Related papers: Martingale representations for diffusion processes…
The connection between forward backward doubly stochastic differential equations and the optimal filtering problem is established without using the Zakai's equation. The solutions of forward backward doubly stochastic differential equations…
We study the large deviations of current-type observables defined for Markov diffusion processes evolving in smooth bounded regions of $\mathbb{R}^d$ with reflections at the boundaries. We derive for these the correct boundary conditions…
A proof of the continuous martingale convergence theorem is provided. It relies on a classical martingale inequality and the almost sure convergence of a uniformly bounded non-negative super-martingale, after a truncation argument.
Parametric estimation for diffusion processes is considered for high frequency observations over a fixed time interval. The processes solve stochastic differential equations with an unknown parameter in the diffusion coefficient. We find…
The diffusion system with time-fractional order derivative is of great importance mathematically due to the nonlocal property of the fractional order derivative, which can be applied to model the physical phenomena with memory effects. We…
Let $M =(M_t)_{t\geq 0}$ be any continuous real-valued stochastic process. We prove that if there exists a sequence $(a_n)_{n\geq 1}$ of real numbers which converges to 0 and such that $M$ satisfies the reflection property at all levels…
We show that propagation speeds in invasion processes modeled by reaction-diffusion systems are determined by marginal spectral stability conditions, as predicted by the marginal stability conjecture. This conjecture was recently settled in…
We attempt to characterize irreversibility of a dynamical system from the existence of different forward and backward mathematical representations depending on the direction of the time arrow. Such different representations have been…
We formulate a martingale problem that describes a diffusion process in a multidimensional Euclidean space with a membrane located on a given smooth surface and having the properties of skewing and delaying. The theorem on the existence of…
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…
In this paper we show that the weak representation property of a semimartingale $X$ with respect to a filtration $\mathbb{F}$ is preserved in the progressive enlargement $\mathbb{G}$ by a random time $\tau$ avoiding $\mathbb{F}$-stopping…
We introduce and analyze a model for the transport of particles or energy in extended lattice systems. The dynamics of the model acts on a discrete phase space at discrete times but has nonetheless some of the characteristic properties of…
In this paper we introduce a variant of Burkholder's martingale transform associated with two martingales with respect to different filtrations. Even though the classical martingale techniques cannot be applied, we show that the discussed…
First, classes of Markov processes that scale exactly with a Hurst exponent H are derived in closed form. A special case of one class is the Tsallis density, advertised elsewhere as nonlinear diffusion or diffusion with nonlinear feedback.…
We solve explicitly the following problem: for a given probability measure mu, we specify a generalised martingale diffusion X which, stopped at an independent exponential time T, is distributed according to mu. The process X is specified…
Existing deterministic variational inference approaches for diffusion processes use simple proposals and target the marginal density of the posterior. We construct the variational process as a controlled version of the prior process and…
In this paper we establish a complete representation theorem for $G$-martingales. Unlike the existing results in the literature, we provide the existence and uniqueness of the second order term, which corresponds to the second order…
We analyze the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the diffusion coefficients may grow faster than linearly and degenerate on the boundaries of…
We attempt to characterize irreversibility of a dynamical system from the existence of different forward and backward mathematical representations depending on the direction of the time arrow. Such different representations have been…
The paper considers the martingale theory in the $G$-framework. A form of Doob's optional sampling is established, which allows to prove the exact analogue of the classical maximal inequality. The obtained results are used to improve the…