The stochastic exponential Zt = exp{Mt − M0 − (1/2)〈M,M〉t} of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale in the case where Mt = R t 0 b(Yu) dWu and Y is a one-dimensional diffusion driven by a Brownian motion W . Furthermore, we provide a necessary and sufficient condition for Z ...

This paper proposes a statistical test of the martingale hypothesis. It can be used to test whether a given time series is a martingale process against certain non-martingale alternatives. The class of alternative processes against which our test has power is very general and it encompasses many nonlinear non-martingale processes which may not be detected using traditional spectrum-based or var...

An equivalent martingale measure selection strategy for discrete time, continuous state, asset price evolution models is proposed. The minimal martingale law is shown to generally fail to produce a probability law in this context. The proposed strategy, termed the Girsanov principle, performs a multiplicative decomposition of asset price movements into a predictable and martingale component wit...

The martingale optimal transport problem is motivated by model-independent bounds for the pricing and hedging exotic options. In the simplest one-period model, the dual formulation of the robust superhedg-ing cost differs from the standard optimal transport problem by the presence of a martingale constraint on the set of coupling measures. The one-dimensional Brenier theorem has a natural exten...

Martingale boosting is a simple and easily understood technique with a simple and easily understood analysis. A slight variant of the approach provably achieves optimal accuracy in the presence of misclassification noise.

We present an elementary treatment of the Optional Decomposition Theorem for continuous semimartingales and general filtrations. This treatment does not assume the existence of equivalent local martingale measure(s), only that of strictly positive local martingale deflator(s).