[Paper Review] Solvability of multidimensional quadratic BSDEs

Asgar Jamneshan, Michael Kupper+1 more|arXiv (Cornell University)|Nov 15, 2016

Stochastic processes and financial applications14 references3 citations

TL;DR

This paper establishes sufficient conditions for the existence and uniqueness of solutions to multidimensional quadratic backward stochastic differential equations (BSDEs), regardless of whether the terminal condition is bounded or unbounded. The key contribution is proving solvability when the terminal condition or the system's dependence is sufficiently small, extending prior results to broader settings.

ABSTRACT

We consider multidimensional quadratic BSDEs with bounded and unbounded terminal conditions. We provide sufficient conditions which guarantee existence and uniqueness of solutions. In particular, these conditions are satisfied if the terminal condition or the dependence in the system are small enough.

Motivation & Objective

• To address the solvability of multidimensional quadratic BSDEs with general terminal conditions, including unbounded ones.
• To identify sufficient conditions ensuring existence and uniqueness of solutions in the multidimensional setting.
• To extend existing results on quadratic BSDEs by relaxing assumptions on the terminal condition and generator dependence.
• To provide a theoretical foundation for applications in mathematical finance and stochastic control involving nonlinear dynamics.

Proposed method

• The authors employ a fixed-point argument in a suitable Banach space to establish existence and uniqueness of solutions.
• They introduce a priori estimates based on exponential martingale techniques and BMO (bounded mean oscillation) theory.
• The method relies on controlling the quadratic growth in the generator through smallness conditions on the terminal condition or the coefficient's dependence.
• A key component is the use of a localization procedure to handle unbounded terminal conditions.
• The analysis extends classical comparison theorems and a priori bounds to the multidimensional case.

Experimental results

Research questions

• RQ1 Under what conditions does a multidimensional quadratic BSDE admit a unique solution when the terminal condition is unbounded?
• RQ2 How can the quadratic growth in the generator be controlled to ensure solvability?
• RQ3 What smallness conditions on the terminal condition or the dependence structure guarantee existence and uniqueness?
• RQ4 Can the results be extended beyond bounded terminal conditions in the multidimensional setting?

Key findings

• Existence and uniqueness of solutions are established for multidimensional quadratic BSDEs with unbounded terminal conditions under appropriate smallness conditions.
• The solutions exist even when the generator exhibits quadratic growth in the gradient component.
• The smallness condition applies to either the terminal condition or the dependence structure of the generator.
• The results generalize previous findings restricted to bounded terminal conditions or scalar equations.