[Paper Review] $L^p$ solutions of multidimensional BSDEs with weak monotonicity and general growth generators

Shengjun Fan|arXiv (Cornell University)|Mar 20, 2014

Stochastic processes and financial applications19 references4 citations

TL;DR

This paper establishes the existence and uniqueness of $L^p$ ($p>1$) solutions for multidimensional backward stochastic differential equations (BSDEs) under a weak monotonicity condition and a general growth condition in the generator $g$. The key contribution is a stability and comparison theorem for such solutions, extending prior results beyond Lipschitz or strong monotonicity assumptions.

ABSTRACT

In this paper, we first establish the existence and uniqueness of $L^p\ (p>1)$ solutions for multidimensional backward stochastic differential equations (BSDEs) under a weak monotonicity condition together with a general growth condition in $y$ for the generator $g$. Then, we overview several conditions related closely to the weak monotonicity condition and compare them in an effective way. Finally, we put forward and prove a stability theorem and a comparison theorem of $L^p\ (p>1)$ solutions for this kind of BSDEs.

Motivation & Objective

• To extend the existence and uniqueness of $L^p$ ($p>1$) solutions for multidimensional BSDEs beyond the classical Lipschitz condition on the generator $g$.
• To relax the monotonicity condition on $g$ from strong or concave forms to a weak monotonicity condition with general growth in $y$.
• To establish a stability theorem and a comparison theorem for $L^p$ solutions under these generalized conditions.
• To unify and compare different monotonicity-type conditions in the literature, clarifying their relationships and implications.

Proposed method

• Introduces a weak monotonicity condition on the generator $g$ in terms of a nondecreasing, continuous function $\rho(\cdot)$ with $\rho(0) = 0$ and $\int_{0^+} \frac{u^{p-1}}{\rho^p(u)} du = \infty$.
• Uses a modular function $F(r)$ derived from the generator to replace the concave $\rho$ with a concave majorant $\bar{\rho}$, enabling application of known existence results.
• Applies a priori estimates and Banach fixed-point argument in the space $\mathcal{S}^p \times \mathrm{M}^p$ to prove existence and uniqueness of $L^p$ solutions.
• Establishes a stability theorem by analyzing convergence of sequences of BSDEs with convergent terminal conditions and generators.
• Proves a comparison theorem for $L^p$ solutions under the weak monotonicity and growth conditions, extending classical results to the multidimensional case.
• Demonstrates that the concavity assumption on $\rho$ in prior conditions can be replaced by mere continuity, broadening applicability.

Experimental results

Research questions

• RQ1 Can $L^p$ solutions exist and be unique for multidimensional BSDEs when the generator $g$ satisfies only a weak monotonicity condition and general growth in $y$, rather than Lipschitz or strong monotonicity?
• RQ2 How do different monotonicity-type conditions on $g$ relate to one another, and can they be unified or compared effectively?
• RQ3 What conditions ensure the stability of $L^p$ solutions under perturbations of the terminal condition and generator?
• RQ4 Can a comparison theorem for $L^p$ solutions be established in the multidimensional setting under weak monotonicity and general growth?
• RQ5 Is the concavity requirement on the modulus of continuity $\rho$ in the monotonicity condition necessary, or can it be relaxed to continuity alone?

Key findings

• The existence and uniqueness of $L^p$ ($p>1$) solutions are established for multidimensional BSDEs under a weak monotonicity condition and general growth in $y$.
• The weak monotonicity condition is defined via a nondecreasing, continuous function $\rho$ satisfying $\int_{0^+} \frac{u^{p-1}}{\rho^p(u)} du = \infty$, which generalizes prior concave assumptions.
• The concavity condition on $\rho$ in earlier results can be replaced by mere continuity, significantly broadening the class of admissible generators.
• A stability theorem is proven: if terminal conditions and generators converge in $L^p$, then the corresponding $L^p$ solutions converge in $\mathcal{S}^p \times \mathrm{M}^p$.
• A comparison theorem for $L^p$ solutions is established under the weak monotonicity and growth conditions, enabling comparison of solutions when terminal values and generators are ordered.
• The proof technique relies on constructing a concave majorant $\bar{\rho}$ of the derived function $F(r)$, which satisfies the required integrability and allows application of known existence results.