Approximate controllability of stochastic differential systems driven by a Lévy process

In this paper, we are concerned with the approximate controllability of stochastic differential systems driven by Teugels martingales associated with a Lévy process. We derive the approximate controllability with the coefficients in the system satisfying some non-Lipschitz conditions, which include classic Lipschitz conditions as special cases. The desired result is established by means of standard Picard’s iteration.     

Optimal portfolio selection in a Lévy market with uncontrolled cash flow and only risky assets

This article considers an investor who has an exogenous cash flow evolving according to a Lévy process and invests in a financial market consisting of only risky assets, whose prices are governed by exponential Lévy processes. Two continuous-time portfolio selection problems are studied for the investor. One is a benchmark problem, and the other is a mean-variance problem. The first problem is solved by adopting the stochastic dynamic programming approach, and the obtained results are extended to the second problem by employing the duality theory. Closed-form solutions of these two problems are derived. Some existing results are found to be special cases of our results.     

A Numerical Method to Approximate Multi-Asset Option Pricing Under Exponential Lévy Model

In this paper, a modification of the original global radial basis functions-based differential quadrature (RBF-DQ) method is set forth and analyzed. The improved RBF-DQ method is applicable to the numerical approximation of solutions of a wide range of partial differential equations with mixed derivative terms. However, it appears to be considerably faster than the original method. In support of this contention, the multi-asset option pricing problems under exponential Lévy framework have been solved numerically by using the proposed method and compared with results obtained via the original RBF-DQ method. For accuracy achieved versus work expended, the improved method performs better.     

Stability in Probability and Inverse Optimal Control of Evolution Systems Driven by Lévy Processes

This paper first develops a Lyapunov-type theorem to study global well-posedness(existence and uniqueness of the strong variational solution)and asymptotic stability in probability of nonlinear stochastic evolution systems(SESs)driven by a special class of Levy processes,which consist of Wiener and compensated Poisson processes.This theorem is then utilized to develop an approach to solve an inverse optimal stabilization problem for SESs driven by Levy processes.The inverse optimal control design achieves global well-posedness and global asymptotic stability of the closed-loop system,and minimizes a meaningful cost functional that penalizes both states and control.The approach does not require to solve a Hamilton-Jacobi-Bellman equation(HJBE).An optimal stabilization of the evolution of the frequency of a certain genetic character from the population is included to illustrate the theoretical developments.