Abstract
We discuss a new type of fully coupled forward-backward stochastic differential equations (FBSDEs) whose coefficients depend on the states of the solution processes as well as their expected values, and we call them fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs). We first prove the existence and the uniqueness theorem of such mean-field FBSDEs under some certain monotonicity conditions and show the continuity property of the solutions with respect to the parameters. Then we discuss the stochastic optimal control problems of mean-field FBSDEs. The stochastic maximum principles are derived and the related mean-field linear quadratic optimal control problems are also discussed.
Highlights
Pardoux and Peng [1] in 1990 first introduced nonlinear classical backward stochastic differential equations (BSDEs)
When σ does not depend on z, z, the meanfield forwardbackward stochastic differential equations (FBSDEs) (8) has a unique adapted solution, but the monotonicity (H3.2) should be weakened as (i) ⟨A(t, λ, λ) − A(t, λ, λ), λ − λ⟩ ≤ −β1|x|2; (ii) ⟨Φ(x, x) − Φ(x, x), G(x − x)⟩ ≥ μ1|x|2; (H3.3) can be weakened as (i) ⟨A(t, λ, λ) − A(t, λ, λ), λ − λ⟩ ≥ β1|x|2; (ii) ⟨Φ(x, x) − Φ(x, x), G(x − x)⟩ ≤ −μ1|x|2, where β1 and μ are given nonnegative constants
We only prove the continuity of the solutions (xα, yα, zα, xα(T)) of mean-field FBSDE (38) at α = 0
Summary
Pardoux and Peng [1] in 1990 first introduced nonlinear classical backward stochastic differential equations (BSDEs). The authors proved the existence and the uniqueness for fully coupled FBSDEs on an arbitrarily given time interval, but they required the diffusion coefficients to be nondegenerate and deterministic. Another one is purely probabilistic continuation method; refer to Hu and Peng [4], Pardoux and Tang [5], Peng and Wu [6], Yong [7], and so on. Wu [19] discussed the stochastic maximum principle for the fully coupled FBSDEs. Recently the methods of mean-field are used in various fields, such as in Finance, Chemistry, and Game Theory.
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