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The objective in stochastic filtering is to reconstruct information about an unobserved (random) process, called the signal process, given the current available observations of a certain noisy transformation of that process. Usually X and Y are modeled by stochastic differential equations driven by a Brownian motion or a jump (or Levy) process. We are interested in the situation where both the state process X and the observation process Y are perturbed by coupled Levy processes. More precisely, L=(L_1,L_2) is a 2--dimensional Levy process in which the structure of dependence is described by a Levy copula. We derive the associated Zakai equation for the density process and establish sufficient conditions depending on the copula and $L$ for the solvability of the corresponding solution to the Zakai equation. In particular, we give conditions of existence and uniqueness of the density process, if one is interested to estimate quantities like P( X(t)>a), where a is a threshold.
In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff] pointed out that it would be natural for $pi_t$, the solution of the stocha
In the paper, we consider nonlinear filtering problems of multiscale systems in two cases-correlated sensor Levy noises and correlated Levy noises. First of all, we prove that the slow part of the origin system converges to the homogenized system in
In this paper, we study almost periodic solutions for semilinear stochastic differential equations driven by L{e}vy noise with exponential dichotomy property. Under suitable conditions on the coefficients, we obtain the existence and uniqueness of bo
In this work we first present the existence, uniqueness and regularity of the strong solution of the tidal dynamics model perturbed by Levy noise. Monotonicity arguments have been exploited in the proofs. We then formulate a martingale problem of Str
This paper deals with linear stochastic partial differential equations with variable coefficients driven by L{e}vy white noise. We first derive an existence theorem for integral transforms of L{e}vy white noise and prove the existence of generalized