Related papers: Path-wise solutions of SDE's driven by Levy proces…
Our aim is to study the existence and uniqueness of the $L^{p}$ - variational solution, with $p>1,$ of the following multivalued backward stochastic differential equation with $p$-integrable data: \[ \left\{ \begin{align*}…
We establish the first existence and uniqueness result for mild solutions of abstract stochastic evolution equations driven by arbitrary cylindrical L\'evy processes in Hilbert spaces. The coefficients are assumed to satisfy global…
We introduce a deformed version of Dyck paths (DDP), where additional to the steps allowed for Dyck paths, 'jumps' orthogonal to the preferred direction of the path are permitted. We consider the generating function of DDP, weighted with…
This paper establishes a Transition Path Theory (TPT) for L\'{e}vy-type processes, addressing a critical gap in the study of the transition mechanism between meta-stabile states in non-Gaussian stochastic systems. A key contribution is the…
Let {X_{t_1,t_2}: t_1,t_2 >= 0} be a two-parameter L\'evy process on R^d. We study basic properties of the one-parameter process {X_{x(t),y(t)}: t \in T} where x and y are, respectively, nondecreasing and nonincreasing nonnegative…
This paper is concerned with the relationship between forward-backward stochastic Volterra integral equations (FBSVIEs, for short) and a system of (non-local in time) path dependent partial differential equations (PPDEs, for short). Due to…
For any real-valued stochastic process $X$ with c\'rdl\'rg paths we define non-empty family of processes which have locally finite total variation, have jumps of the same order as the process $X$ and uniformly approximate its paths on…
A new method of solution is proposed for solution of the wave equation in one space dimension with continuously-varying coefficients. By considering all paths along which information arrives at a given point, the solution is expressed as an…
Let $X$ be a real L\'evy process and let $\Xpos $ be the process conditioned to stay positive. We assume that $ 0 $ is regular for $(-\infty, 0)$ and $(0, +\infty) $ with respect to $X$. Using elementary excursion theory arguments, we…
We construct a pathwise integration theory, associated with a change of variable formula, for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of $p$-th variation along a sequence of time…
Several two-boundary problems are solved for a special L\'{e}vy process: the Poisson process with an exponential component. The jumps of this process are controlled by a homogeneous Poisson process, the positive jump size distribution is…
We study the trajectories of a solution $X_t$ to an It\^o stochastic differential equation in $\Rm^d$, as the process passes between two disjoint open sets, $A$ and $B$. These segments of the trajectory are called transition paths or…
We study one-dimensional stochastic integral equations with non-smooth dispersion coefficients, and with drift components that are not restricted to be absolutely continuous with respect to Lebesgue measure. In the spirit of Lamperti, Doss…
Let $X$ be a regular one-dimensional transient diffusion and $L^y$ be its local time at $y$. The stochastic differential equation (SDE) whose solution corresponds to the process $X$ conditioned on $[L^y_{\infty}=a]$ for a given $a\geq 0$ is…
In this paper, we prove two main results. The first one is to give a new condition for the existence of two-parameter $p, q$-variation path integrals. Our condition of locally bounded $p,q$-variation is more natural and easy to verify than…
We prove the Ito-Tanaka formula and the existence of pathwise stochastic integrals for a wide class of Gaussian processes. Motivated by financial applications, we define the stochastic integrals as forward-type pathwise integrals introduced…
In this paper, we investigate ergodicity in total variation of the process $X_t$, related to a L\'evy-driven stochastic differential equation with unbounded coefficients, and describe the speed of convergence to the respective invariant…
Let $\{D(s), s \geq 0 \}$ be a L\'evy subordinator, that is, a non-decreasing process with stationary and independent increments and suppose that $D(0) = 0$. We study the first-hitting time of the process $D$, namely, the process $E(t) =…
We study a one-dimensional stochastic differential equation driven by a stable L\'evy process of order $\alpha$ with drift and diffusion coefficients $b,\sigma$. When $\alpha\in (1,2)$, we investigate pathwise uniqueness for this equation.…
Path decomposition is performed to analyze the pre-supremum, post-supremum, post-infimum and the intermediate processes of a spectrally negative Levy process taken up to an independent exponential time T as motivated by the aim of finding…