[Paper Review] Reflected BSDEs when the obstacle is not right-continuous and optimal stopping

Miryana Grigorova, Peter Imkeller+3 more|arXiv (Cornell University)|Apr 23, 2015

Stochastic processes and financial applicationsEconomics, Econometrics and Finance15 references3 citations

TL;DR

This paper establishes existence and uniqueness of solutions to reflected backward stochastic differential equations (RBSDEs) with obstacles that are right upper-semicontinuous but not necessarily right-continuous, using tools from optimal stopping, Mertens decomposition, and generalized It’s formula. It further links these RBSDEs to optimal stopping problems under f-conditional expectations, characterizing the value function and proving existence of optimal stopping times under additional regularity conditions.

ABSTRACT

In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is established by using some tools from the general theory of processes such as Mertens decomposition of optional strong (but not necessarily right-continuous) supermartingales, some tools from optimal stopping theory, as well as an appropriate generalization of It{ô}'s formula due to Gal'chouk and Lenglart. In the second part of the paper, we provide some links between the RBSDE studied in the first part and an optimal stopping problem in which the risk of a financial position $ξ$ is assessed by an $f$-conditional expectation $\mathcal{E}^f(\cdot)$ (where $f$ is a Lipschitz driver). We characterize the "value function" of the problem in terms of the solution to our RBSDE. Under an additional assumption of left upper-semicontinuity on $ξ$, we show the existence of an optimal stopping time. We also provide a generalization of Mertens decomposition to the case of strong $\mathcal{E}^f$-supermartingales.

Motivation & Objective

• To extend the theory of reflected BSDEs to obstacles that are right upper-semicontinuous but not necessarily right-continuous.
• To establish existence and uniqueness of solutions in appropriate Banach spaces under these relaxed continuity assumptions.
• To connect the RBSDE solution to an optimal stopping problem where risk is assessed via f-conditional expectation.
• To prove existence of an optimal stopping time under an additional left upper-semicontinuity condition along stopping times.
• To generalize Mertens decomposition to strong E^f-supermartingales and establish a comparison principle for the RBSDEs.

Proposed method

• Utilizes tools from optimal stopping theory, including the characterization of essential suprema in dynamic risk measurement.
• Applies Mertens decomposition of strong optional supermartingales to handle non-right-continuous processes.
• Employs a generalized It’s formula due to Gal’chouk and Lenglart for strong optional semimartingales.
• Uses the general theory of stochastic processes, particularly the optional decomposition and predictable compensators.
• Applies the theory of f-conditional expectations and E^f-supermartingales to model dynamic risk measures.
• Constructs solutions via a penalization approach and proves convergence in appropriate Banach spaces.

Experimental results

Research questions

• RQ1 Can RBSDEs be solved when the obstacle is not right-continuous, but only right upper-semicontinuous?
• RQ2 How does the solution to such an RBSDE relate to an optimal stopping problem with f-conditional expectation?
• RQ3 Under what conditions does an optimal stopping time exist for the risk-minimization problem under f-expectation?
• RQ4 Can Mertens decomposition be generalized to strong E^f-supermartingales?
• RQ5 What is the role of the nondecreasing process in maintaining the solution above the obstacle when the obstacle lacks right-continuity?

Key findings

• The RBSDE with a right upper-semicontinuous obstacle admits a unique solution in appropriate Banach spaces, even when the obstacle is not right-continuous.
• The value function of the optimal stopping problem is characterized as the first component of the unique solution to the RBSDE with driver f and obstacle ξ.
• An optimal stopping time exists when the process ξ is left upper-semicontinuous along stopping times.
• An ε-optimal stopping time exists in the general case without the left upper-semicontinuity assumption.
• A generalization of Mertens decomposition is established for strong E^f-supermartingales, extending classical results to nonlinear expectations.
• A comparison principle is proven for the RBSDEs studied, ensuring order preservation under appropriate conditions.