An introduction to the basic consepts and ideas in nonlinear filtering theory is given. We consider the system (signal) dX_t = a(X_t)dt + b(X_t)dB_t (observations) dY_t = h(X_t)dt + dW_t the problem is now to find the best (L2) estimate for X_t given the information contained in observations up to time t. The linear version of this problem was solved in 1960 by Kalman and Bucy. The nonlinear problem is essensially more difficult. Two main approaches has been used. The first is based on martingale theory and the innovation process and leads to the Fujisaki, Kallianpur and Kunita equation (FKK) describing the conditional density for X_t. The second approach uses a transformation of measue method (Girsanovs Theorem) and leads to the Zakai equation. The talk will give an introduction to the theory, consider specially the Zakai equation and also some simple examples.

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Helge Holden <holden@math.ntnu.no>

2000-08-30 11:47