Related papers: On It\^o formulas for jump processes
In this paper, we derive comparison results for terminal values of $d$-dimensional special semimartingales and also for finite-dimensional distributions of multivariate L\'{e}vy processes. The comparison is with respect to nondecreasing,…
We provide a suitable framework for the concept of finite quadratic variation for processes with values in a separable Banach space $B$ using the language of stochastic calculus via regularizations, introduced in the case $B= \R$ by the…
Moderate deviation principles for stochastic differential equations driven by a Poisson random measure (PRM) in finite and infinite dimensions are obtained. Proofs are based on a variational representation for expected values of positive…
Pure-jump L\'evy processes are popular classes of stochastic processes which have found many applications in finance, statistics or machine learning. In this paper, we propose a novel family of self-decomposable L\'evy processes where one…
We present a new method for simulating Markovian jump processes with time-dependent transitions rates, which avoids the transformation of random numbers by inverting time integrals over the rates. It relies on constructing a sequence of…
We construct a pathwise calculus for functionals of integer-valued measures and use it to derive an martingale representation formula with respect to a large class of integer-valued random measures. Using these results, we extend the…
This paper proposes a novel test for simultaneous jumps in a bivariate It\^o semimartingale when observation times are asynchronous and irregular. Inference is built on a realized correlation coefficient for the jumps of the two processes…
We develop the functional It\^o/path-dependent calculus with respect to fractional Brownian motion with Hurst parameter $H> \frac{1}{2}$. Firstly, two types of integrals are studied. The first type is Stratonovich integral, and the second…
In this article, we establish integration by parts formulas for the solutions of McKean-Vlasov stochastic differential equations with jumps under elliptic coefficients. The derived formulas accommodate both derivatives with respect to…
Bayesian inference for Markov jump processes (MJPs) where available observations relate to either system states or jumps typically relies on data-augmentation Markov Chain Monte Carlo. State-of-the-art developments involve representing MJP…
We give a short overview of recent results on a specific class of Markov process: the Piecewise Deterministic Markov Processes (PDMPs). We first recall the definition of these processes and give some general results. On more specific cases…
This thesis develops a new framework for modelling price processes in finance, such as an equity price or foreign exchange rate. This can be related to the conventional Ito calculus-based framework through the time integral of a price's…
We consider the stochastic ranking process with space-time dependent jump rates for the particles. The process is a simplified model of the time evolution of the rankings such as sales ranks at online bookstores. We prove that the joint…
In this paper we prove a version of the Berman-Konsowa principle for reversible Markov jump processes on Polish spaces. The Berman-Konsowa principle provides a variational formula for the capacity of a pair of disjoint measurable sets.…
We define a fractional Ito stochastic integral with respect to a randomly scaled fractional Brownian motion via an $S$-transform approach. We investigate the properties of this stochastic integral, prove the Ito formula for functions of…
We study the connection between PDEs and L\'{e}vy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to…
We treat the class of universal Markov processes on the d-dimensional Euklidean space which do not depend on random. For these, as well as for several subclasses, we prove criteria whether a function f, defined on the positive half-line,…
IIn this paper, we study a partially observed progressive optimal control problem of forward-backward stochastic differential equations with random jumps, where the control domain is not necessarily convex, and the control variable enter…
Stochastic modelling of fatigue (and other material's deterioration), as well as of cumulative damage in risk theory, are often based on compound sums of independent random variables, where the number of addends is represented by an…
The article is devoted to the expansions of iterated Ito stochastic integrals based on generalized multiple Fourier series converging in the sense of norm in the space $L_2([t, T]^k),$ $k\in\mathbb{N}.$ The method of generalized multiple…