In this article, we study the existence and uniqueness of the strong pathwise solution of stochastic Navier-Stokes equation with Itô-Lévy noise. Nonlinear filtering problem is formulated for the recursive estimation of conditional expectation of the flow field given back measurements of sensor output data. The corresponding Fujisaki-Kallianpur-Kunita and Zakai equations describing the time evolution of the nonlinear filter are derived. Existence and uniqueness of measure-valued solutions are proven for these filtering equations.
In this paper, we study solvability of the local mild solution of stochastic Navier‐Stokes equation with jump noise in Lp‐spaces. This research work has mainly carried out by exploiting some interesting works of Tosio Kato.
The objective in stochastic filtering is to reconstruct the 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 Lévy) process. We are interested in the situation where both the state process X and the observation process Y are perturbed by coupled Lévy processes. More precisely, L = (L1, L2) is a 2-dimensional Lévy process in which the structure of dependence is described by a Lévy 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.Primary 60H15, 35R30, 60K35; secondary 46G10, 28B05
In this paper we consider a stochastic counterpart of Tosio Kato's quasi-linear partial differential equations and prove the existence and uniqueness of mild solutions.
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