Stopped Doob inequality for p-th moment, 0 <p<∞, stochastic convolution integrals

Abstract

We give stopped Doob inequalities for p-th moment, 0<p<∞, of stochastic convolution integrals ∫_(0,t] U(t,s)φ _{s ⁻} dM _s in a Hilbert space, where M is a Hilbert space-valued cadlag square integrable martingale, φ is an operator-valued predictable process and U(t,s) is a contraction-type evolution operator. We also generalize the previous results and try to get smaller constants.

¹This research was partially supported by Sharif University of Technology.

Keywords:

• Stochastic convolution integral
• Stopped Doob inequality
• Contraction-type evolution operator and martingale
• 60H10
• 34F05.

Acknowledgment

The first author is grateful to Professor N.U. Ahmed for very stimulating attention to this work.

Notes

¹This research was partially supported by Sharif University of Technology.