Singular control and optimal stopping of memory mean-field processes

arXiv:1802.05527v1 [math.OC] 15 Feb 2018

Singular control and optimal stopping of memory mean-field processes Nacira Agram1,2, Achref Bachouch1, Bernt Øksendal1 and Frank Proske1 15 February 2018 Abstract By a memory mean-field stochastic differential equation (MMSDE) we mean a stochastic differential equation (SDE) where the coefficients depend on not just the current state X(t) at time t, but also on previous values (history/memory) Xt := {X(t − s)}s∈[0,δ] , as well as the law M (t) := L(X(t)) and its history/memory Mt = {M (t − s)}s∈[0,δ] . Similarly, we call the equation (time-)advanced and abbreviate it to AMSDE if we in the above replace Xt by the forward segment X t := {X(t + s)}s∈[0,δ] and replace Mt by M t := {M (t + s)}s∈[0,δ] . In the same way we define the reflected advanced mean-field backward SDE (AMBSDE). We study the following topics related MMSDEs/AMBSDEs: • We prove the existence and uniqueness of the solutions of some reflected AMBSDEs, • we give sufficient and necessary conditions for an optimal singular control of an MMSDE with partial information, and • we deduce a relation between the optimal singular control of a MMSDE, and the optimal stopping of such processes.

MSC(2010): 60H10, 60HXX, 93E20, 93EXX, 46E27, 60BXX. Keywords: Memory mean-field stochastic differential equation; reflected advanced meanfield backward stochastic differential equation; singular control; optimal stopping. 1

Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N–0316 Oslo, Norway. Emails: [email protected], [email protected], [email protected], [email protected]. 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. 2 University Mohamed Khider, Biskra, Algeria.

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1

Introduction

Let (Ω, F , P) be a given probability space with filtration F = (Ft )t≥0 generated by a 1dimensional Brownian motion B = B(t, ω); (t, ω) ∈ [0, T ] × Ω. Let G = {Gt }t≥0 be a given subfiltration of F = (Ft )t≥0 , in the sense that Gt ⊂ Ft for all t. The purpose of this paper is to study the following concepts and problems, and the relation between them. For simplicity of notation we deal only with the 1-dimensional case. • Topic 1: Optimal singular control of memory mean-field stochastic differential equations: Consider the following mean-field memory singular controlled system, with a state process X(t) = X ξ (t) and a singular control process ξ(t), of the form   dX(t) = b(t, X(t), Xt , M(t), Mt , ξ(t), ω)dt + σ(t, X(t), Xt , M(t), Mt , ξ(t), ω)dB(t) +λ(t, ω)dξ(t), t ∈ [0, T ]  X(t) = α(t); t ∈ [−δ, 0]. (1.1) where Xt = {X(t − s)}0≤s≤δ , (the memory segment of X(t)) M(t) = L(X(t)) (the law of X(t)), Mt = {M(t − s)}0≤s≤δ , (the memory segment of M(t)). We assume that our control process ξ(t) is R-valued right-continuous G-adapted process, and t 7→ ξ(t) is increasing (non-decreasing) with ξ(0− ) = 0, and such that the corresponding state equation has a unique solution X with ω 7→ X(t, ω) ∈ L2 (P) for all t. The set of such processes ξ is denoted by Ξ. The performance functional is assumed to be of the form RT J(ξ) = E[ 0 f (t, X(t), Xt , M(t), Mt , ξ(t), ω)dt + g(X(T ), M(T ), ω) RT + 0 h(t, X(t), ω)dξ(t)]; ξ ∈ Ξ .

For simplicity we will in the following suppress the ω in the notation. We may interpret these terms as follows: The state X(t) may be regarded as the value at time t of, e.g. a fish population. The control process ξ(t) models the amount harvested up to time t, the coefficient λ(t) is the unit price of the amount harvested, f is a profit rate, g is a bequest or salvage value function, and h is a cost rate for the use of the singular control ξ. The σ-algebra Gt represents the amount of information available to the controller at time t. The problem we consider, is the following: 2

Problem 1.1 Find an optimal control ξˆ ∈ Ξ such that ˆ = sup J(ξ) . J(ξ)

(1.2)

ξ∈Ξ

This problem turns out to be closely related to the following topic: • Topic 2: Reflected mean-field backward stochastic differential equations We study reflected mean-field advanced BSDEs where at any time t the driver F may depend on future information of the solution processes. More precisely, for a given driver F , a given threshold process S(t) and a given terminal value R we consider the following type of reflected BSDEs in the unknown processes Y, Z, K: RT (i)Y (t) = R + t F (s, Y (s), Z(s), E[Y s |Fs ], E[Z s |Fs ], L(Y s , Z s ))ds RT +K(T ) − K(t) − t Z(s)dB(s), 0 ≤ t ≤ T, (ii)Y (t) ≥ S(t), 0 ≤ t ≤ T, RT (iii) 0 (Y (t) − S(t))dK c (t) = 0 a.s. and △K d (t) = −△Y (t)1{Y (t− )=S(t− )} a.s., (iv)Y (t) = R, t ≥ T, (v)Z(t) = 0, t > T. Here L(Y s , Z s ) is the joint law of paths (Y s , Z s ), and for a given positive constant δ we have put Y t := {Y (t + s)}s∈[0,δ] and Z t := {Z(t + s)}s∈[0,δ] (the (time)-advanced segment). This problem is connected to the following: • Topic 3: Optimal stopping and its relation to the problems above. For t ∈ [0, T ] let T[t,T ] denote the set of all F-stopping times τ with values in [t, T ]. Suppose (Y, Z, K) is a solution of the reflected AMBSDE in Topic 2 above. (i) Then for t ∈ [0, T ] the process Y (t) is the solution of the optimal stopping problem n R τ Y (t) = ess sup E[ t F (s, Y (s), Z(s), E[Y s |Fs ], E[Z s |Fs ], L(Y s , Z s ))ds τ ∈T[t,T ]

o + S(τ )1τ K(t)} ∧ T. (iii) In particular, if we choose t = 0 we get that τˆ0 : = inf{s ∈ [0, T ], Y (s) ≤ S(s)} ∧ T = inf{s ∈ [0, T ], K(s) > 0} ∧ T solves the optimal stopping problem Rτ Y (0) = sup E[ 0 F (s, Y (s), Z(s), E[Y s |Fs ], E[Z s |Fs ], L(Y s , Z s ))ds τ ∈T[0,T ]

+ S(τ )1τ T, 7

where Y s = (Y (s + r))r∈[0,δ] , Z s = (Z(s + r))r∈[0,δ] , the terminal condition R ∈ L2 (Ω, FT ), the driver F : [0, T ] × Ω × R2 ×L2 × L2 × M0,δ −→ R is Ft -progressively measurable and we have denoted by K c and K d the continuous and discontinuous parts of K respectively. We may remark here that in order to guarantee adaptedness, the time-advanced terms are given under conditional expectation with respect to Fs . Our result can be regarded as an extension of the existing results on advanced BSDEs of Peng & Yang [18], Øksendal et al [16], Jeanblanc et al [12] and we refer here to the paper by Quenez and Sulem [19] on reflected BSDE for c`adl`ag obstacle. To obtain the existence and the uniqueness of a solution, we make the following set of assumptions: • For the driver F, we assume (i) There exists a constant c ∈ R such that |F (·, 0, 0, 0, 0, L(0, 0))| ≤ c, where L(0, 0) is the Dirac measure with mass at zero. F (ii) There exists a constant CLip ∈ R such that, for t ∈ [0, T ],

|F (t, y1, z1 , y2, z2 , L(y2, z2 )) − F (t, y1′ , z1′ , y2′ , z2′ , L(y2′ , z2′ ))|2 F ≤ CLip {|y1 − y1′ |2 + |z1 − z1′ |2 + ||y2 − y2′ ||2L2 + ||z2 − z2′ ||2L2

+ ||L(y2, z2 ) − L(y2′ , z2′ )||2M0,δ )}, for all y1 , z1 , y1′ , z1′ ∈ R, y2 , z2 , y2′ , z2′ ∈ L2 , L(y2 , z2 ), L(y2′ , z2′ ) ∈ M0,δ . • For the barrier S, we assume: (iii) The barrier S is nondecreasing, F-adapted, c`adl`ag process satisfying E[ sup |S(t)|2 ] < ∞. t∈[0,T ]

(iv) Y (t) ≥ S(t), 0 ≤ t ≤ T . • For the local time K, we assume: (v) K is a nondecreasing F-adapted c`adl`ag process with K(0− ) = 0, such that S(t))dK c (t) = 0 a.s. and △K d (t) = −△Y (t)1{Y (t− )=S(t− )} a.s.

RT 0

(Y (t) −

Theorem 3.1 (Existence and Uniqueness) Under the above assumptions (i)-(v), the reflected AMBSDE (3.1) has a unique solution (Y, Z, K) ∈ S 2 × L2 × Ξ.

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Proof. For t ∈ [0, T ] and for all β > 0, we define the Hilbert space H2β to be the set of all (Y, Z) ∈ S 2 × L2 , equipped with the norm R T +δ ||(Y, Z)||2H2 := E[ 0 eβt (Y 2 (t) + Z 2 (t))dt] . β

Define the mapping Φ : H2β →H2β by Φ(y, z) = (Y, Z) where (Y, Z) ∈S 2 ×L2 (⊂ L2 × L2 ) is defined by  RT Y (t) = R + F (s, y(s), z(s), E[y s |Fs ], E[z s |Fs ], L(y s, z s ))ds  t  RT  +K(T ) − K(t) − t Z(s)dB(s), 0 ≤ t ≤ T,  Y (t) = R, t ≥ T,   Z(t) = 0, t > T,

To prove the theorem, it suffices to prove that Φ is a contraction mapping in H2β under the norm || · ||H2β for large enough β. For two arbitrary elements (y1 , z1 , k1 ) and (y2 , z2 , k2 ), we denote their difference by (e y , ze, e k) = (y1 − y2 , z1 − z2 , k1 −, k2 ) .

Applying Itˆo formula for semimartingale, we get RT E[ 0 eβt (β Ye 2 (t) + Ze2 (t))dt] RT = 2E[ 0 eβt Ye (t){F (t, y1(t), z1 (t), E[y1t |Ft ], E[z1t |Ft ], L(y1t , z1t )) − F (t, y2 (t), z2 (t), E[y2t |Ft ], E[z2t |Ft ], L(y2t , z2t ))}dt] RT RT + 2E[ eβt Ye (t)dK 1 (t)] − 2E[ eβt Ye (t)dK 2 (t)]. 0

0

We have that

Ye (t)dK 1,c (t) = (Y 1 (t) − S(t))dK 1,c (t) − (Y 2 (t) − S(t))dK 1,c (t) = −(Y 2 (t) − S(t))dK 1,c (t) ≤ 0 a.s., and by symmetry, we have also Ye (t)dK 2,c (t) ≥ 0 a.s. For the discontinuous case, we have as well Ye (t)dK 1,d (t) = (Y 1 (t) − S(t))dK 1,d (t) − (Y 2 (t) − S(t))dK 1,d (t) = −(Y 2 (t) − S(t))dK 1,d (t) ≤ 0 a.s.,

and by symmetry, we have also Ye (t)dK 2,d (t) ≥ 0 a.s. By Lipschitz assumption, standard estimates, it follows that RT E[ 0 eβt (β Ye 2 (t) + Ze2 (t))dt]

RT ≤ 8ρC 2 E[ 0 eβt Ye 2 (t)dt] RT Rδ 2 1 E[ 0 eβt (e y 2(t) + ze2 (t) + 0 (e y (t + r) + ze2 (t + r))dr)dt] . + 2ρ 9

By the change of variable s = t + r, gives RT Rδ 2 E[ 0 eβt 0 (e y (t + r) + ze2 (t + r))dr)dt] R T R t+δ 2 ≤ E[ 0 eβt t (e y (s) + ze2 (s))ds)dt]

Fubini’s theorem

RT

Rδ 2 eβt 0 (e y (t + r) + ze2 (t + r))dr)dt] R T +δ R s ≤ E[ 0 ( s−δ eβt dt)(e y 2(s) + ze2 (s)))ds] R T +δ ≤ E[ 0 eβs (e y 2(s) + ze2 (s)))ds].

E[

0

Consequently, by choosing β = 1 + 8ρC 2 , we have R T +δ RT y 2(t) + ze2 (t))dt] . E[ 0 eβt (Ye 2 (t) + Ze2 (t))dt] ≤ ρ1 E[ 0 eβt (e

e = 0 for t > T , we get Since Ye (t) = Z(t)

e 22 ≤ ||(Ye , Z)|| H β

1 ρ

||(e y, ze)||2H2 . β

For ρ> 1, we get that Φ is a contraction on H2β .

4



Reflected AMBSDEs and optimal stopping under partial information

In this section we recall a connection between reflected AMBSDEs and optimal stopping problems under partial information. Definition 4.1 Let F : Ω × [0, T ] × R2 × L2 × L2 × M0,δ → R be a given function. Assume that: • F is G-adapted and |F (t, 0, 0, 0, 0, L(0, 0))| < c. • S(t) is a given F-adapted c`adl`ag nondecreasing process such that E[ sup (S(t))2 ] < ∞. t∈[0,T ]

• The terminal value R ∈ L2 (Ω, FT ) is such that R ≥ S(T ) a.s. We say that a G-adapted triplet (Y, Z, K) is a solution of the reflected AMBSDE with driver F , terminal value R and the reflecting barrier S(t) under the filtration G, if the following hold: 1. E[

RT 0

|F (s, Y (s), Z(s), E[Y s |Fs ], E[Z s |Fs ], L(Y s , Z s ))|2 ds] < ∞, 10

2. Z(t) is a G − martingale, 3. Y (t) = R +

RT t

or, equivalently, Y (t) = E[R +

F (s, Y (s), Z(s), E[Y s |Fs ], E[Z s |Fs ], L(Y s , Z s ))ds RT RT − t dK(s) − t dZ(s), t ∈ [0, T ] ,

RT t

F (s, Y (s), Z(s), E[Y s |Fs ], E[Z s |Fs ], L(Y s , Z s ))ds RT − t dK(s)|Gt ], t ∈ [0, T ] ,

4. K(t) is nondecreasing, G-adapted, c`adl`ag and K(0− ) = 0, 5. Y (t) ≥ S(t) a.s., t ∈ [0, T ], RT 6. 0 (Y (t) − S(t))dK(t) = 0 a.s.

The following result is essentially due to El Karoui et al [11]. See also Øksendal & Sulem [15] and Øksendal & Zhang [17]. Theorem 4.2 For t ∈ [0, T ] let T[t,T ] denote the set of all G-stopping times τ : Ω 7→ [t, T ]. Suppose (Y, Z, K) is a solution of the reflected AMBSDE above. (i) Then Y (t) is the solution of the optimal stopping problem Rτ Y (t) = ess sup {E[ t F (s, Y (s), Z(s), Y s , Z s , L(Y s , Z s ))ds τ ∈T[t,T ]

+S(τ )1τ K(t)} ∧ T.

11

(iii) In particular, if we choose t = 0 we get that τˆ0 : = inf{s ∈ [0, T ], Y (s) ≤ S(s)} ∧ T = inf{s ∈ [0, T ], K(s) > 0} ∧ T, solves the optimal stopping problem Rτ Y (0) = sup E[ 0 F (s, Y (s), Z(s), E[Y s |Fs ], E[Z s |Fs ], L(Y s , Z s ))ds τ ∈T[0,T ]

+ S(τ )1τ T and RT E[ 0 Y 2 (t)dt] < ∞ a.s. • Let G(t, x ¯) = Gx¯ (t, ·) : [0, T ] × L2 7→ R be a bounded linear functional on L2 for each t, uniformly bounded in t.Then the map RT Y 7→ E[ 0 < Gx (t), Yt > dt]; Y ∈ L20

is a bounded linear functional on L20 . Therefore, by the Riesz representation theorem there exists a unique process denoted by G∗x¯ (t) ∈ L20 such that RT RT E[ 0 < Gx (t), Yt > dt] = E[ 0 G∗x¯ (t)Y (t)dt] (5.4)

for all Y ∈ L20 .

• Similarly we see that if G : M0 7→ R is a bounded linear functional on M0 , then by Lemma 2.2 we have | < G, M(T ) > |2 ≤ ||G||2||M(T )||2M0 ≤ CE[X 2 (T )]. Hence the map X 7→< G, L(X) >;

X ∈ L2 (P)

is a bounded linear functional on L2 (P). Therefore by the Riesz representation theorem there exists a random variable G∗ ∈ L2 (P) such that < G, M(T ) >= E[G∗ X(T )]. • In the same way we see that if Gm (t, m) : [0, T ] × M0 7→ R is a bounded linear functional on M0 for each t, uniformly bounded in t, then by Lemma 2.2 we see that the map RT X(·) 7→ 0 < Gm (t), M(t) > dt; M(t) = L(X(t)) is a bounded linear functional on L20 , as follows: RT RT | 0 < Gm (t), M(t) > dt|2 ≤ T 0 ||Gm (t)||2 ||M(t)||2 dt RT ≤ const. sup ||Gm(t)||2 E[ 0 X 2 (t)dt] ≤ const.||X||2L2 . 0

t∈[0,T ]

Therefore, by the Riesz representation theorem there exists a unique process denoted by G∗m (t) ∈ L20 such that RT RT < Gm (t), M(t) > dt = E[ 0 G∗m (t)X(t)dt], (5.5) 0 for all X ∈ L20 .

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• Proceeding as above, we also see that if Gm¯ (t, ·) : [0, T ] × M0 7→ R is a bounded linear functional on M0 for each t, uniformly bounded in t, then the map RT X(·) 7→ 0 < Gm¯ (t), Mt > dt; Mt = L(X)t

is a bounded linear functional on L20 . Therefore, there exists a unique process denoted ∗ 2 by Gm ¯ (t) ∈ L0 such that RT RT E[ 0 < Gm¯ (t), Mt > dt] = E[ 0 G∗m¯ (t)X(t)dt] (5.6) for all X ∈ L20 .

We illustrate these operators by some auxiliary results. Lemma 5.2 Consider the case when Gx¯ (t, ·) =< F, · > p(t). Then Gx∗¯ (t) :=< F, pt >

(5.7)

satisfies (5.4), where pt := {p(t + r)}r∈[0,δ] . Proof. We must verify that if we define G∗x¯ (t) by (5.7), then (5.4) holds. To this end, choose Y ∈ L20 and consider RT

RT t < F, p > Y (t)dt = < F, {p(t + r)}r∈[0,δ] > Y (t)dt 0 0 RT R T +r = 0 < F, {Y (t)p(t + r)}r∈[0,δ] > dt =< F, { r Y (u − r)p(u)du}r∈[0,δ] > RT RT =< F, { 0 Y (u − r)p(u)du}r∈[0,δ] = 0 < F, Yu > p(u)du RT = 0 < ∇x¯ G(u), Yu > du. 

x) is the averaging Example 5.1 (i) For example, if a ∈ R[0,δ] is a bounded function and F (¯ operator defined by Rδ F (¯ x) =< F, x ¯ >= 0 a(s)x(s)ds when x¯ = {x(s)}s∈[0,δ] , then

< F, pt >=

Rδ 0

a(r)p(t + r)dr.

(ii) Similarly, if t0 ∈ [0, δ] and G is evaluation at t0 , i.e. G(¯ x) = x(t0 ) when x¯ = {x(s)}s∈[0,δ] , 14

then < G, pt >= p(t + t0 ). We now have the machinery to start working on Problem (5.1). c be the set of all random measures on [0, T ]. Define the (singular) Hamiltonian Let M c H : [0, T ] × R × L2 × M0 × M0,δ × Ξ × R × R 7→ M

as the following random measure:

dH(t) = dH(t, x, x ¯, m, m, ¯ ξ, p, q) = H0 (t, x, x¯, m, m, ¯ ξ, p, q)dt + {λ(t)p + h(t, x)}dξ(t) ,

(5.8)

H0 (t, x, x¯, m, m, ¯ ξ, p, q) := f (t, x, x¯, m, m, ¯ ξ) + b(t, x, x¯, m, m, ¯ ξ)p + σ(t, x, x¯, m, m, ¯ ξ)q.

(5.9)

where

Here m denotes a generic value of the measure M(t). We assume that H0 and h are Fr´echet differentiable (C 1 ) in the variables x, x¯, m, m, ¯ ξ. We associate to this problem the following singular BSDE in the adjoint variables (p, q) :

where

 0 (t) + E(∇∗x¯ H0t |Ft ) + ∇∗m H0 (t) + E(∇∗m¯ H t |Ft )]dt dp(t) = −[ ∂H  ∂x    ∂h (t)dξ(t) + q(t)dB(t); 0 ≤ t ≤ T, − ∂x ∂g  p(t) = ∂x (T ) + ∇∗m g(T ); t ≥ T,    q(t) = 0; t > T, ∂H0 (t) ∂x ∂g (T ) ∂x

= =

(5.10)

∂H0 (t, X(t), Xt , M(t), Mt , p(t), q(t)), ∂x ∂g (X(T ), M(T )), ∂x

and similarly with the other terms. Remark 5.3 Partial result on existence and uniqueness of the BSDE (5.10) with singular drift but without mean-field terms, can be found in Dahl and Øksendal [9].

5.2

A sufficient maximum principle for singular mean field control with partial information

We proceed to state a sufficient maximum principle (a verification theorem) for the singular mean-field control problem described by (5.1) - (5.3).

15

Theorem 5.4 (Sufficient maximum principle for mean-field singular control ) Let ˆ ξˆ ∈ Ξ be such that the system of (5.1) and (5.10) has a solution X(t), pˆ(t), qˆ(t) and set ˆ ˆ M (t) = L(X(t)). Suppose the following conditions hold: • (The concavity assumptions) The functions R × L2 × M0 × M0,δ × Ξ ∋ (x, x¯, m, m, ¯ ξ) → dH(t, x, x ¯, m, m, ¯ ξ, pˆ(t), qˆ(t)) and R × M0 ∋ (x, m) → g(x, m) are concave for all t ∈ [0, T ] and almost all ω ∈ Ω. (5.11) • (Conditional variational inequality) For all ξ ∈ Ξ we have ˆ E[dH(t)|Gt ] ≤ E[dH(t)|G t ], i.e.

ˆ E[H0 (t)|Gt ]dt + E[λ(t)ˆ p(t) + h(t)|G t ]dξ(t) ˆ ˆ ˆ 0 (t)|Gt ]dt + E[λ(t)ˆ ≤ E[H p(t) + h(t)|G t ]dξ(t),

(5.12)

where the inequality is interpreted in the sense of inequality between random measures in M. ˆ is an optimal control for J(ξ). Then ξ(t) Proof.

Choose ξ ∈ Ξ and consider ˆ = I1 + I2 + I3 , J(ξ) − J(ξ)

where RT I1 = E[ 0 {f (t) − fˆ(t)}dt], I2 = E[g(T ) − gˆ(T )], RT ˆ ˆ I3 = E[ h(t)dξ(t) − h(t)d ξ(t)].

(5.13)

0

By the definition of H we get RT ˆ 0 (t) − p(t)(b(t) − ˆb(t)) − q(t)(σ(t) − σ I1 = E[ 0 {H0 (t) − H ˆ (t))}dt ˆ ˆ − (h(t)dξ(t) − h(t)d ξ(t))].

(5.14)

Next, by concavity of g we have ∂g ˆ ))+ < ∇m g(T ), M(T ) − M(T ˆ ) >] (T )(X(T ) − X(T I2 ≤ E[ ∂x ∂g ˆ ))] = E[p(T )(X(T ) − X(T ˆ ))] (T ) + ∇m g ∗ (T ))(X(T ) − X(T = E[( ∂x RT ˆ = E[ 0 p(t)((b(t) − ˆb(t))dt + λ(t)(dξ(t) − dξ(t)) RT ∂H0 ˆ (t) + E[∇∗ H t |Ft ] + ∇∗ H0 (t) + E[∇∗ H t |Ft ]) − {(X(t) − X(t))( 0

∂x

x ¯

0

m

ˆ ˆ ∂h (t)dξ(t)dt]. + (σ(t) − σ ˆ (t))q(t)}dt − (X(t) − X(t) ∂x 16

m ¯

(5.15)

Note that RT

RT ˆ ˆ t > dt], ∇x∗¯ H0t (t)(X(t) − X(t))dt] = E[ 0 < ∇x¯ H0 (t), Xt − X 0 ˆ ˆ (t) >], E[∇∗m H0 (t)(X(t) − X(t))] = E[< ∇m H0 (t), M(t) − M RT ∗ t RT ˆ ˆ E[ 0 ∇m ¯ H0 (t), Mt − Mt > dt]. ¯ H0 (t)(X(t) − X(t))dt] = E[ 0 < ∇m

E[

Substituting these equations into (5.15), and then adding (5.13),(5.14) and (5.15) we get, by concavity of dH, ˆ = I1 + I2 + I3 J(ξ) − J(ξ) RT ˆ 0 (t) − ∂ Hˆ 0 (t)(X(t) − X(t))− ˆ ˆ 0 (t), Xt − X ˆt > ≤ E[ 0 {H0 (t) − H < ∇x¯ H ∂x ˆ 0 (t), M(t) − M ˆ (t) > − < ∇m¯ H ˆ 0 (t), Mt − M ˆ t >}dt − < ∇m H RT R ˆ − ∂ hˆ (t)(X(t) − X(t)) ˆ + T λ(t)ˆ ˆ ˆ + 0 {h(t) − h(t) >}dξ(t) p(t)(dξ(t) − dξ(t))] ∂x 0 ˆ0 R T ∂H ˆ ˆ ˆ (t)(ξ(t) − ξ(t))dt + (λ(t)ˆ p(t) + h(t, X(t))(dξ(t) − dξ(t))] ≤ E[ 0 ∂ξ R T ∂ Hˆ 0 ˆ ˆ ˆ (ξ(t) − ξ(t))dt + (λ(t)ˆ p(t) + h(t, X(t))(dξ(t) − dξ(t))] ≤ 0, = E[ 0 ∂ξ(t)

ˆ ˆt, M ˆ (t), M ˆ t , ξ, pˆ(t), qˆ(t)). since ξ = ξˆ maximizes the random measure dH(t, X(t), X From the above result we can deduce the following sufficient variational inequalities.



Theorem 5.5 (Sufficient variational inequalities) Suppose that H0 does not depend on ξ,i.e. that ∂H0 = 0, ∂ξ and that the following variational inequalities hold: ˆ (i) E[λ(t)ˆ p(t) + h(t, X(t))|G t ] ≤ 0, ˆ ˆ (ii) E[λ(t)ˆ p(t) + h(t, X(t))|G t ]dξ(t) = 0.

(5.16) (5.17)

Then ξˆ is an optimal singular control. Proof.

Suppose (5.16) - (5.17) hold. Then for ξ ∈ Ξ we have ˆ ˆ ˆ E[λ(t)ˆ p(t) + h(t, X(t))|G p(t) + h(t, X(t))|G t ]dξ(t) ≤ 0 = E[λ(t)ˆ t ]dξ(t).

Since H0 does not depend on ξ, it follows that (5.12) hold. ˆ Note that the following processes ηi (s), i = 1, 2, 3 belong to V(ξ): η1 (s) := α(ω)χ[t,T ] (s), where α > 0 is Gt -measurable, for all t ∈ [0, T ], ˆ η2 (s) := ξ(s), ˆ η3 (s) := −ξ(s), s ∈ [0, T ]. 17



5.3

A necessary maximum principle for singular mean-field control

In the previous section we gave a verification theorem, stating that if a given control ξˆ satisfies (5.11)-(5.12), then it is indeed optimal for the singular mean-field control problem. We now establish a partial converse, implying that if a control ξˆ is optimal for the singular mean-field control problem, then it is a conditional critical point for the Hamiltonian. For ξ ∈ Ξ, let V(ξ) denote the set of G-adapted processes η of finite variation such that there exists ε = ε(ξ) > 0 satisfying ξ + aη ∈ Ξ for all a ∈ [0, ε].

(5.18)

Then for ξ ∈ Ξ and η ∈ V(ξ) we have, by our smoothness assumptions on the coefficients, lim 1 (J(ξ + aη) − J(ξ)) a→0+ a RT = E[ 0 { ∂f (t)Z(t)+ < ∇x¯ f (t), Z¯t ∂x

(5.19)

> ¯ t >}dt + < ∇m f (t), DM(t) > + < ∇m¯ f (t), Mt , ξ(t), D M R T ∂h RT ∂g (T )Z(T )+ < ∇m g(T ), DM(T ) > + 0 ∂x (t)Z(t)dξ(t) + 0 h(t)dη(t)], + ∂x where

and

Z(t) := Zη (t) := lima→0+ a1 (X (ξ+aη) (t) − X (ξ) (t)) (ξ+aη) (ξ) − Xt ) Zt := Zt,η := lima→0+ a1 (Xt

(5.20)

DM(t) := Dη M(t) := lima→0+ a1 (M (ξ+aη) (t) − M (ξ) (t)), (ξ+aη) (ξ) − Mt ). DMt := Dη Mt := lima→0+ a1 (Mt

(5.21)

Then  ∂b dZ(t) = [ ∂x (t)Z(t)+ < ∇x¯ b(t), Zt > + < ∇m b(t), DM(t) > + < ∇m¯ b(t), DMt >    ∂b + ∂ξ (t)η(t)]dt + [ ∂σ (t)Z(t)+ < ∇x¯ σ(t), Zt > + < ∇m σ(t), DM(t) > ∂x ∂b (t)η(t)]dB(t) + λ(t)dη(t) ; + < ∇m¯ σ(t), DMt > + ∂ξ    Z(0) = 0 , and similarly with dZt, dDM(t) and dDMt . We first state and prove a basic step towards a necessary maximum principle. Proposition 5.6 Let ξ ∈ Ξ and choose η ∈ V(ξ).Then d J(ξ da

Proof.

+ aη)|a=0 = E[

R T ∂H0 0

∂ξ

(t)η(t)dt +

RT 0

{λ(t)p(t) + h(t)}dη(t)].

(5.22)

Let ξ ∈ Ξ and η ∈ V(ξ). Then we can write d J(ξ da

+ aη)|a=0 = A1 + A2 + A3 + A4 , 18

(5.23)

where RT A1 = E[ 0 { ∂f (t)Z(t)+ < ∇x¯ f (t), Zt > + < ∇m f (t), DM(t) > + < ∇m¯ f (t), DMt >}dt], ∂x R T ∂f A2 = E[ 0 ∂ξ (t)η(t)dt], ∂g A3 = E[ ∂x (T )Z(T )+ < ∇m g(T ), DM(T ) >]

∂g (T ) + ∇∗m g(T ))Z(T )] = E[p(T )Z(T )], = E[( ∂x R T ∂h A4 = E[ 0 ∂x (t)Z(t)dξ(t) + h(t)dη(t)].

By the definition of H0 we have RT ∂b 0 (t) − ∂x (t)p(t) − ∂σ (t)q(t)}dt A1 = E[ 0 Z(t){ ∂H ∂x ∂x RT + 0 < ∇x¯ H0 (t) − ∇x¯ b(t)p(t) − ∇x¯ σ(t)q(t), Zt > dt RT + 0 < ∇m H0 (t) − ∇m b(t)p(t) − ∇m σ(t)q(t), DM(t) > dt RT + 0 < ∇m¯ H0 (t) − ∇m¯ b(t)p(t) − ∇m¯ σ(t)q(t), DMt >}dt],

and

A2 = E[

RT 0

0 { ∂H (t) − ∂ξ

∂b (t)p(t) ∂ξ

−

(5.24)

∂σ (t)q(t)}η(t)dt]. ∂ξ

By the terminal condition of p(T ) (see (5.10)) and then by the Itˆo formula we have A3 = E[p(T )Z(T )] RT RT = E[ 0 p(t)dZ(t) + 0 Z(t)dp(t) RT (t)Z(t)+ < ∇x¯ σ(t), Z(t) > + < ∇m σ(t), DM(t) > + 0 q(t){ ∂σ ∂x

(5.25)

(t)η(t)}dt + < ∇m¯ σ(t), DM(t) > + ∂σ ∂ξ

RT ∂b = E[ 0 p(t){ ∂x (t)Z(t)+ < ∇x¯ b(t), Zt > + < ∇m b(t), DM(t) >

∂b (t)η(t)}dt + < ∇m¯ b(t), DMt > + ∂ξ RT + 0 q(t){ ∂σ (t)Z(t)+ < ∇x¯ σ(t), Zt > + < ∇m σ(t), DM(t) > ∂x

(t)η(t)}dt + < ∇m¯ σ(t), DMt > + ∂σ ∂ξ RT RT 0 + 0 p(t)λ(t)dη(t) + 0 Z(t)(−{ ∂H (t) + E(∇x∗¯ H0t |Ft ) ∂x R T ∂h (t)Z(t)dξ(t)]. + ∇∗m H0 (t) + E(∇∗m¯ H0t |Ft )})dt − 0 ∂x

Combining (5.23)-(5.25) and using (5.4), (5.5) and (5.6), we get (5.22).



Theorem 5.7 (Necessary maximum principle for mean-field singular control) Suppose ξˆ ∈ Ξ is optimal, i.e. satisfies (5.3). Suppose that ∂H0 ∂ξ

= 0.

19

Then the following variational inequalities hold: (i)

E[λ(t)ˆ p(t) + h(t)|Gt ] ≤ 0 for all t ∈ [0, T ] a.s. and ˆ ˆ (ii) E[λ(t)ˆ p(t) + h(t)|G t ]dξ(t) = 0 for all t ∈ [0, T ] a.s.

Proof.

(5.26) (5.27)

From Proposition (5.6) we have, since ξˆ is optimal, 0≥

d J(ξˆ + da

RT ˆ aη)|a=0 = E[ 0 {λ(t)ˆ p(t) + h(t)}dη(t)],

(5.28)

ˆ for all η ∈ V(ξ). If we choose η to be a pure jump process of the form P η(s) = 0 0 is Gs -measurable for all s, then η ∈ V(ξ) ˆ E[{λ(t)ˆ p(t) + h(t)}α(t i )] ≤ 0 for all ti a.s. Since this holds for all such η with arbitrary ti , we conclude that ˆ E[λ(t)ˆ p(t) + h(t)|G t ] ≤ 0 for all t ∈ [0, T ] a.s.

(5.29)

ˆ and then to η2 := ξˆ ∈ V(ξ) ˆ we get, for all t ∈ [0, T ], Finally, applying (5.28) to η1 := ξˆ ∈ V(ξ) ˆ ˆ E[λ(t)ˆ p(t) + h(t)|G t ]dξ(t) = 0 for all t ∈ [0, T ] a.s. With (5.29) and (5.30) the proof is complete.

6

(5.30) 

Application to optimal stopping

From now on, let us assume, in addition to ∂H0 = 0, ∂ξ that λ(t) = −λ0 where λ0 > 0, and G = F.

(6.1) (6.2)

Then, dividing by λ0 in (5.26) - (5.27) we get 1 (i) pˆ(t) ≥ ˆh(t)) for all t ∈ [0, T ] a.s. and λ0 n 1ˆ o ˆ (ii) pˆ(t) − h(t) dξ(t) = 0 for all t ∈ [0, T ] a.s. λ0 20

(6.3) (6.4)

Comparing with (3.1), we see that (6.3)-(6.4), together with the singular BSDE (5.10) for ˆ constitutes an advanced mean-field reflected BSDE related to the type p = pˆ, q = qˆ, ξ = ξ, discussed in Section 3 above, with S(t) =

1ˆ h(t), λ0

(6.5)

and Y (t) := pˆ(t), Z(t) := qˆ(t), ˆ ∂h ˆ dK(t) := (t)dξ(t). ∂x

(6.6) (6.7) (6.8)

We summarize what we have proved as follows: Theorem 6.1 Suppose ξˆ is an optimal control for the singular control problem (5.1) - (5.3), ˆ ˆ t , M(t), ˆ ˆ t . Define S, Y, Z, K as in (6.5), (6.6), with corresponding optimal processes X(t), X M ˆ (6.8). Then X together with (Y, Z, K) solve the following forward-backward memoryadvanced mean-field singular reflected system: ˆ • (i) Forward mean-field memory singular SDE in X:  ˆ ˆ ˆt, M ˆ (t), M ˆ t )dt = b(t, X(t), X  dX(t) ˆ ˆ ˆ t , M(t), ˆ ˆ t )dB(t) − λ0 dξ(t); +σ(t, X(t), X M  X(t) = α(t), t ∈ [−δ, 0],

t ∈ [0, T ]

(6.9)

ˆ • (ii) Advanced reflected BSDE in (Y, Z, K) (for given X(t)):   Hˆ 0 ˆ t |Ft] + ∇∗ H ˆ 0 (t) + E[∇∗ H ˆ t |Ft ] dt  (t) + E[∇∗x¯ H dY (t) = − ∂∂x  0 m m ¯ 0      −dK(t) + Z(t)dB(t); t ∈ [0, T ], Y (t) ≥ S(t); t ∈ [0, T ],    [Y (t) − S(t)]dK(t) = 0; t ∈ [0, T ],     ∂g Y (T ) = ∂x (T ) + ∇∗m g(T ).

6.1

Connection to optimal stopping of memory mean-field SDE

If we combine the results above, we get Theorem 6.2 Suppose ξˆ is an optimal control for the singular control problem (5.1) - (5.3), ˆ ˆ t , M(t), ˆ ˆ t and adjoint processes pˆ(t), qˆ(t). Put with corresponding optimal processes X(t), X M R=

∂g (T ) + ∇∗m g(T ). ∂x 21

(6.10)

Let S(t), (Y (t), Z(t), K(t)) be as above and define ˆ ˆ (t), X ˆt, M ˆ t , Y (t), Z(t), Y t , Z t ) F (t) := F (t, X(t), M ˆ0 ∂H ∗ ˆt ˆ t |Ft ] + ∇∗ H ˆ := (t) + E[∇∗x¯ H 0 m 0 (t) + E[∇m ¯ H0 |Ft ]. ∂x

(6.11)

(i) Then, for each t ∈ [0, T ] , Y (t) is the solution of the optimal stopping problem n R o τ Y (t) = ess sup E[ t F (s)ds + S(τ )1τ K(t)} ∧ T. (iii) In particular, if we choose t = 0 we get that τˆ0 : = inf{s ∈ [0, T ], Y (s) ≤ S(s)} ∧ T = inf{s ∈ [0, T ], K(s) > 0} ∧ T solves the optimal stopping problem Rτ Y (0) = sup E[ 0 F (s)ds + S(τ )1τ