Abstract
We consider the convergence of a continuous-time Markov chain approximation X^h, h>0, to an R^d-valued Levy process X. The state space of X^h is an equidistant lattice and its Q-matrix is chosen to approximate the generator of X. In dimension one (d=1), and then under a general sufficient condition for the existence of transition densities of X, we establish sharp convergence rates of the normalised probability mass function of X^h to the probability density function of X. In higher dimensions (d>1), rates of convergence are obtained under a technical condition, which is satisfied when the diffusion matrix is non-degenerate.
Highlights
Discretization schemes for stochastic processes are relevant both theoretically, as they shed light on the nature of the underlying stochasticity, and practically, since they lend themselves well to numerical methods
We study the rate of convergence of a weak approximation of an Rd-valued (d ∈ N) Lévy process X by a continuous-time Markov chain (CTMC)
Markov chain approximations for densities of Lévy processes be viewed as expectations of realvalued functions against the marginals of the processes, and are in general hard to study
Summary
Discretization schemes for stochastic processes are relevant both theoretically, as they shed light on the nature of the underlying stochasticity, and practically, since they lend themselves well to numerical methods. In particular, constitute a rich and fundamental class with applications in diverse areas such as mathematical finance, risk management, insurance, queuing, storage and population genetics etc. (see e.g. [22])
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