In this paper, we study extended backward stochastic Volterra integral equations (EBSVIEs, for short). We establish the well-posedness under weaker assumptions than the literature, and prove a new kind of regularity property for the solutions. As an application, we investigate, in the open-loop framework, a time-inconsistent stochastic recursive control problem where the cost functional is defined by the solution to a backward stochastic Volterra integral equation (BSVIE, for short). We show that the corresponding adjoint equations become EBSVIEs, and provide a necessary and sufficient condition for an open-loop equilibrium control via variational methods.
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
In this paper, we formulate and investigate the notion of causal feedback strategies arising in linearquadratic control problems for stochastic Volterra integral equations (SVIEs) with singular and nonconvolution-type coefficients. We show that there exists a unique solution, which we call the causal feedback solution, to the closed-loop system of a controlled SVIE associated with a causal feedback strategy. Furthermore, introducing two novel equations named a Type-II extended backward stochastic Volterra integral equation and a Lyapunov-Volterra equation, we prove a duality principle and a representation formula for a quadratic functional of controlled SVIEs in the framework of causal feedback strategies.
In this paper, we provide variation of constants formulae for linear (forward) stochastic Volterra integral equations (SVIEs, for short) and linear Type-II backward stochastic Volterra integral equations (BSVIEs, for short) in the usual Itô's framework. For these purposes, we define suitable classes of stochastic Volterra kernels and introduce new notions of the products of adapted L 2 -processes. Observing the algebraic properties of the products, we obtain the variation of constants formulae by means of the corresponding resolvent. Our framework includes SVIEs with singular kernels such as fractional stochastic differential equations. Also, our results can be applied to general classes of SVIEs and BSVIEs with infinitely many iterated stochastic integrals. The duality principle between generalized SVIEs and generalized BSVIEs is also proved.
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