[Paper Review] Viscosity Solution for Optimal Stopping Problems of Feller Processes

Suhang Dai, Olivier Menoukeu Pamen|arXiv (Cornell University)|Mar 10, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance19 references4 citations

TL;DR

This paper establishes the existence and uniqueness of viscosity solutions to the Hamilton-Jacobi-Bellman (HJB) equation for optimal stopping problems driven by general Feller processes, without assuming a specific form for the generator. By leveraging Feller semigroup properties and the penalty method, it proves the value function is the unique viscosity solution, resolving a conjecture from prior work and extending applicability to processes like diffusions with piecewise coefficients and semi-Markov processes.

ABSTRACT

We study an optimal stopping problem when the state process is governed by a general Feller process. In particular, we examine viscosity properties of the associated value function with no a priori assumption on the stochastic differential equation satisfied by the state process. Our approach relies on properties of the Feller semigroup. We present conditions on the state process under which the value function is the unique viscosity solution to an Hamilton-Jacobi-Bellman (HJB) equation associated with a particular operator. More specifically, assuming that the state process is a Feller process, we prove uniqueness of the viscosity solution which was conjectured in [26]. We then apply our results to study viscosity property of optimal stopping problems for some particular Feller processes, namely diffusion processes with piecewise coefficients and semi-Markov processes. Finally, we obtain explicit value functions for optimal stopping of straddle options, when the state process is a reflected Brownian motion, Brownian motion with jump at boundary and regime switching Feller diffusion, respectively (see Section 8).

Motivation & Objective

• To establish the existence and uniqueness of viscosity solutions for optimal stopping problems when the underlying process is a general Feller process.
• To resolve the conjecture in [26] regarding uniqueness of viscosity solutions in the context of Feller semigroups.
• To provide a general analytical framework applicable to a broad class of Markov processes where the generator is not explicitly a differential operator.
• To extend viscosity solution theory to processes such as diffusions with piecewise coefficients and semi-Markov processes, which are underexplored in optimal stopping literature.
• To derive explicit value functions for straddle options under specific Feller diffusions, including reflected Brownian motion and regime-switching diffusions.

Proposed method

• Utilizes the penalty method to approximate the value function via smooth functions, avoiding direct reliance on differential operators.
• Employs the Feller semigroup structure to define the infinitesimal generator of the extended process on the one-point compactification space.
• Applies perturbation theory to transform one-dimensional semi-Markov processes into two-dimensional Markov processes for analysis.
• Establishes that the value function is the unique viscosity solution to the HJB equation by proving the generator’s core property and domain equivalence.
• Uses the extended generator $\tilde{\mathcal{L}}$ on $\mathcal{C}(\mathsf{E}_\partial)$, defined via the limit of the semigroup action, to characterize the HJB operator.
• Relies on the core property of $D(\tilde{\mathcal{G}})$ in $D(\tilde{\mathcal{L}})$ to ensure convergence and well-posedness of the viscosity solution.

Experimental results

Research questions

• RQ1 Under what conditions is the value function of an optimal stopping problem for a Feller process the unique viscosity solution to the associated HJB equation?
• RQ2 Can the uniqueness of viscosity solutions be established in the absence of a differential generator, relying only on semigroup properties?
• RQ3 How can the penalty method be adapted to ensure convergence to the value function in the context of general Feller processes?
• RQ4 What are the viscosity properties of optimal stopping problems for diffusions with piecewise coefficients and semi-Markov processes?
• RQ5 Can explicit value functions be derived for straddle options under specific Feller diffusions, such as reflected Brownian motion or regime-switching diffusions?

Key findings

• The value function is the unique viscosity solution to the HJB equation for optimal stopping problems of Feller processes, resolving a conjecture from [26].
• The infinitesimal generator $\tilde{\mathcal{L}}$ of the extended process satisfies $D(\tilde{\mathcal{L}}) = \{ w \in \mathcal{C}(\mathsf{E}_\partial) \mid (w - w(\partial))|_\mathsf{E} \in D(\mathcal{L}) \}$, establishing domain equivalence.
• The restriction of $\tilde{\mathcal{L}}$ to $D(\tilde{\mathcal{G}})$ is well-defined and forms a core for $\tilde{\mathcal{L}}$, ensuring convergence of approximations.
• For diffusions with piecewise coefficients, the value function is a viscosity solution to an HJB equation with a specific operator, as shown in Corollaries 7.9 and 7.10.
• For semi-Markov processes, the value function is the unique viscosity solution after transforming the process into a two-dimensional Markov process via perturbation theory.
• Explicit value functions are derived for straddle options under reflected Brownian motion, Brownian motion with boundary jumps, and regime-switching Feller diffusions in Section 8.