[Paper Review] Representation theorems for generators of Reflected BSDEs with continuous and linear-growth generators

Shiqiu Zheng, Shoumei Li|arXiv (Cornell University)|Apr 1, 2014

Stochastic processes and financial applications14 references3 citations

TL;DR

This paper establishes a local representation theorem for generators of reflected backward stochastic differential equations (RBSDEs) with continuous and linear-growth generators, extending known results from standard BSDEs. The key contribution is a representation that incorporates the increasing process $K$ and holds in a localized space, enabling a general converse comparison theorem and new properties for RBSDEs under mild conditions.

ABSTRACT

In this paper, we establish a local representation theorem for generators of reflected backward stochastic differential equations (RBSDE), whose generators are continuous with linear growth. It generalizes some known representation theorems for generators of backward stochastic differential equations (BSDE). As some applications, a general converse comparison theorem for RBSDE is obtained and some properties of RBSDE are discussed.

Motivation & Objective

• To extend representation theorems for BSDE generators to the reflected BSDE (RBSDE) setting, where solutions are constrained by an obstacle.
• To address the challenge that RBSDE solutions are restricted by the obstacle $L_t$, which complicates direct generalization of BSDE representation results.
• To establish a local representation theorem for generators that are continuous and satisfy linear growth in $(y,z)$, incorporating the increasing process $K$.
• To apply the representation theorem to derive a general converse comparison theorem for RBSDEs under weaker conditions than prior work.
• To investigate structural properties of RBSDEs, such as self-financing and zero-interest conditions, using the new representation framework.

Proposed method

• Utilizes a localization method to handle the obstacle constraint in RBSDEs, enabling the derivation of local representation results.
• Applies the representation theorem in a localized space, where the generator $g$ is compared via stopped processes and local solutions.
• Employs the comparison principle for RBSDEs in the localized setting, relying on the continuity and linear growth of $g$.
• Uses stopping times $\tau$ to construct local solutions $(y_t^\tau, z_t^\tau, k_t^\tau)$ that satisfy the RBSDE up to $\tau$, enabling comparison of generators.
• Relies on the fact that $g(t,0,0)=0$ under assumption (A3), which simplifies the analysis of trivial solutions.
• Applies Corollary 3.4 and Remark 1 to show that generator comparison is only possible locally, not globally, distinguishing RBSDEs from standard BSDEs.

Experimental results

Research questions

• RQ1 Can a representation theorem for generators of RBSDEs be established when the generator is continuous and satisfies linear growth in $(y,z)$?
• RQ2 How does the presence of the increasing process $K$ and the obstacle $L_t$ affect the structure of the representation theorem compared to standard BSDEs?
• RQ3 Can the representation theorem be used to derive a converse comparison theorem for RBSDEs under weaker conditions than Lipschitz continuity?
• RQ4 What conditions ensure that a constant solution $Y_t = y$ exists for RBSDEs with terminal condition $y$ and obstacle $L_t$?
• RQ5 How do self-financing and zero-interest conditions relate to the vanishing of the generator $g$ at specific points?

Key findings

• A local representation theorem for generators of RBSDEs is established under continuity and linear growth conditions, with the result depending on the increasing process $K$.
• The representation theorem shows that generator comparison is only possible in a localized space, not globally, due to the obstacle constraint.
• A general converse comparison theorem for RBSDEs is obtained, valid when generators are continuous and linear-growth in $(y,z)$, extending prior results requiring Lipschitz continuity.
• The self-financing condition holds if and only if $g(t,0,0) = 0$ $P$-a.s. for almost every $t$, under the assumption $\sup_t L_t < 0$.
• The zero-interest condition holds if and only if $g(t,y,0) = 0$ $P$-a.s. for all $y \geq C$ and almost every $t$, provided $\sup_t L_t \leq C$.
• A general condition is derived: $g(t,\eta_t,0) = 0$ $P$-a.s. for all $\eta \geq L$ and a.e. $t$ if and only if there exists a solution to the RBSDE with constant path $Y_s = \eta_t$ on $[t,\sigma_t]$.