[Paper Review] A unified approach to well-posedness of type-I backward stochastic Volterra integral equations

Camilo Hernández, Dylan Possamaï|arXiv (Cornell University)|Jul 23, 2020

Stochastic processes and financial applicationsEconomics, Econometrics and Finance58 references17 citations

TL;DR

This paper introduces a unified framework for proving the well-posedness of multidimensional type-I backward stochastic Volterra integral equations (BSVIEs) by transforming them into an equivalent infinite-dimensional system of standard backward SDEs. The key contribution is establishing the equivalence between the solution of the BSVIE and the solution of this extended system, along with a representation formula via a Hamilton-Jacobi-Bellman-type PDE, which enables a game-theoretic analysis of time-inconsistent stochastic control problems.

ABSTRACT

We study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite dimensional system of standard backward SDEs and establish its well-posedness, and we show that it is equivalent to that of a type-I backward stochastic Volterra integral equation. We also establish a representation formula in terms of non-linear semilinear partial differential equation of Hamilton-Jacobi-Bellman type. As an application, we consider the study of time-inconsistent stochastic control from a game-theoretic point of view. We show the equivalence of two current approaches to this problem from both a probabilistic and an analytic point of view.

Motivation & Objective

• To develop a general and unified method for establishing the well-posedness of multidimensional type-I backward stochastic Volterra integral equations (BSVIEs).
• To establish an equivalence between type-I BSVIEs and an infinite-dimensional system of standard backward SDEs.
• To derive a representation formula for the solution in terms of a nonlinear semilinear PDE of Hamilton-Jacobi-Bellman type.
• To apply the framework to time-inconsistent stochastic control problems from a game-theoretic perspective.
• To show the equivalence between probabilistic and analytic approaches to equilibrium solutions in time-inconsistent control.

Proposed method

• The authors introduce an infinite-dimensional system of standard backward SDEs that is equivalent to the original type-I BSVIE.
• They prove the well-posedness of this infinite-dimensional system under suitable integrability and Lipschitz conditions on the generator and terminal condition.
• The solution of the BSVIE is shown to be equivalent to the solution of the infinite-dimensional SDE system via a transformation involving adapted processes and stochastic integrals.
• A representation formula is derived by linking the solution to a Hamilton-Jacobi-Bellman-type PDE through a verification argument.
• The framework is applied to time-inconsistent control by showing equivalence between the probabilistic equilibrium strategy and the analytic solution of the HJB equation.
• A contraction mapping argument is used to prove existence and uniqueness, with exponential weighting and Gronwall-type estimates to control the growth of differences in solutions.

Experimental results

Research questions

• RQ1 Can a unified approach be developed to establish the well-posedness of multidimensional type-I BSVIEs?
• RQ2 Is there an equivalent infinite-dimensional system of standard backward SDEs that captures the solution structure of a type-I BSVIE?
• RQ3 How can the solution of a type-I BSVIE be represented via a nonlinear PDE of Hamilton-Jacobi-Bellman type?
• RQ4 What is the connection between the probabilistic and analytic approaches to time-inconsistent stochastic control?
• RQ5 Can the equivalence between the game-theoretic equilibrium and the solution of the HJB equation be rigorously established within this framework?

Key findings

• The infinite-dimensional system of standard backward SDEs is well-posed under standard integrability and Lipschitz conditions, ensuring existence and uniqueness of solutions.
• The solution of the type-I BSVIE is equivalent to the solution of the infinite-dimensional SDE system, establishing a new characterization of BSVIE solutions.
• A representation formula is derived that links the solution of the BSVIE to a nonlinear semilinear PDE of Hamilton-Jacobi-Bellman type.
• The framework enables a rigorous equivalence between the probabilistic and analytic approaches to time-inconsistent stochastic control problems.
• The contraction mapping argument with exponential weighting ensures uniqueness and stability, with the contraction constant bounded by $ C/c $, which becomes small for large $ c $.
• The diagonal process $ v_t^t $ does not need to be in $ \mathbb{H}^2 $, relaxing a common technical assumption in the analysis of BSVIEs.