Nonexistence of strong nonanticipating solutions to stochastic DEs: implications for functional DEs, filtering, and control

By applying a known n.a.s. condition for a weak solution of a stochastic equation to be strong, we generalize B.S. Cirel'son's example of a stochastic DE without a causal solution to whole classes of similar equations. Then the nonexistence of a strong nonanticipating solution to the stochastic DE(1) d[zeta] = [beta](t, [zeta])dt + dw, w Wiener, is established for a set of drifts [beta] uniformly dense in the bounded causal drifts, and also for a set of drifts "pointwise" L2-dense in the drifts of linear growth. These results may be interpreted as saying that in a topological sense the DE (1) with a reasonable causal drift almost never has a causal solution. The various applications of these methods to applied problems are mostly negative: in filtering, we show that the innovations conjecture is almost always false; for functional DEs we show that the causal bounded "right-hand sides" [beta] for which xt - [integral operator] [beta](s, x)ds = yt has at least two solutions for a countable set of continuous y, are dense in the uniform topology; in control theory, we exhibit problems whose state equation has no causal solution for any admissible control law.