Suppose that a real valued process X is given as a solution to a stochastic differential equation. Then, for any twice continuously differentiable function f , the Feynman-Kac formula gives a condition for f(t,X) to be a local martingale. We generalize the Feynman-Kac formula in two main ways. First, it is extended to nondifferentiable functions. Second, the process X is not required to satisfy...

In this paper we give a characterization of minimal distance martingale measures with respect to f-divergence distances in a general semimartin-gale market model. We provide necessary and suucient conditions for minimal distance martingale measures and determine them explicitly for exponential L evy processes with respect to several classical distances. It is shown that the minimal distance mar...

Using the martingale approach we find sufficient conditions for exponential boundedness of first passage times over a level for ergodic first order autoregressive sequences (AR(1)). Further, we prove a martingale identity to be used in obtaining explicit bounds for the expectation of first passage times.

We give sufficient criteria for the Doléans-Dade exponential of a stochastic integral with respect to a counting process local martingale to be a true martingale. The criteria are sufficiently weak to be useful and verifiable, as illustrated by several non-trivial examples, without introducing artificial constraints. In particular, they make it possible to construct nonexplosive point processes...

We show that there is a one-to-one correspondence between diffusions and the solutions of the Skorokhod Embedding Problem due to Bertoin and Le-Jan. In particular, the minimal embedding corresponds to a ‘minimal local martingale diffusion’, which is a notion we introduce in this article. Minimality is closely related to the martingale property. A diffusion is minimal if it it minimises the expe...

The contract is described and market examples given. Essential theoretical developments are introduced and cited chronologically. The principles and techniques of hedging and unique pricing are illustrated for the two simplest nontrivial examples: the classical Black-Scholes/Merton/Margrabe exchange option model brought somewhat upto-date from its form three decades ago, and a lesser exponentia...

A result is found which is similar to BDG-inequalities, but in the framework of exponential (non moderate) Orlicz spaces. A special class of such spaces is introduced and its properties are discussed with respect to probability measures, whose densities are connected by an exponential model. Acknowledgement: Thanks are due to Prof. M. Mania (Georgian Academy of Sciences) for the discussions and...

The goal of this paper is to present a novel application of a recent and useful martingale inequality. As an illustration, we prove an essentially sharp bound for the probability that a random graph contains significantly more triangles than expected.

An analogue of the Fourier transform will be introduced for all square integrable continuous martingale processes whose quadratic variation is deterministic. Using this transform we will formulate and prove a stochastic Heisenberg inequality.