[Paper Review] An extended existence result for quadratic BSDEs with jumps with application to the utility maximization problem

Marie Amélie Morlais|ArXiv.org|Sep 2, 2008

Stochastic processes and financial applications8 references3 citations

TL;DR

This paper establishes a new existence result for quadratic backward stochastic differential equations (BSDEs) with jumps under a general Lévy measure (infinite activity), extending prior work that required finite Lévy measure. The method relies on dynamic programming and auxiliary BSDEs to solve the exponential utility maximization problem under portfolio constraints in a jump-diffusion model with discontinuous filtration.

ABSTRACT

In this study, we consider the exponential utility maximization problem in the context of a jump-diffusion model. To solve the problem, we rely on the dynamic programming principle and we derive from it a quadratic BSDE with jumps. Since this quadratic BSDE is driven both by a Wiener process and by a Poisson random measure having a Levy measure with infinite mass, our main task consists in establishing a new existence result for the specific BSDE introduced.

Motivation & Objective

• To extend the existence theory of quadratic BSDEs with jumps beyond the finite Lévy measure assumption.
• To solve the exponential utility maximization problem under portfolio constraints in a discontinuous filtration.
• To characterize the value process and optimal strategies via a specific quadratic BSDE with jumps.
• To establish a correspondence between solutions of an auxiliary BSDE and the original BSDE for arbitrary bounded terminal conditions.
• To provide explicit dynamic characterization of the value function and optimal investment strategies.

Proposed method

• Derive a quadratic BSDE with jumps from the dynamic programming principle applied to the utility maximization problem.
• Introduce an auxiliary BSDE with a modified generator to handle the existence problem under bounded terminal conditions.
• Prove existence of solutions for the auxiliary BSDE using a fixed-point argument under a norm constraint on the terminal condition.
• Construct a solution for the original BSDE via an explicit approximation scheme for arbitrary bounded terminal variables.
• Establish a one-to-one correspondence between solutions of the auxiliary and original BSDEs.
• Use the solution of the original BSDE to characterize the value process and optimal strategies via the martingale optimality principle.

Experimental results

Research questions

• RQ1 Can the existence of solutions to quadratic BSDEs with jumps be established when the Lévy measure has infinite mass?
• RQ2 How can the dynamic programming principle be applied to utility maximization under portfolio constraints in a discontinuous filtration?
• RQ3 What is the relationship between solutions of an auxiliary BSDE and the original quadratic BSDE with jumps?
• RQ4 How can the value process and optimal strategies be explicitly characterized in terms of the BSDE solution?
• RQ5 What conditions ensure the existence of optimal investment strategies in a jump-diffusion model with exponential utility?

Key findings

• A new existence result is established for quadratic BSDEs with jumps when the Lévy measure is infinite, extending previous results that required finite Lévy measure.
• The solution to the original BSDE exists for any bounded terminal condition, achieved through an explicit construction using an auxiliary BSDE.
• The value process of the utility maximization problem is characterized as the solution to the original quadratic BSDE with jumps.
• An optimal and admissible strategy exists and is characterized by a pointwise minimization of a quadratic expression involving the BSDE's Z and U components.
• The optimal strategy satisfies the condition π* ∈ argminπ∈C(½α|πσ − (Z + θ/α)|² + |U − πβ|α) P-a.s. for all t.
• The value function is explicitly given by V(x) = −exp(−α(x − Y₀)), where Y₀ is the initial value of the BSDE solution.