Iterative approximations and hard bounds for stochastic process with jumps

Abstract

Markov modulation and stochastic processes with regime-switching and jumps have been widely employed in various fields of application, such as finance, economics, information and computer sciences, operations research, healthcare, and bio-medicines, whereas the additional modeling flexibility comes at the cost of demanding computation and complex inference procedure. We establish a novel theoretical framework in which weak approximation can be conducted in an iterative and convergent manner for a large class of multivariate inhomogeneous Markov modulation and stochastic differential equations with regime-switching and jumps of general time-state dependent intensity. The proposed iteration scheme is built on a sequence of approximate solutions, each of which makes use of a jump (or switching) time of the underlying dynamics as an information relay point in passing the past on to a previous iteration step to fill in the missing information on the unobserved trajectory ahead. We prove that the proposed iteration scheme is convergent and can be represented in a similar form to Picard iterates under the probability measure with its jump (and switching) component suppressed. On the basis of the approximate solution at each iteration step, we construct upper and lower bounding functions that are convergent towards the true solution as the iterations proceed. We provide illustrative examples so as to examine our theoretical findings and demonstrate the effectiveness of the proposed theoretical framework and the resulting iterative weak approximation scheme.