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Vector measures induced by stochastic processes, especially martingales, have been discussed by several authors ([2], [3], [5]), primarily in the context of stochastic integration. Our purpose here is to invetigate some of their properties, including differentiation and the Radon-Nikodym property. Our approach combines methods drawn from the existing literature with several new techniques.
In [1] the existence of strong solutions of one-dimensional SDEs with coefficients depending only on the present was established. The proof was based on the Ito change-of-variables formula applied to a specially chosen function. However that function was not sufficiently smooth to enable the application of the Ito formula immediately. The aim of this paper is (1) to get Ito's formula for "bad"...
Let (Wt) = (Wtl, Wtl, ..., Wtd), d≧2, t≧0, denote a (standard) d-dimensional Brownian motion and let x(t), A(t) be measurable functions from [0, ∞] into ℍd and sd, sd the space of dxd skew-symmetric matrices, respectively which are bounded on every interval [0, T], T>0. Define the stochastic process (LtA,x) by the stochastic integral being defined in the...
We consider a nonlinear filtering problem with observations of the mixed type. With reference to such a problem we discuss the concept of robustness, describe an approximation approach and show its robustness properties.
Let (Xt, t∈[0,1]) be a diffusion process. Suppose we observe X0, X1, and (Yt, t∈[0,1]), where Yt is a noisy observation of (Xs, s∈[o,t]). We caracterise the conditional law of Xt (for t∈]0,1[), given these data, by means of a pair of stochastic PDEs.
Langevin-type equations are used as models for diffusions in force fields in many physical situations. In particular, the mean first passage time of such a process over a potential barrier, or the time of transition from one stable state to another, is the physical quantity that determines chemical reactions rates (Arrhenius' law), diffusion tensors for atomic migration in crystals, ionic conductivity,...
The adaptive stochastic filtering problem for Gaussian processes is considered. The selftuning-synthesis procedure is used to derive two algorithms for this problem. Almost sure convergence for the parameter estimate and the filtering error will be established. The convergence analysis is based on an almost-supermartingale convergence lemma that allows a stochastic Lyapunov like approach.
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