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A Malliavin Calculus Approach to Backward Stochastic Volterra Integral Equations

Qian Lei, Chi Seng Pun

TL;DR

This work tackles general backward stochastic Volterra integral equations (BSVIEs) with nonlinear dependence on the solution (Y) and martingale integrand (Z), including their diagonal terms. By integrating Malliavin calculus, it provides a robust framework that renders the diagonal processes well-defined and yields a probabilistic representation of nonlocal parabolic PDEs through a nonlocal Feynman–Kac formula. It establishes existence, uniqueness, and regularity of solutions in carefully designed Banach spaces that capture t- and Malliavin-differentiability, and extends the results to non-uniform Lipschitz settings. The paper also demonstrates applications to Markovian BSVIEs with a PDE link and to time-inconsistent stochastic control, notably deriving dynamically optimal mean-variance policies in stochastic volatility models. Overall, the results broaden the applicability of BSVIEs in finance and provide a foundation for deep-learning solvers for high-dimensional nonlocal PDEs and related TIC problems, with myopic and intertemporal hedging identified through diagonal and martingale components respectively.

Abstract

In this paper, we establish existence, uniqueness, and regularity properties of the solutions to multi-dimensional backward stochastic Volterra integral equations (BSVIEs), whose (possibly random) generator reflects nonlinear dependence on both the solution process and the martingale integrand component of the adapted solutions, as well as their diagonal processes. The well-posedness results are developed with the use of Malliavin calculus, which renders a novel perspective in tackling with the challenging diagonal processes while contrasts with the existing methods. We also provide a probabilistic interpretation of the classical solutions to the counterpart semi-linear partial differential equations through the explicit adapted solutions of BSVIEs. Moreover, we formulate with BSVIEs to explicitly characterize dynamically optimal mean-variance portfolios for various stochastic investment opportunities, with the myopic investment and intertemporal hedging demands being identified as two diagonal processes of BSVIE solutions.

A Malliavin Calculus Approach to Backward Stochastic Volterra Integral Equations

TL;DR

This work tackles general backward stochastic Volterra integral equations (BSVIEs) with nonlinear dependence on the solution (Y) and martingale integrand (Z), including their diagonal terms. By integrating Malliavin calculus, it provides a robust framework that renders the diagonal processes well-defined and yields a probabilistic representation of nonlocal parabolic PDEs through a nonlocal Feynman–Kac formula. It establishes existence, uniqueness, and regularity of solutions in carefully designed Banach spaces that capture t- and Malliavin-differentiability, and extends the results to non-uniform Lipschitz settings. The paper also demonstrates applications to Markovian BSVIEs with a PDE link and to time-inconsistent stochastic control, notably deriving dynamically optimal mean-variance policies in stochastic volatility models. Overall, the results broaden the applicability of BSVIEs in finance and provide a foundation for deep-learning solvers for high-dimensional nonlocal PDEs and related TIC problems, with myopic and intertemporal hedging identified through diagonal and martingale components respectively.

Abstract

In this paper, we establish existence, uniqueness, and regularity properties of the solutions to multi-dimensional backward stochastic Volterra integral equations (BSVIEs), whose (possibly random) generator reflects nonlinear dependence on both the solution process and the martingale integrand component of the adapted solutions, as well as their diagonal processes. The well-posedness results are developed with the use of Malliavin calculus, which renders a novel perspective in tackling with the challenging diagonal processes while contrasts with the existing methods. We also provide a probabilistic interpretation of the classical solutions to the counterpart semi-linear partial differential equations through the explicit adapted solutions of BSVIEs. Moreover, we formulate with BSVIEs to explicitly characterize dynamically optimal mean-variance portfolios for various stochastic investment opportunities, with the myopic investment and intertemporal hedging demands being identified as two diagonal processes of BSVIE solutions.
Paper Structure (11 sections, 10 theorems, 61 equations, 4 tables)

This paper contains 11 sections, 10 theorems, 61 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Well-posedness of Solutions of BSVIEs
  3. Norms and Banach Spaces
  4. Banach Spaces for BSVIE Solutions
  5. Differentiation of BSVIE Solutions
  6. Existence and uniqueness of BSVIE solutions
  7. Extensions to Non-Uniformly Lipschitz Cases
  8. Markovian BSVIEs and PDEs
  9. Time-Inconsistent Stochastic Control Problems
  10. Dynamic Mean-Variance Portfolio Selection
  11. Conclusion

Key Result

Lemma 2.1

If $\int^T_0|Z(s)|^2ds$ is integrable, then If $Z\in S^\textsc{BMO}_\mathbb{F}(0,T;\mathbb{R}^{k\times n})$ then one has automatically that $Z\in L^p_{\mathbb{F}}(0,T;\mathbb{R}^{k\times n})$ for all $p\geq 2$. Furthermore, the inclusion $L^\infty_\mathbb{F}\subsetneq S^{\textsc{BMO}}_\mathbb{F}\subsetneq \cap_{p>1}L^{p}_\mathbb{F}$ holds.

Theorems & Definitions (20)

  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Corollary 1
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 1: Locally Lipschitz BSVIEs
  • Lemma 3.1
  • ...and 10 more