Conditional Reflected Backward Stochastic Differential Equations with Two Barriers
Hanwu Li
TL;DR
The paper develops a theory for doubly conditional reflected BSDEs with two barriers under partial information, proving existence and uniqueness under the Mokobodski condition and linking the Y-process to Dynkin games with partial information via the conditional expectation $\mathsf{E}[Y_t|\mathcal{G}_t]$. A key contribution is representing the finite-variation term $K$ through a Skorokhod problem on a time-dependent interval and solving via contraction, with a penalization approach discussed in the appendix. The conditional expectation of the solution is shown to equal the value of a Dynkin game, and an affine-driver case yields an auxiliary saddle-point representation using a linear multiplier $\Gamma$, providing a fixed-coefficient game formulation. The results are then applied to optimal switching under partial information, characterizing optimal stopping times via conditional payoff thresholds and establishing that the maximal expected profit is captured by a two-process BSDE system $(Y^1,Y^2)$, with explicit switching rules.
Abstract
In this paper, we study the doubly conditional reflected backward stochastic differential equations (BSDEs), where constraints are made on the conditional expectation of the first component of the solution with respect to a general subfiltration. With the help of the Skorokhod problem on a time-dependent interval and the Dynkin game in a general framework, we establish the existence and uniqueness result under the Mokobodski condition for the obstacles. The relation between the conditional expectation of the solution and the value function of a certain Dynkin game with partial information is obtained. As a by-product, we obtain a weaker version of the comparison theorem. Finally, we provide an application to the starting and stopping problem in reversible investments under partial information.