Hiroshi Kunita

On square integrable martingales

Theory of real and time continuous martingales has been developed recently by P. Meyer [8, 9]. Let be a square integrable martingale on a probability space P . He showed that there exists an increasing process ‹X› t such that

First Order Stochastic Partial Differential Equations

Publisher Summary This chapter discusses the first order stochastic partial differential equations. The chapter studies the Cauchy problem of the first order stochastic partial differential equations of the parabolic type. The method of stochastic characteristic curve has been proposed to construct a solution of a suitable linear stochastic partial differential equation of first order. The chapter defines the first order stochastic partial differential equation rigorously and then introduces the associated stochastic characteristic equation. The proof of the existence and uniqueness of local solutions associated with a given initial condition is given. The chapter discusses quasi-linear, semi-linear and linear equation as special cases. The existence of the maximal or global solution is shown. The chapter also focuses on the regularity of Ito integral and the Stratonovich integral.

Ergodic Properties of Nonlinear Filtering Processes

Let x t be a temporally homogenous Markov process with state space S, called a system process in this paper. Suppose that we want to observe the sample path x t , but what we can observe is a stochastic process Y t of the form

Cauchy problem for stochastic partial differential equations arizing in nonlinear filtering theory

Y_t = \int_0^t {h\left( {x_s } \right)dt + N_t ,}

Stochastic differential equations with jumps and stochastic flows of diffeomorphisms

(0.1) where h is a continous function on S and N t is a standard Brownian motion independent of x s . The filtering of the system based on the observation data Y t is defined by a conditional distribution

The Stability and Approximation Problems in Nonlinear Filtering Theory

\pi _t \left( A \right) = P\left( {x_t \in \left. A \right|\mathcal{G}_t } \right),