A martingale estimator for the expected discounted warranty cost process of a minimally repaired coherent system under its component level observation is proposed. Its asymptotic properties are also presented using the Martingale Central Limit Theorem.

The preservation of the semi-martingale property in a progressive enlargement of filtration has been studied by many authors. Most of them focus on progressive enlargement with a honest time, allowing for semi-martingale invariance and simple decomposition formulas. However, times allowing for semi-martingale invariance in initial enlargements preserve as well this property in progressive enlar...

Some classes of increment martingales, and the corresponding localized classes, are studied. An increment martingale is indexed by R and its increment processes are martingales. We focus primarily on the behaviour as time goes to −∞ in relation to the quadratic variation or the predictable quadratic variation, and we relate the limiting behaviour to the martingale property. Finally, integration...

We show that the method of Kipnis and Varadhan (Comm. Math. Phys. 104 (1986) 1) to construct a martingale approximation to an additive functional of a stationary ergodic Markov process via the resolvent is universal in the sense that a martingale approximation exists if and only if the resolvent representation converges. A sufficient condition for the existence of a martingale approximation is ...

In this paper we introduce the notion of α-martingale as the fractional derivative of order α of a continuous local martingale, where α ∈ (−12 , 1 2), and we show that it has a nonzero finite variation of order 2 1+2α , under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of Lévy’s characterization theorem for the f...

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in [3, 24]. In this paper, we extend the one-dimensional Brenier’s theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions correspon...

A martingale transform T , applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp Ap bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale t...

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 ...

The main result of this paper is that a martingale evolution can be chosen for LIBOR such that, by appropriately fixing the drift, all LIBOR interest rates have a common market measure. LIBOR is described using a quantum field theory model, and a common measure is seen to emerge naturally for such models. To elaborate how the martingale for the LIBOR belongs to the general class of numeraires f...

In this paper we use a class of stochastic functional Kolmogorov-type model with jumps to describe the evolutions of population dynamics. By constructing a special Lyapunov function, we show that the stochastic functional differential equation associated with our model admits a unique global solution in the positive orthant, and, by the exponential martingale inequality with jumps, we dis...