Recurrence and Ergodicity for A Class of Regime-Switching Jump Diffusions

This work develops asymptotic properties of a class of switching jump diffusion processes. The processes under consideration may be viewed as a number of jump diffusion processes modulated by a random switching mechanism. The underlying processes feature in the switching process depends on the jump diffusions. In this paper, conditions for recurrence and positive recurrence are derived. Ergodicity is examined in detail. Existence of invariant probability measures is proved.

Properties of Switching Jump Diffusions: Maximum Principles and Harnack Inequalities

This work examines a class of switching jump diffusion processes. The main effort is devoted to proving the maximum principle and obtaining the Harnack inequalities. Compared with the diffusions and switching diffusions, the associated operators for switching jump diffusions are non-local, resulting in more difficulty in treating such systems. Our study is carried out by taking into consideration of the interplay of stochastic processes and the associated systems of integro-differential equations.

The Stochastic Solution to a Cauchy Problem for Degenerate Parabolic Equations

We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally Holder continuous with exponent $deltain (0, 1]$, the stochastic solution, which represents the price of a European option, is shown to be a classical solution to the Cauchy problem. This improves the standard requirement $deltage 1/2$. Uniqueness results, including a Feynman-Kac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth, it is characterized as the limit of an approximating smooth stochastic solutions. In deriving the main results, we discover a new, probabilistic proof of Kotanis criterion for martingality of a one-dimensional diffusion in natural scale.

Indifference Pricing of American Option Underlying Illiquid Stock under Exponential Forward Performance

This work focuses on the indifference pricing of American call option underlying a non-traded stock, which may be partially hedgeable by another traded stock. Under the exponential forward measure, the indifference price is formulated as a stochastic singular control problem. The value function is characterized as the unique solution of a partial differential equation in a Sobolev space. Together with some regularities and estimates of the value function, the existence of the optimal strategy is also obtained. The applications of the characterization result includes a derivation of a dual representation and the indifference pricing on employee stock option. As a byproduct, a generalized Itos formula is obtained for functions in a Sobolev space.