On the regularity of American options with regime-switching uncertainty

We study the regularity of the stochastic representation of the solution of a class of initial-boundary value problems related to a regime-switching diffusion. This representation is related to the value function of a finite-horizon optimal stopping problem such as the price of an American-style option in finance. We show continuity and smoothness of the value function using coupling and time-change techniques. As an application, we find the minimal payoff scenario for the holder of an American-style option in the presence of regime-switching uncertainty under the assumption that the transition rates are known to lie within level-dependent compact sets.Comment: 22 pages, to appear in Stochastic Processes and their Application

Regularity of the Optimal Stopping Problem for Jump Diffusions

The value function of an optimal stopping problem for jump diffusions is known to be a generalized solution of a variational inequality. Assuming that the diffusion component of the process is nondegenerate and a mild assumption on the singularity of the L\'{e}vy measure, this paper shows that the value function of this optimal stopping problem on an unbounded domain with finite/infinite variation jumps is in $W^{2,1}_{p, loc}$ with $p\in(1, \infty)$. As a consequence, the smooth-fit property holds.Comment: To Appear in the SIAM Journal on Control and Optimizatio

A note on the continuity of free-boundaries in finite-horizon optimal stopping problems for one dimensional diffusions.

We provide sufficient conditions for the continuity of the free-boundary in a general class of finite-horizon optimal stopping problems arising, for instance, in finance and economics. The underlying process is a strong solution of a one-dimensional, time-homogeneous stochastic differential equation (SDE). The proof relies on both analytic and probabilistic arguments and is based on a contradiction scheme inspired by the maximum principle in partial differential equations theory. Mild, local regularity of the coefficients of the SDE and smoothness of the gain function locally at the boundary are required

On two-sided controls of a linear diffusion

siirretty Doriast

Finite Horizon Time Inhomogeneous Singular Control Problem of One-dimensional Diffusion via Dynkin Game

The Hamilton-Jacobi-Bellman equation (HJB) associated with the time inhomogeneous singular control problem is a parabolic partial differential equation, and the existence of a classical solution is usually difficult to prove. In this paper, a class of finite horizon stochastic singular control problems of one dimensional diffusion is solved via a time inhomogeneous zero-sum game (Dynkin game). The regularity of the value function of the Dynkin game is investigated, and its integrated form coincides with the value function of the singular control problem. We provide conditions under which a classical solution to the associated HJB equation exists, thus the usual viscosity solution approach is avoided. We also show that the optimal control policy is to reflect the diffusion between two time inhomogeneous boundaries. For a more general terminal payoff function, we showed that the optimal control involves a possible impulse at maturity