A recursive representation for decoupling time-state dependent jumps from jump-diffusion processes

TL;DR: A recursive representation fully decouples jumps from a large class of multivariate inhomogeneous stochastic differential equations with jumps of general time-state dependent unbounded intensity.

Abstract: AbstractWe establish a recursive representation that fully decouples jumps from a large class of multivariate inhomogeneous stochastic differential equations with jumps of general time-state dependent unbounded intensity, not of Lévy-driven type that essentially benefits a lot from independent and stationary increments. The recursive representation, along with a few related ones, are derived by making use of a jump 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 recursive representations are convergent exponentially fast in the limit, and can be represented in a similar form to Picard iterates under the probability measure with its jump component suppressed. On the basis of each iterate, we construct upper and lower bounding functions that are also convergent towards the true solution as the iterations proceed. We provide numerical results to justify our theoretical findings.Keywords: Jump-diffusion processestime-state dependent jump ratePicard iterationpartial integro-differential equationsfirst exit times2020 Mathematics Subject Classifications: 91B3060G5165M1565N15 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was partially supported by JSPS Grants-in-Aid for Scientific Research 20K22301 and 21K03347.