Mean-field backward stochastic differential equations and applications

In this paper we study the mean-field backward stochastic differential equations (mean-field bsde) of the formdY (t) = −f(t, Y (t), Z(t), K(t, ·), E[ϕ(Y (t), Z(t), K(t, ·))])dt + Z(t)dB(t) + R R0 K ...


Introduction
Optimal control of mean-field stochastic differential equation has been studied by a number of researchers lately. To make things more precise let us explain the situation on optimal control of stochastic systems of the following type: dX(t) = b(t, X(t), L(X(t)), u(t))dt + σ(t, X(t), L(X(t)), u(t))dB(t), X(0) = x 0 with the performance f (x(t), u(t), L(X(t)))dt + h(X(T ), L(X(t))) , where B is the standard d-dimensional Brownian motion, L(X(t)) denotes the probability law of the state X(t) at time t and b, σ, f , h are some properly defined function. We refer to Anderson and Djehiche [5], [3], Lasry & Lions [17], Carmona & Delarue [10], [9] and Agram & Øksendal [1], [4] for some discussion. In particular, Pham & Wei [21] have introduced a dynamic programming approach by using a randomised stopping method.
Due to the presence of the law L(X(t)) in the equation and in the performance functional, the process X(t) is no longer Markovian and it is more effective to use the Pontryagin maximum principle to solve the above mean field stochastic control problem, which will give a mean field backward stochastic differential equation (mean field bsde).
To limit ourselves, we shall only deal with the case that the law L(X(t)) appears in the its simplest form of expectation (see equation (1.2) below). This simplest mean-field bsde also represents interesting models in finance, for example models of risk measures and recursive utilities. (1.2) The notation and conditions will be explained in details in Section 3. The purpose of this paper is the following.
(1) To prove new existence and uniqueness results for the above mean-field bsde.
(2) To give an explicit formula for the solution when the equation is linear.
(3) To prove a comparison principle for a type of mean-field bsde different from those in [8] and under weaker assumptions on the driver. (4) To apply the obtained results to study a mean-field recursive utility optimization problem in finance.

Malliavin calculus
In this section we give a brief summary of Malliavin calculus for processes driven by Brownian motion and compensated Poisson random measures. For more details we refer to [11], [15], [19] and [26].

(2.4)
We define the Malliavin derivative as D = (D 1 r , D 2 ρ,ζ ) (where D 1 denotes the partial Malliavin derivative with respect to the Brownain motion and D 2 denotes the partial Malliavin derivative with respect to the compensated Poisson process) as follows (2.5) We define and When there is confusion, we shall also omit the superscript and write D r = D 1 r and D t,ζ = D 2 t,ζ . We give some examples of Malliavin derivatives.
, for some given initial value p(0) (determined by the bsde (2.8). The Malliavin derivatives of p(t) are given, for all r < t (if r > t then D r p(t) = 0), by D r p(t) = t r D r f (s)ds − t r q(s)dB(s) + q(r), and D r,ζ p(t) = t r D r,ζ f (s)ds + t r R0 D r,ζ r(s, ζ)Ñ (ds, dζ) + r(r, ζ). Now let r → t − , then for a.a. t, q(t) = D t p(t), and for a.a. t and ζ, r(t, ζ) = D t,ζ p(t).
(ii) If F = exp( T 0 f (s)dB(s)), taking the conditional expectation, we get , then by the chain rule for Malliavin derivatives of processes driven by compensated Poisson random measures, we get , then by the chain rule for Malliavin derivatives of processes driven by compensated Poisson random measures, we get 3. Mean-field BSDE's 3.1. Existence and uniqueness of the solution. We define the following spaces for the solution triplet: • S 2 consists of the F-adapted càdlàg processes Y : Ω × [0, T ] → R, equipped with the norm We consider the following mean-field bsde: where ξ ∈ L 2 (Ω, F T ) is called the terminal condition and f is the generator.
Strictly speaking it is Y (s−) in the above equation but for simplicity we will drop the minus from now on. To obtain the existence and uniqueness of a solution we make the following set of assumptions.
Remark 3.4. In the above theorem if we take d = 3, ϕ i (x 1 , x 2 , x 3 ) = x i for i = 1, 2, 3, we see that the following mean-field bsde has a unique solution Linear mean-field bsde. In this section, we shall find the closed formula corresponding to the linear mean-field bsde of the form where the coefficients α 1 (t), α 2 (t), β 1 (t), β 2 (t), η 1 (t, ·), η 2 (t, ·) are given deterministic functions; γ(t) is a given F-adapted process and ξ ∈ L 2 (Ω, F T ) is a given F T measurable random variable. Applying a result from Øksendal and Sulem [20] or Quenez and Sulem [22]), the above linear mean-field bsde (3.3) can be written as follows.
Summarizing, we have the following theorem.

A comparison theorem for mean-field bsde's
In this section we are interested in a subclass of mean-field bsde. Our idea is to use Picard iteration. So first, we shall prove a convergence result for the Picard iteration.

Picard iteration.
To be able to prove the comparison theorem for mean-field bsde, we consider a mean field bsde and with driver allowed only to depend on the expectation of Y (t) and independent of the expectations of Z(t) and K(t, ζ), as follows  (i) Here g : Ω × [0, T ] × R 2 × L 2 ν × R → R is F-adapted and satisfies the Lipschitz assumption in the sense that |g(t, y, z, k, y) − g(t, y ′ , z ′ , k ′ , y ′ )| ≤ C(|y − y ′ | + |z − z ′ | + k − k ′ L 2 (ν) + |y − y ′ |), for all y, z, y, y ′ , z ′ , y ′ ∈ R, k, k ′ ∈ L 2 ν .
The following result is a consequence of Theorem 3.3 with d = 1 and ϕ(x) = x : Theorem 4.2. Under the above Assumption 4.1, the mean-field bsde (4.1) admits a unique solution (Y, Z, K) ∈ S 2 × L 2 × H 2 ν . To prove a comparison theorem, we need the following convergence to hold: where ξ and g are supposed to satisfy Assumption 4.1. We assume that for all n ≥ 1, the triplet (Y n , Z n , K n ) satisfies Y n (t) = ξ +  where Y n−1 (s) is known. Thus, the following convergence holds where Y n (t) is knowing. Defineḡ i (t, y, z, k) := g i (t, y, z, k, E[Y n i (t)]), then Y n+1 We have by our assumptions that g 1 (t, y, z, k) ≥ḡ 2 (t, y, z, k), for each t ≥ 0.
By the comparison theorem for BSDE with jumps e.g. Theorem 2.3 in Royer [25], it follows that Y n+1 (t) for all t ≥ 0. By our convergence result 4.3, we conclude that

Mean-field recursive utility
We consider in this section a mean-field recursive utility process Y (t), defined to be the first component of the solution triplet (Y, Z, K) of the following mean-field bsde: We denote by U, the set of all consumption processes. For each π(t) ∈ U, the driver g : Ω × [0, T ] × R 2 × L 2 ν × R 2 × L 2 ν × U → R and the terminal value ξ satisfies assumptions (I). Suppose that (y, z, k, y, z, k, π) → g(t, y, z, k,ȳ, z, k, π) is concave for each t ∈ [0, T ]. The driver represents the instantaneous utility at time t of the consumption rate π(t) ≥ 0, such that E[ T 0 |g(t, 0, 0, 0, 0, π(t))| 2 dt] < ∞, for all t ∈ [0, T ] .
We call a process π(t) a consumption rate process if π(t) is predictable and π(t) ≥ 0 for each t P-a.s. Then Y (t) = Y g (0) is called a mean-field recursive utility process of the consumption π(·), and the number U (π) = Y g (0) is called the total mean-field recursive utility of π(·). This is an extension to mean-field (and jumps) of the classical recursive utility concept of Duffie and Epstein [12]. See also Duffie and Zin [13], Kreps and Parteus [16], El Karoui et al [14], Øksendal and Sulem [20] and Agram and Røse [2] and the reference their in. Finding the consumption rateπ which maximizes its total mean-field recursive utility is an interesting problem in mean-field stochastic control.