We establish a functional large deviation principle and a functional moderate deviation principle for Markov-modulated risk models with reinsurance by constructing an exponential martingale approach. Lundberg’s estimate of the ruin time is also presented.

We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates.

Over the past decade there has been a flurry of new concentration of measure inequalities; we refer the reader to [4] for an in-depth survey, or [2, 3, 5] for some more recent advances. In [2] the martingale difference method was employed in a novel way to obtain a general concentration inequality for dependent random variables, with respect to the (unweighted) Hamming metric. At the core of th...

The TCP window size process can be modeled as a piecewise-deterministic Markov process that increases linearly and experiences downward jumps at Poisson times. We present a transient analysis of this window size process. Our main result is the Laplace transform of the transient moments. Formulae for the integer and fractional moments are derived, as well as an explicit characterization of the s...

We discuss the difference between locally risk-minimizing and delta hedging strategies for exponential Lévy models, where delta hedging strategies in this paper are defined under the minimal martingale measure. We give firstly model-independent upper estimations for the difference. In addition we show numerical examples for two typical exponential Lévy models: Merton models and variance gamma m...

This paper is concerned with the parameter estimation problem for a class of diffusion process from discrete observations. The approximate likelihood function given by using Riemann sum and an Itˆo to integrals in continuous-time function. consistency maximum estimator asymptotic normality error are proved applying martingale moment inequality, Holder’s Chebyshev B-D-G inequality uniform ergodi...

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 ...

We show existence and uniqueness of solutions stochastic path-dependent differential equations driven by cadlag martingale noise under joint local monotonicity coercivity assumptions on the coefficients with a bound in terms supremum norm. In this set-up, usual proof using ordinary Gronwall lemma together Burkholder-Davis-Gundy inequality seems impossible. order to solve problem, we prove new q...