[Paper Review] $L^p$ solution of backward stochastic differential equations driven by a marked point process

Fulvia Confortola|arXiv (Cornell University)|Nov 30, 2016

Stochastic processes and financial applications13 references4 citations

TL;DR

This paper establishes existence and uniqueness of $L^p$ solutions, for $p > 1$, to backward stochastic differential equations (BSDEs) driven by a marked point process on a bounded time interval. It proves the solution can be approximated by finite systems of deterministic ODEs and applies the theory to prove existence of an optimal control and representation of the value function in non-Markovian point process control problems with $L^p$ integrability conditions.

ABSTRACT

We obtain existence and uniqueness in L^p, p>1 of the solutions of a backward stochastic differential equations (BSDEs for short) driven by a marked point process, on a bounded interval. We show that the solution of the BSDE can be approximated by a finite system of deterministic differential equations. As application we address an optimal control problems for point processes of general non-Markovian type and show that BSDEs can be used to prove existence of an optimal control and to represent the value function.

Motivation & Objective

• To extend the $L^p$-theory for backward SDEs driven by marked point processes beyond the $L^2$ framework.
• To establish existence and uniqueness of solutions in $L^p$ spaces for $p > 1$ under $L^p$ summability and Lipschitz conditions on the generator.
• To develop an approximation scheme using finite-jump processes and show convergence with error estimates.
• To apply the BSDE framework to solve optimal control problems for general non-Markovian point processes.
• To prove existence of an optimal control and represent the value function via the BSDE solution.

Proposed method

• Uses a fixed-point argument in weighted $L^p$ spaces to prove existence and uniqueness of the solution to the BSDE.
• Employs the martingale representation theorem under the assumption that jump times are totally inaccessible.
• Constructs an approximation scheme by truncating the point process to finite jumps and proves convergence to the true solution.
• Derives error estimates for the approximation that depend on the distribution of the last jump time.
• Reduces the approximating BSDEs to systems of deterministic ODEs when an additional technical assumption (A2) holds.
• Applies the BSDE solution to represent the value function in a stochastic optimal control problem with general non-Markovian dynamics.

Experimental results

Research questions

• RQ1 Can existence and uniqueness of $L^p$ solutions be established for BSDEs driven by a marked point process when $p > 1$?
• RQ2 How can the solution to such a BSDE be approximated by finite-dimensional systems of ODEs?
• RQ3 Can the BSDE framework be used to prove existence of an optimal control in non-Markovian point process control problems?
• RQ4 What $L^p$ summability and Lipschitz conditions are sufficient for the existence of a solution?
• RQ5 How can the value function in a stochastic control problem be represented using the solution of the associated BSDE?

Key findings

• The BSDE has a unique solution in $L^p_eta$ for $p > 1$ under a $L^p$ summability condition on the final condition $\xi$ and generator $f$, and a uniform Lipschitz condition on $f$.
• The solution can be approximated by a sequence of BSDEs driven by finite-jump processes, with an error estimate depending on the distribution of the last jump time.
• Under Assumption A2, the approximating BSDEs reduce to finite systems of deterministic ODEs, enabling numerical computation.
• The value function in the optimal control problem is represented as $Y_0$, the initial value of the BSDE solution, under $L^p$ integrability and Lipschitz conditions.
• The optimal control $u^*$ is given by $\underline{u}^Z$, where $Z$ is the $Z$-component of the BSDE solution, and $Y_0 = \inf_{u \in \mathcal{A}} J(u)$.
• Sufficient conditions for the existence of a measurable selector $\underline{u}^Z$ are provided, including compactness and continuity of the intensity and cost functions.