Large deviations for the optimal filter of nonlinear dynamical systems driven by Lévy noise

Nonlinear filtering of stochastic differential equations driven by correlated Lévy noises

“…That is, our result is more general. Besides, note that in [12] three authors showed uniquenesses of the Kushner-Stratonovich and Zakai equations by the same method. However, they did not give the clear definitions of solutions and uniquenesses for the two equations.…”

On partially observed jump diffusions II. The filtering density

“…The same method is also used in [1] to prove uniqueness of measure-valued solutions for the Zakai equation in the case where the signal is a diffusion process, the observation contains a jump term and the coefficients are time-independent, globally Lipschitz, except for the observation drift term, which contains a time dependence, but is bounded and globally Lipschitz. The approach from [13] is extended in [14] to partially observed jump diffusions when the Wiener process in the observation process Y is independent of the Wiener process in the unobserved process, to prove, in particular, the existence of the conditional density in L 2 , if the initial conditional density exists, belongs to L 2 , the coefficients are bounded Lipschitz functions, the coefficients of the random measures in the unobservable process are differentiable in x and satisfy a condition in terms of their Jacobian. In [17] and [18] the filtering equations for fairly general filtering models with partially observed jump diffusions are obtained and studied, but the existence of the conditional density (in L 2 ) is proved only in [17], in the special case when the equation for the unobserved process is driven by a Wiener process and an α-stable additive Lévy process, ρ " 0, the coefficients b and σ are bounded functions of x P R d , b has bounded first order derivatives, σ has bounded derivatives up to second order and B " Bpt, x, yq is a bounded Lipschitz function in z " px, yq.…”