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We use the abstract method of (local) martingale problems in order to give criteria for convergence of stochastic processes. Extending previous notions, the formulation we use is neither restricted to Markov processes (or semimartingales),…

We develop a stochastic calculus that makes it easy to capture a variety of predictable transformations of semimartingales such as changes of variables, stochastic integrals, and their compositions. The framework offers a unified treatment…

For a series of Markov processes we prove stochastic duality relations with duality functions given by orthogonal polynomials. This means that expectations with respect to the original process (which evolves the variable of the orthogonal…

A new derivation method of duality relations in stochastic processes is proposed. The current focus is on the duality between stochastic differential equations and birth-death processes. Although previous derivation methods have been based…

This paper considers binomial approximation of continuous time stochastic processes. It is shown that, under some mild integrability conditions, a process can be approximated in mean square sense and in other strong metrics by binomial…

We show the existence of superprocesses in a random medium with location dependent branching. Technically, we make use of a duality relation to establish the uniqueness of the martingale problem and to obtain the moment formulas.

In the development of stochastic integration and the theory of semimartingales, Markov processes have been a constant source of inspiration. Despite this historical interweaving, it turned out that semimartingales should be considered the…

Comparison results for Markov processes w.r.t. function class induced (integral) stochastic orders have a long history. The most general results so far for this problem have been obtained based on the theory of evolution systems on Banach…

We use the two-time scale subordination in order to describe dynamical processes in continuous media with a long-term memory. Our consideration touches two physical examples in detail. First we study a temporal evolution of the species…

This paper presents a simulation-based framework for sequential inference from partially and discretely observed point process (PP's) models with static parameters. Taking on a Bayesian perspective for the static parameters, we build upon…

We introduce a multistable subordinator, which generalizes the stable subordinator to the case of time-varying stability index. This enables us to define a multifractional Poisson process. We study properties of these processes and…

We introduce and study a multiparameter Poisson process (MPP). In a particular case, it is observed that the MPP has a unique representation. Its subordination with the multivariate subordinator and inverse subordinator are studied in…

We give a stochastic extension of the Brane Calculus, along the lines of recent work by Cardelli and Mardare. In this presentation, the semantics of a Brane process is a measure of the stochastic distribution of possible derivations. To…

We provide a systematic study of the notion of duality of Markov processes with respect to a function. We discuss the relation of this notion with duality with respect to a measure as studied in Markov process theory and potential theory…

An approach for the description of stochastic systems is derived. Some of the variables in the system are studied forward in time, others backward in time. The approach is based on a perturbation expansion in the strength of the coupling…

A two-type continuous-state branching process in varying environments is constructed as the pathwise unique solution of a system of stochastic equations driven by time-space noises, where the pathwise uniqueness is derived from a comparison…

In this paper we present a novel method for learning hierarchical representations of Markov decision processes. Our method works by partitioning the state space into subsets, and defines subtasks for performing transitions between the…

A stochastic calculus is given for processes described by stochastic integrals with respect to fractional Brownian motions and Rosenblatt processes somewhat analogous to the stochastic calculus for It\^{o} processes. These processes for…

A multiplicative stochastic process with the lower bound lognormally distributed is investigated. For the process, the model is constructed, and its distribution function (involving four parameters) and the related statistical properties…

We construct a family of self-similar Markov martingales with given marginal distributions. This construction uses the self-similarity and Markov property of a reference process to produce a family of Markov processes that possess the same…