Near-martingale property of anticipating stochastic integration

A stochastic process Xt is called a near-martingale with respect to a filtration {Ft} if E[Xt|Fs] = E[Xs|Fs] for all s ≤ t. It is called a near- submartingale with respect to {Ft} if E[Xt|Fs] ≥ E[Xs|Fs] for all s ≤ t. Near-martingale property is the analogue of martingale property when the Itô integral is extended to non-adapted integrands. We prove that Xt is a near-martingale (near-submartingale) if and only if E[Xt|Ft] is a martingale (near-submartingale, respectively). Doob-Meyer decomposition theorem is extended to near-submartingale. We study stochastic differential equations with anticipating initial conditions and obtain a relationship between such equations and the associated stochastic differential equations of the Itô type. © 2018 International Society for Bayesian Analysis