A Footnote to the Papers which Prove the Nonexistence of Finite Dimensional Filters

Abstract

Let the signal be defined by dx_t = m(x_t)dt + σ(x_t)dw_t and observed via dy_t = h(x_t)dt + dν_t where w and ν are independent Brownian motions. The filtering problem is the problem of determining the conditional distribution of x_t conditioned on y_θ, 0 ≤ Θ ≤ t and a basic question is that of existence of finite dimensional filters, i.e., when can the filtering problem be solved by a set of finite dimensional stochastic differential equations driven by the observation y_t. Following the results on the nonexistence of finite dimensional filters, the question arises whether there exists a set of finite dimensional equations driven by y_t which determines the conditional moment.