Algebraic structure of multiple stochastic integrals with respect to brownian motions and poisson processes

Abstract

In this paper, we extend the result of Gaines [4] to show that Lyndon set forms an algebraicbasis for the set of multiple stochastic integrals with respect to Brownian motions and Poissonprocesses, so that any multiple stochastic integral can be expressed as a polynomial of Lyndonbasis. The results for Philip Hall basis is similar. From the computational point of view, theadoption of either of the bases has the same effect on reducing much computing work of thenumerical approximation for stochastic differential equations of jump-diffusion type

Keywords:

• Brownian motions
• Poisson processes
• multiple stochastic integrals
• shuffle algebra
• Lyndon basis
• Chen series
• Philip Hall basis AMS Subject Classification (1991): 60H05, 60J65, 60G65, 17B66

^*Research supported in part by HKRGC Grant CPHK 232'93F

^*Research supported in part by HKRGC Grant CPHK 232'93F

Notes

^*Research supported in part by HKRGC Grant CPHK 232'93F