown (p^{1}, q^{1}, r^{1})∈ S^{2}

K×L^{2}_{K}×L^{2}ν,K is given by

dp^{1}(t) =−[∇_{m}H(t) +E(∇_{m}H^{t}|F_{t})]dt+q^{1}(t)dB(t) +R

R0r^{1}(t, ζ)Ne(dt, dζ);t ∈[0, T],
p^{1}(t) =∇_{m}h(X(T), M(T));t ≥T,

q^{1}(t) = 0;t > T,
r^{1}(t,·) = 0;t > T.

(4.14) {p1}

where ∇m¯H^{t} is defined in the similar way as∇x¯H^{t} above, i.e. by the property that
RT

0 Γ_{m}_{¯}(t)M(t)dt=RT

0 <∇_{m}H(t), M_{t} > dt, (4.15) {eq4.6}
for all M ∈L^{2}_{0}.

Advanced backward stochastic differential equations (absde) have been studied by Peng and Yang [?] in the Brownian setting and for the jump case, we refer to Øksendal et al [?], Øksendal and Sulem [?]. It was also extended to the context of enlargement progressive of filtration by Jeanblanc et al in [?].

When Agram and Røse [?] used the maximum principle to study optimal control of mean- field delayed sfde (??), 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 ub ∈ U^{ad} with corresponding solu-
tions Xb ∈ S^{2}, (pb^{0},qb^{0},br^{0}) ∈ S^{2} ×L^{2} ×L^{2}ν and (pb^{1},bq^{1},br^{1}) ∈ S^{2}

K×L^{2}_{K} ×L^{2}ν,K of the forward

and backward stochastic differential equations (??), (??) and(??)respectively. For arbitrary u∈ U, put

H(t) :=H(t,X(t),b Xb_{t},Mc(t),Mc_{t}, u(t), u_{t},pb^{0}(t),bq^{0}(t),br^{0}(t,·),pb^{1}(t)), (4.16)
H(t) :=b H(t,X(t),b Xb_{t},Mc(t),Mc_{t},bu(t),bu_{t},pb^{0}(t),bq^{0}(t),br^{0}(t,·),pb^{1}(t)). (4.17)
Suppose that

• (Concavity) The functions

(x, x, m, m, u, u) 7→ H(t, x, x, m, m, u, u,pb^{0},qb^{0},br^{0}(·),pb^{1}),

(x, m) 7→ h(x, m),

are concave P-a.s. for each t∈[0, T].

• (Maximum condition)

E[H(t)|Gb _{t}] = sup

u∈U^{ad}

E[H(t)|G_{t}], (4.18) {maxQ}
P-a.s. for each t∈[0, T].

Then ubis an optimal control for the problem (??).

Proof. By considering a sequence of stopping times converging upwards toT, we see tghat we may assume that all the dB− and ˜N− integrals in the following are martingales and hence have expectation 0. We refer to the proof of Lemma 3.1 in [?] for details.

We want to prove that J(u) ≤J(u) for allb u∈ U^{ad}. Application of definition (??) gives
for fixed u∈ U^{ad} that

J(u)−J(u) =b I_{1}+I_{2}, (4.19) {J}
where

I_{1} = E[RT

0 {`(t)−b`(t)}dt],

I2 = E[h(X(T), M(T))−h(X(Tb ),Mc(T))], with

`(t) :=`(t,X(t),b Xb_{t},Mc(t),Mc_{t}, u(t), u_{t}), (4.20)

`(t) :=b `(t,X(t),b Xb_{t},Mc(t),Mc_{t},bu(t),bu_{t}). (4.21)
and similarly with b(t),bb(t) etc. later.

Applying the definition of the Hamiltonian (??), we get
I_{1} = E[RT

0 {H(t)−H(t)b −pb^{0}(t)eb(t)−bq^{0}(t)σ(t)e

−R

R0br^{0}(t, ζ)˜γ(t, ζ)ν(dζ)−<pb^{1}(t),Mf^{0}(t)>}dt], (4.22) {I1}

whereeb(t) = b(t)−bb(t) etc., and

Mf^{0}(t) = ^{df}^{M}_{dt}^{(t)} = _{dt}^{d}(M(t)−Mc(t)).

Using concavity of h and the definition of the terminal values of the absfde (??) and (??), we get

I_{2} ≤ E[^{∂}_{∂x}^{b}^{h}(T)X(Te ) +∇_{m}bh(T)Mf(T)]

= E[pb^{0}(T)X(Te )+<pb^{1}(T),Mf(T)>]. (4.23) {I2}
Applying the Itˆo formula to pb^{0}Xe and pb^{1}Mf, we have

E[pb^{0}(T)X(Te )] = E[RT

0 pb^{0}(t)dX(t) +e RT

0 X(t)de pb^{0}(t) +RT

0 qb^{0}(t)eσ(t)dt
+RT

0

R

R0br^{0}(t, ζ)eγ(t, ζ)ν(dζ)dt]

= E[RT

0 pb^{0}(t)eb(t)dt−RT
0

∂Hb

∂x(t)X(t)dte −RT

0 E(∇xHc^{t}|Ft)X(t)dte
+RT

0 qb^{0}(t)σ(t)dte +RT
0

R

R0br^{0}(t, ζ)γe(t, ζ)ν(dζ)dt],

(4.24) {.10}

and

E[<pb^{1}(T),Mf(t)>]

=E[RT

0 <pb^{1}(t), dfM(t)> dt+RT

0 <M(t), df pb^{1}(t)> dt] (4.25)

=E[RT

0 <pb^{1}(t),Mf^{0}(t)> dt−RT

0 <∇_{m}H(t),b Mf(t)> dt −RT

0 E(∇_{m}Hc^{t}|F_{t})Mf(t)dt],
(4.26)

where we have used that thedB(t) andNe(dt, dζ) integrals have mean zero. On substituting _{{}_{.11}_{}}
(??),(??) and (??) into (??), we obtain

J(u)−J(bu) ≤E[RT

0 {H(t)−H(t)b −RT 0

∂Hb

∂x(t)X(t)dte −RT

0 ∇_{x}Hc^{t}X(t)}dte

−RT

0 <∇_{m}H(t),b Mf(t)> dt−RT

0 ∇_{m}Hc^{t}Mf(t)dt].

SinceX(t) = 0 for alle t∈[−δ,0] and for allt > T we see that Xe ∈L^{2}_{0} and therefore by (??),
we have

RT

0 ∇_{x}_{¯}Hb^{t}X(t)dte =RT

0 <∇_{x}_{¯}H(t),b Xe_{t} > dt. (4.27) {estpath}

Similar considerations give RT

0 ∇mHb^{t}Mf(t)dt =RT

0 <∇mH(t),b Mft> dt. (4.28) {stpath2}
By the assumption thatH is concave and that the processu isG_{t}-adapted, we therefore get

J(u)−J(u)b ≤E[RT

0 {^{∂}_{∂u}^{H}^{b}(t)eu(t)+ <∇_{u}H(t),b eu_{t}>}dt]

=E[RT

0 E(^{∂}_{∂u}^{H}^{b}(t)u(t)+e <∇uH(t),b eut>|Gt)dt]

=E[RT

0 {E[^{∂}_{∂u}^{H}^{b}(t)|G_{t}]u(t)+e <E[∇_{u}H(t)|Gb _{t}],ue_{t}>}dt]≤0.

For the last inequality to hold, we use that E[H(t)|Gb _{t}] has a maximum at bu(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.

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:

• Wheneveru∈ 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 variablesα, the process
π(t) = α1_{(t}_{0}_{,T}_{]}(t),

belongs toU^{ad}.

In general, if K^{u}(t) is a process depending on u, we define the operator D on K by
DK^{u}(t) :=D^{π}K^{u}(t) = d

dλK^{u+λπ}(t)|_{λ=0}, (4.29)
whenever the derivative exists.

Define the derivative process Z(t) by

Z(t) := DX(t) := _{dλ}^{d}X^{u+λπ}|λ=0.
Using matrix notation, note that Z(t) satisfies the equation

dZ(t) = (∇b(t))^{T} (Z(t), Z_{t}, DM(t), DM_{t}, π(t), π_{t})dt
+(∇σ(t))^{T} (Z(t), Z_{t}, DM(t), DM_{t}, π(t), π_{t})B(t)
+R

R0(∇γ(t, ζ))^{T} (Z(t), Z_{t}, DM(t), DM_{t}, π(t), π_{t}, ζ)Ne(dt, dζ); t ∈[0, T],
Z(t) = 0; t∈[−δ,0],

(4.30) {dervz}

where (∇b)^{T} = (^{∂b}_{∂x},∇_{x}b,∇_{m}b,∇_{m}b,_{∂u}^{∂b},∇_{u}b)^{T}, (·)^{T} denotes matrix transposed and we mean
by ∇_{x}b(t)Z_{t}, (respectively ∇_{m}b(t)DM_{t}) the action of the operator ∇_{x}b(t) (∇_{m}b(t)) on
the segment Z_{t} = {Z(t + s)}_{s∈[−δ,0]} (DM_{t} = {DM(t +s)}_{s∈[−δ,0]}) i.e., < ∇_{x}b(t), Z_{t} >

(<∇_{m}b(t), DM_{t}>) and similar considerations forσ and γ.

Theorem 4.6 (Necessary maximum principle) Let bu ∈ U^{ad} with corresponding solu-
tions Xb ∈ S^{2} and (pb^{0},qb^{0},br^{0})∈ S^{2}×L^{2}×L^{2}ν and (pb^{1},bq^{1},br^{1})∈ S_{K}^{2} ×L^{2}_{K}×L^{2}ν,K of the forward
and backward stochastic differential equations (??) and (??)− (??) respectively, with the
corresponding derivative process Zb∈ S^{2} given by (??). Then the following, (i) and (ii), are
equivalent:

(i) For all bounded π∈ U^{ad}

d

dλJ(ˆu+λπ)|_{λ=0} = 0.

(ii)

E[(^{∂H}_{∂u}(t) +∇_{u}H_{t})|G_{t}]_{u=ˆ}_{u} = 0 for all t∈[0, T).

Proof. Before starting the proof, let us first clarify some notation: Note that

∇_{m} < p^{1}_{1}(t),_{dt}^{d}m > =< p^{1}_{1}(t),_{dt}^{d}(·)>,
and hence

<∇_{m} < p^{1}_{1}(t),_{dt}^{d}m >, DM(t)> =< p^{1}_{1}(t),_{dt}^{d}DM(t)> =< p^{1}_{1}(t), DM^{0}(t)> =p^{1}_{1}(t)DM^{0}(t).

Also, note that

dDM(t) =DM^{0}(t)dt. (4.31)

Assume that (i) holds. Then

0 = _{dλ}^{d}J(u+λπ)|_{λ=0}

=E[RT

0 {(∇`(t))^{T} (Z(t), Z_{t}, DM(t), DM_{t}, π(t), π_{t})}dt
+^{∂h}_{∂x}(T)Z(T) +∇_{m}h(T)DM(T)].

Hence, by the definition of H (??) and the terminal values of the absfde p^{0}(T) and p^{1}(T),
we have

0 = _{dλ}^{d}J(u+λπ)|_{λ=0}

=E[RT

0 {(∇H(t))^{T} (Z(t), Z_{t}, DM(t), DM_{t}, π(t), π_{t})

−p^{0}(t)(∇b(t))^{T} (Z(t), Z_{t}, DM(t), DM_{t}, π(t), π_{t})

−q^{0}(t)(∇σ(t))^{T} (Z(t), Z_{t}, DM(t), DM_{t}, π(t), π_{t})

−R

R0r^{0}(t, ζ)(∇γ(t, ζ))^{T} (Z(t), Z_{t}, DM(t), DM_{t}, π(t), π_{t})ν(dζ)}dt]

−RT

0 p^{1}(t)DM^{0}(t)dt+p^{0}(T)Z(T) +p^{1}(T)DM(T)].

Applying Itˆo formula to both p^{0}Z and p^{1}DM, we get
E[p^{0}(T)Z(T)] =E[RT

0 p^{0}(t)dZ(t) +RT

0 Z(t)dp^{0}(t) + [p^{0}, Z]_{T}]

=E[RT

0 p^{0}(t)(∇b(t))^{T} (Z(t), Zt, DM(t), DMt, π(t), πt)dt

−RT

0 {^{∂H}_{∂x}(t) +∇_{x}H^{t}}Z(t)dt
+RT

0 q^{0}(t)(∇σ(t))^{T} (Z(t), Z_{t}, DM(t), DM_{t}, π(t), π_{t})dt
+RT

0

R

R0r^{0}(t, ζ)(∇γ(t, ζ))^{T} (Z(t), Z_{t}, DM(t), DM_{t}, π(t), π_{t})ν(dζ)dt],

and

E[p^{1}(T)DM(T)] =E[RT

0 p^{1}(t)DM^{0}(t)dt+RT

0 DM(t)dp^{1}(t)]

=E[RT

0 p^{1}_{1}(t)DM^{0}(t)dt−RT

0 {∇_{m}H(t) +∇_{m}H^{t}}DM(t)dt].

Proceeding as in (??)−(??), we obtain RT

0 ∇_{x}H^{t}Z(t)dt =RT

0 <∇_{x}H(t), Z_{t}> dt,
RT

0 ∇mH^{t}DM(t)dt =RT

0 <∇mH(t), DMt> dt.

Combining the above, we get

0 =E[RT

0 (^{∂H}_{∂u}(t)π(t) +h∇_{u}H(t), π_{t}i)dt]. (4.32) {h_pi}

Now choose π(t) = α1_{(t}_{0}_{,T}_{]}(t), where α = α(ω) is bounded and G_{t}_{0}-measurable and t_{0} ∈
[0, T). Then π_{t}=α{1_{(t}_{0}_{,T]}(t+s)}s∈[−δ,0] and (??) gives

0 =E[RT t0

∂H

∂u(t)αdt+RT

t0 h∇_{u}H(t), αi {1_{(t}_{0}_{,T}_{]}(t+s)}s∈[−δ,0]dt].

Differentiating with respect tot_{0}, we obtain

E[(^{∂H}_{∂u}(t0) +∇uHt0)α] = 0,
Since this holds for all such α, we conclude that

E[(^{∂H}_{∂u}(t_{0}) +∇_{u}H_{t}_{0})|G_{t}_{0}] = 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.

### 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 [?], Anderson and Djehiche [?]

and Røse [?].

Consider the state equation X^{π}(t) =X(t) on the form
dX(t) =X(t−δ)π(t)[b_{0}(t)dt+σ_{0}(t)dB(t) +R

R0γ_{0}(t, ζ)Ne(dt, dζ)];t∈[0, T],

X(t) =ξ(t);t∈[−δ,0], (5.1) {w}

for some bounded deterministic function ξ(t);t ∈ [−δ,0]. We assume that the admissible
processes are c`adl`ag processes inL^{2}(Ω,[0, T]), that are adapted to the filtrationF_{t} and such

that a unique solution exists. The coefficientsb_{0}, σ_{0} andγ_{0} >−1 are supposed to be bounded
F-adapted processes with

|b0(t)|>0 and σ_{0}^{2}(t) +R

R0γ_{0}^{2}(t, ζ)ν(dζ)>0 a.s. for all t.

We want to find an admissible portfolioπ(t) which maximizes

J(π) = E[− ^{1}_{2}(X(T)−a)^{2}], (5.2) {p}

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}(·)) = πG(x)(b_{0}p^{0}+σ_{0}q^{0}+R

R0γ_{0}(ζ)r^{0}(ζ)ν(dζ)), (5.3) {h}

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}×L^{2}×L^{2}ν is the adjoint
process which satisfies

dp^{0}(t) = −E[π(t+δ)(b_{0}(t+δ)p^{0}(t+δ) +σ_{0}(t+δ)q^{0}(t+δ)
+R

R0γ0(t+δ, ζ)r^{0}(t+δ, ζ)ν(dζ))|Ft]dt+q^{0}(t)dB(t)
+R

R0r^{0}(t, ζ)Ne(dt, dζ);t∈[0, T],
p^{0}(t) =−(X(T)−a);t≥T,

q^{0}(t) =r^{0}(·) = 0;t > T.

(5.5) {ap}

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

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

0 = ^{∂}_{∂π}^{H}^{b}(t,Xc_{t},π(t),b pb^{0}(t),qb^{0}(t),br^{0}(t,·)) (5.6) {nc}

=X(tb −δ)(b_{0}(t)pb^{0}(t) +σ_{0}(t)qb^{0}(t) +R

R0γ_{0}(t, ζ)br^{0}(t, ζ)ν(dζ)).

So we search for a candidate bπ satisfying
0 =b_{0}(t)pb^{0}(t) +σ_{0}(t)qb^{0}(t) +R

R0γ_{0}(t, ζ)rb^{0}(t, ζ)ν(dζ), for all t. (5.7) {pi}

This gives the following adjoint equation:

dpb^{0}(t) = qb^{0}(t)dB(t) +R

R0rb^{0}(t, ζ)Ne(dt, dζ);t∈[0, T],
pb^{0}(t) =−(X(T)−a);t≥T,

qb^{0}(t) =br^{0}(·) = 0;t > T.

(5.8) {ap1}

We start by guessing that pb^{0} has the form

pb^{0}(t) = ϕ(t)X(t) +b ψ(t) (5.9) {p^}

for some deterministic functions ϕ, ψ∈C^{1}[0, T] with

ϕ(T) = −1, ψ(T) = a. (5.10) {v}

Using the Itˆo formula to find the integral representation ofpb^{0} and comparing with the adjoint
equation (??), we find that the following three equations need to be satisfied:

0 = ϕ^{0}(t)X(t) +b ψ^{0}(t) +ϕ(t)X(tb −δ)bπ(t)b_{0}(t), (5.11) {d}

qb^{0}(t) = ϕ(t)X(tb −δ)π(t)σb _{0}(t), (5.12) {di}

br^{0}(t, ζ) = ϕ(t)X(tb −δ)bπ(t)γ0(t, ζ). (5.13) {j}

Assuming thatX(t)b 6= 0 P×dt-a.e. andϕ(t)6= 0 for each t, we find from equation (??) that bπ needs to satisfy

π(t) =b −^{ϕ}^{0}^{(t)}^{X(t)+ψ}^{b} ^{0}^{(t)}

ϕ(t)X(t−δ)bb _{0}(t).

Now inserting the expressions for the adjoint processes (??), (??) and (??) into (??), the following equation need to be satisfied:

0 =b0(t)[ϕ(t)X(t) +b ψ(t)] +ϕ(t)X(tb −δ)π(t)b σ^{2}_{0}(t) +R

R0γ_{0}^{2}(t, ζ)ν(dζ)
.

This means that the control bπ also needs to satisfy

bπ(t) = − ^{b}^{0}^{(t)[ϕ(t)}^{X(t)+ψ(t)]}^{b}

[σ^{2}_{0}(t)+R

R0γ_{0}^{2}(t,ζ)ν(dζ)]ϕ(t)X(t−δ)b . (5.14) {pih}

By comparing the two expressions for bπ, we find that
b^{2}_{0}(t)[ϕ(t)X(t) +b ψ(t)]

= (σ_{0}^{2}(t) +R

R0γ^{2}_{0}(t, ζ)ν(dζ))[ϕ^{0}(t)X(t) +b ψ^{0}(t)]. (5.15) {co}

Now define

Λ(t) := _{σ}2 ^{b}^{2}^{0}^{(t)}
0(t)+R

R0γ_{0}^{2}(t,ζ)ν(dζ). (5.16) {lam}

Then from equation (??), we need to have

ϕ^{0}(t)−Λ(t)ϕ(t) = 0,
ψ^{0}(t)−Λ(t)ψ(t) = 0.

Together with the terminal values (??), these equations have the solution ϕ(t) = −exp(−RT

t Λ(s)ds), ψ(t) = aexp(−RT

t Λ(s)ds).

Then from equation (??) we can compute

bπ(t) =

b0(t)

Xb(t)−ψ(t) ϕ(t)

σ_{0}^{2}(t)+R

R0γ_{0}^{2}(t,ζ)ν(dζ))X(t−δ)b = Λ(t)

X(t)−b ψ(t) ϕ(t)

b0(t)X(t−δ)b = ^{Λ(t)}

b0(t)X(t−δ)b (X(t)b −a).

Now, with our choice of π, the corresponding state equation is the solution ofb
( dX(t) =b _{b}^{Λ(t)}

0(t)(X(t)b −a)[b_{0}(t)dt+σ_{0}(t)dB(t) +R

R0γ_{0}(t, ζ)Ne(dt, dζ)];t∈[0, T],
X(t)b =x_{0}(t);t∈[−δ,0].

(5.17) {wp}

PutY(t) =X(t)b −a, then

dY(t) = Y(t)[Λ(t)b_{0}(t)dt+_{b}^{Λ(t)}

0(t)σ_{0}(t)dB(t) +R

R0

Λ(t)

b0(t)γ_{0}(t, ζ)Ne(dt, dζ)]. (5.18) {lw}

The linear equation (??) has the following explicit solution Y(t) = Y(0) exp[Rt

0 Λ(s)b0(s)ds+Rt 0

Λ(s)

b0(s)σ0(s)dB(s) +Rt 0

R

R0

Λ(s)

b0(s)γ0(s, ζ)Ne(ds, dζ)].

So ifY(0)>0 thenY(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(tb −δ) > 0 for all t ≥ 0 and the solution bπ ∈ U^{ad} of the mean-variance
portfolio problem (??) is given in feedback form as

π(t) =b ^{Λ(t)}

b0(t)Xb(t−δ)(X(t)b −a),

where X(t)b and Λ(t) are given by equations (??) and (??) respectively.

### 5.2 A linear-quadratic (LQ) problem with memory

We now consider a linear-quadratic control problem for a controlled system X(t) = X^{u}(t)
driven by a distributed delay, of the form

dX(t) = [R0

−δa(s)X(t+s)ds+u(t)]dt+α_{0}dB(t) +R

R0β_{0}(ζ)Ne(dt, dζ);t∈[0, T],
X(t) =ξ(t);t ∈[−δ,0],

(5.19) {f}

where ξ(·) and a(·) are given bounded deterministic functions, α_{0} is a given constant, β_{0} is
a given function from R0 intoR with

R

R0β0(ζ)ν(dζ)<∞,

and u∈ U^{ad} is our control process. We want to minimize the expected value of X^{2}(T) with
a minimal average use of energy, measured by the integralE[RT

0 u^{2}(t)dt], i.e. the performance
functional is of the quadratic type

J(u) =−^{1}_{2}E[X^{2}(T) +RT

0 u^{2}(t)dt].