Small-Time Solvability of a Flow of Forward–Backward Stochastic Differential Equations

Abstract

Motivated from time-inconsistent stochastic control problems, we introduce a new type of coupled forward-backward stochastic systems, namely, flows of forward-backward stochastic differential equations. They are systems consisting of a single forward SDE and a continuum of BSDEs, which are defined on different time-intervals and connected via an equilibrium condition. We formulate a notion of equilibrium solutions in a general framework and prove small-time well-posedness of the equations. We also consider discretized flows and show that their equilibrium solutions approximate the original one, together with an estimate of the convergence rate. satisfies lim inf ǫ↓0 J t (u t,ǫ,v ;x t ) − J t (û;x t ) ǫ ≥ 0 for any t ∈ [0, T ) and control v, where u t,ǫ,v is the "spike variation" ofû at time t with respect to v, namely, u t,ǫ,v s := v s if s ∈ [t, t + ǫ) and u t,ǫ,v s:=û s otherwise. Then, by a version of the stochastic maximum principle (see [3,5]), a subgame perfect Nash equilibrium strategŷ u must satisfy the relationfor any t ∈ [0, T ) and v ∈ U. Here, the function H is the Hamiltonian defined by

Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach

“…Henceû (·) is an equilibrium control. The equality in (16) is an immediate consequence of the probabilistic representations (17)- (18). This completes the proof.…”

“…Invoking this into (20), we obtain (17). Similarly, we proof Equality (18). Define for each 1 ≤ i ≤ n,…”

“…Loosely speaking, they are systems consisting of a single forward SDE and a flow of backward stochastic differential equations (BSDEs) plus a minimum condition called the equilibrium law. The unique (global) solvability of this type of systems remains a challenging open problem except for some special cases; we refer the readers to Hamaguchi [18] for a detailed discussion.…”

Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems

“…Recently, as a generalization of BSVIEs, Wang [21] introduced EBSVIEs, and investigated the Feynman-Kac formula for a non-local quasilinear parabolic partial differential equation (PDE, for short). A similar equation was considered by Hamaguchi [7]. The author established the well-posedness of a "flow of forward-backward stochastic differential equations" over small time horizon, which is a coupled system of a stochastic differential equation (SDE, for short) and an EBSVIE.…”

“…Note that the coefficients of EBSVIEs ( 32) and ( 33) are not continuous with respect t in the pointwise sense, and hence they are beyond the literature [21]. Alternatively, we can easily check that they satisfy the weak continuity assumption (7) and any other conditions in Assumption 1 for any p ≥ 2; see Remark 3.1. Moreover, by Assumption 4 (iv), we see that the coefficients of the first-order adjoint equation ( 32) satisfy Assumption 2.…”