Abstract: This is an expository review paper elaborating on the proof of the martingale functional central limit theorem (FCLT). This paper also reviews tightness and stochastic boundedness, highlighting one-dimensional criteria for tightness used in the proof of the martingale FCLT. This paper supplements the expository review paper Pang, Talreja and Whitt (2007) illustrating the “martingale m...

We consider the martingale problem related to the solution of an SDE on the line. It is shown that the solution of this martingale problem can be approximated by solutions of the corresponding time-discrete martingale problems under some conditions. This criterion is especially expedient for establishing the convergence of population processes to SDEs. We also show that the criterion yields a w...

A strict local martingale is a local martingale which is not a martingale. There are few explicit examples of “naturally occurring” strict local martingales with jumps available in the literature. The purpose of this paper is to provide such examples, and to illustrate how they might arise via filtration shrinkage, a phenomenon we would contend is common in applications such as filtering, contr...

For multiple testing problems, Benjamini and Hochberg (1995) proposed the false discovery rate (FDR) as an alternative to the family-wise error rate (FWER). Since then, researchers have provided many proofs to control the FDR under different assumptions. Storey et al. (2004) showed that the rejection threshold of a BH step-up procedure is a stopping time with respect to the reverse filtration g...

Let X be a continuous adapted process for which there exists an equivalent local martingale measure (ELMM). The minimal martingale measure P̂ is the unique ELMM for X with the property that local P -martingales strongly orthogonal to the P -martingale part of X are also local P̂ -martingales. We prove that if P̂ exists, it minimizes the reverse relative entropy H(P |Q) over all ELMMs Q for X. A co...

We investigate exponential stock models driven by tempered stable processes, which constitute a rich family of purely discontinuous Lévy processes. With a view of option pricing, we provide a systematic analysis of the existence of equivalent martingale measures, under which the model remains analytically tractable. This includes the existence of Esscher martingale measures and martingale measu...

I fhI is the respective Haar coefficient, and σ(I) = ±1. This operator, which we denote by Tσ, is a dyadic martingale transform. The martingale transform is bounded as an operator on L(R,C). We want to find a condition on matrix weights, U and V , that implies that all martingale transforms are uniformly bounded as operators from L(R,C, V ) to L(R,C, U) where L(R,C, V ) is the space of function...