A Sharp and Strict L p -Inequality for Stochastic Integrals

Abstract

Let (Ω, F, P) be a complete probability space and (F t )t ≥ 0 a nondecreasing right-continuous family of sub-σ-fields of F where F 0 contains all A ∈ F with P(A) = 0. Suppose that M = (M t )t ≥ 0 is a real martingale adapted to (F t )t ≥ 0 such that almost all of the paths of M are right-continuous on [0, ∞) and have left limits on (0, ∞). Let V = (V t )t ≥ 0 be a predictable process with values in [−1, 1] and denote by N= V∙ M the stochastic integral of V with respect to M: N is an adapted right-continuous process with left limits on (0, ∞) such that $${N_t} = \int_{\left[ {0,\,t} \right]} {{V_s}\,d{M_s}} \,a.s.$$