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Anticipated backward stochastic Volterra integral equations and their applications to nonzero-sum stochastic differential games

Bixuan Yang, Tiexin Guo

TL;DR

The paper advances the theory of anticipated backward stochastic Volterra integral equations (ABSVIEs) by handling generators with both pointwise time-advanced and average time-advanced terms, establishing well-posedness and Malliavin-regularity of adapted M-solutions, and proving a comparison theorem. It then leverages a duality principle between ABSVIEs and stochastic delay Volterra integral equations (SDVIEs) to derive a Pontryagin-type maximum principle for nonzero-sum stochastic differential games and to obtain a Nash equilibrium for a linear-quadratic SDVIE game. The results extend the BSVIE framework to anticipative settings and provide a structured approach to analyze nonzero-sum stochastic differential games with delays, including potential Feynman-Kac-type connections. This work lays theoretical groundwork for applications in stochastic control and mathematical finance where delays and anticipations are intrinsic.

Abstract

In [J. Wen, Y. Shi, Stat. Probab. Lett. 156 (2020) 108599] the authors first introduced a kind of anticipated backward stochastic Volterra integral equations (anticipated BSVIEs, for short). By virtue of the duality principle, it is found in this paper that the anticipated BSVIEs can be applied to the study of stochastic differential games. Naturally, in order to develop the relevant theories and applications of BSVIEs, in this paper we deeply investigate a more general class of anticipated BSVIEs whose generator includes both pointwise time-advanced functions and average time-advanced functions. In theory, the well-posedness and the comparison theorem of anticipated BSVIEs are established, and some regularity results of adapted M-solutions are proved by applying Malliavin calculus, which cover the previous results for BSVIEs. Further, using linear anticipated BSVIEs as the adjoint equation, we present the maximum principle for the nonzero-sum differential game system of stochastic delay Volterra integral equations (SDVIEs, for short) for the first time. As one of the applications of the principle, a Nash equilibrium point of the linear-quadratic differential game problem of SDVIEs is obtained.

Anticipated backward stochastic Volterra integral equations and their applications to nonzero-sum stochastic differential games

TL;DR

The paper advances the theory of anticipated backward stochastic Volterra integral equations (ABSVIEs) by handling generators with both pointwise time-advanced and average time-advanced terms, establishing well-posedness and Malliavin-regularity of adapted M-solutions, and proving a comparison theorem. It then leverages a duality principle between ABSVIEs and stochastic delay Volterra integral equations (SDVIEs) to derive a Pontryagin-type maximum principle for nonzero-sum stochastic differential games and to obtain a Nash equilibrium for a linear-quadratic SDVIE game. The results extend the BSVIE framework to anticipative settings and provide a structured approach to analyze nonzero-sum stochastic differential games with delays, including potential Feynman-Kac-type connections. This work lays theoretical groundwork for applications in stochastic control and mathematical finance where delays and anticipations are intrinsic.

Abstract

In [J. Wen, Y. Shi, Stat. Probab. Lett. 156 (2020) 108599] the authors first introduced a kind of anticipated backward stochastic Volterra integral equations (anticipated BSVIEs, for short). By virtue of the duality principle, it is found in this paper that the anticipated BSVIEs can be applied to the study of stochastic differential games. Naturally, in order to develop the relevant theories and applications of BSVIEs, in this paper we deeply investigate a more general class of anticipated BSVIEs whose generator includes both pointwise time-advanced functions and average time-advanced functions. In theory, the well-posedness and the comparison theorem of anticipated BSVIEs are established, and some regularity results of adapted M-solutions are proved by applying Malliavin calculus, which cover the previous results for BSVIEs. Further, using linear anticipated BSVIEs as the adjoint equation, we present the maximum principle for the nonzero-sum differential game system of stochastic delay Volterra integral equations (SDVIEs, for short) for the first time. As one of the applications of the principle, a Nash equilibrium point of the linear-quadratic differential game problem of SDVIEs is obtained.
Paper Structure (6 sections, 7 theorems, 132 equations, 1 figure)

This paper contains 6 sections, 7 theorems, 132 equations, 1 figure.

Table of Contents

  1. Introduction and motivation
  2. Preliminaries
  3. Well-posedness of ABSVIEs
  4. Comparison theorem of ABSVIEs
  5. Regularity of adapted M-solutions to ABSVIEs
  6. Nonzero-sum stochastic differential games

Key Result

Lemma 1

Suppose that Assumptions $(A1)$-$(A2)$ hold for the special $g$ in an appropriate form, and for any $(\varphi (\cdot), \eta(\cdot, \cdot))\in \mathcal{M}^2 [0,T+K]$, there exists a unique adapted M-solution $(Y(\cdot),Z(\cdot,\cdot))\in\mathcal{M}^2 [0,T+K]$ solving the following special ABSVIE: and the following estimate holds: Moreover, for $i=1,2$, if $(\varphi_i (\cdot), \eta_i (\cdot, \cdot

Theorems & Definitions (10)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Example 1
  • Proposition 1
  • Remark 1