Volume 23, Issue 2

In this paper, we study a one-dimensional random motion with switching process having an Erlang-3 distribution for the sojourn times. We obtain the solution of the associated hyperbolic PDE of sixth order through algebraic techniques.

We are concerned with the discretization of a solution of a forward-backward stochastic differential equation (FBSDE) with a jump process depending on the Brownian motion. In this paper, we study the cases of Lipschitz generators and the generators with a quadratic growth with respect to the variable z . We propose a recursive scheme based on a general existence result given in the companion paper [Journal of Theoretical Probability 27 (2014), 683–724] and we study the error induced by the time discretization. We prove the convergence of the scheme when the number of time steps n goes to infinity. Our approach allows to get a convergence rate similar to that of schemes of Brownian FBSDEs.

In this paper, we show that a non-local operator of certain type extends to the generator of a strong Markov process, admitting the transition probability density. For this transition probability density we construct the intrinsic upper and lower bounds, and prove some smoothness properties. Some examples are provided.

Asymptotic analysis of the problem of large deviations for random evolutions with independent increments in the circuit of Lévy approximation with split and double merging is carried out. Large deviations for random evolutions in the circuit of Lévy approximation with split and double merging are determined by the exponential generator for the jumping process with independent increments.