Optimal quantization methods for filtering and applications to finance.

Summary We develop a numerical solution approach to grid filtering, using optimal quantization results for random variables. We implement two filter computation algorithms using 0-order and 1-order approximation techniques. We propose implementable versions of these algorithms and study the behavior of the error of the approximations as a function of the quantizer size based on the stationarity property of optimal quantizers. We position this grid approach in relation to the Monte Carlo particle approach through the comparison of the two methods and their experimentation on different state models. In a second part, we focus on the advantage of quantization for the preprocessing of offline data to develop a filtering algorithm by quantization of the observations (and the signal). The error is also studied and a convergence rate is established as a function of the quantizer size. Finally, the quantization of the filter as a random variable is studied in order to solve an American option pricing problem in a market with unobserved stochastic volatility. All results are illustrated through numerical examples.