Markov chain approximations for transition densities of Lévy processes

Abstract

We consider the convergence of a continuous-time Markov chain approximation X h , h > 0, to an R d -valued Lévy 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.2000 Mathematics Subject Classification. 60G51.

Markov chain approximations to scale functions of Lévy processes

“…In summary, then, h is chosen so small as to guarantee that, for all h ∈ (0, h ): (i) µ − µ h ≥ 0 and (ii) ψ h is the Laplace exponent of some CP process X h , which is also a CTMC with state space Z h (note that in [30,Proposition 3.9] it is shown h can indeed be so chosen, viz. point…”

“…Suppose we seek to prove that f ≥ 0 converges to 0 no faster than g > 0, i.e. that (This principle was also applied in [30] to establish sharpness of the stated rates of convergence there. )…”

“…(3) The proof of Theorem 1.1 consists of studying the differences of the integral representations of the scale functions. The integrands, however, decay only according to some power law, making the analysis much more involved than was the case in [30], where the corresponding decay was exponential. In particular, one cannot, in the pure-jump case, directly apply the integral triangle inequality.…”

On finite difference schemes for partial integro-differential equations of Lévy type

“…In this estimate, the constant N (δ) depends on δ and blows up as δ → 0 at a rate of ν({|z| ≥ δ}), which is a consequence of the fact that the integro-differential operator is treated as a first/zero order operator away from the ball (−δ, δ). In a similar manner in [14], δ is a function of h, and the corresponding convergence rate for the spatial approximation is of order hκ(h/2) where κ(δ) := (−1,1)\(−δ,δ) |z|ν(dz). If then for example the Lévy measure has a density of the form |z| −(2+α) for some α ∈ (0, 1), then the convergence is of order h (1−α) , which can be very slow, depending on α.…”

“…Equations of this form are of importance, since are satisfied by certain functionals of jump-diffusion Markov processes, that are known to be of interest in mathematical finance (for further reading on the subject we refer to [1]). Finite difference schemes for equations of this form have previously been studied in [2], [6] and [14]. In these articles the integro-differential operator is either truncated, or approximated by a second order difference operator in a neighborhood around the origin of radius δ > 0, and the remaining operator (the integral over {|z| ≥ δ}) is treated as a zero/first order operator.…”