We formulate the generalized master equation for a class of continuous-time random walks in the presence of a prescribed deterministic evolution between successive transitions. This formulation is exemplified by means of an advection-diffusion and a jump-diffusion scheme. Based on this master equation, we also derive reaction-diffusion equations for subdiffusive chemical species, using a mean-f...

In this paper we consider Markov-modulated diffusion risk reserve processes. Using diffusion approximation we show the relation to classical Markov-modulated risk reserve processes. In particular we derive a representation for the adjustment coefficient and prove some comparison results. Among others we show that increasing the volatility of the diffusion increases the probability of ruin.

We provide a general method to analyze the asymptotic properties of a variety of estimators of continuous time diffusion processes when the data are not only discretely sampled in time but the time separating successive observations may possibly be random. We introduce a new operator, the generalized infinitesimal generator, to obtain Taylor expansions of the asymptotic moments of the estimator...

We consider the geometric Markov renewal processes as a model for a security market and study this processes in a diffusion approximation scheme. Weak convergence analysis and rates of convergence of ergodic geometric Markov renewal processes in diffusion scheme are presented. We present European call option pricing formulas in the case of ergodic, double-averaged, and merged diffusion geometri...

We consider a stochastic model of congestion control that represents the randomly varying number of flows present in a network where bandwidth is shared fairly amongst elastic document transfers. We focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. We first describe a fluid model (or functional law of large numbers approx...

For positive diffusions, we construct split-step second-order weak approximations preserving the positivity property. For illustration, we apply the construction to some popular stochastic differential equations such as Verhulst, CIR, and CKLS equations.

The survival problem for a diffusing particle moving among random traps is considered. We introduce a simple argument to derive the quenched asymptotics of the survival probability from the Lifshitz tail effect for the associated operator. In particular, the upper bound is proved in fairly general settings and is shown to be sharp in the case of the Brownian motion among Poissonian obstacles. A...

In this work, diffusion processes in thermally connected cylindrical fibers with weakly guiding and circular cross-section, that is, telecommunication fibers, have been presented. There have been discussed diffusion distributions of the core dopant of fibers spliced at T ≈ 2000 °C. Gaussian approximations of the core dopant concentration distribution and refractive index in the connecting area ...

L = 1 2∇ · a∇ + b · ∇ on D ⊆ R and whose spatially dependent branching term is of the form β x z − α x z2, x∈D where β satisfies a very general condition and α > 0. In the case that α and β are bounded from above, we show that the measurevalued process can also be obtained as a limit of approximating branching particle systems. We give criteria for extinction/survival, recurrence/transience of ...