Remarks on the finite energy condition in additive white noise filtering (Q800882)

The author considers the filtering problem where the signal process x(t) is given by \(dx(t)=m(x(t))dt+\sigma (x(t)db(t)\), \(x(0)=x_ 0\), and the observation process is given by \(dy(t)=h(x(t))dt+dw(t)\), \(y(0)=0\), for \(0\leq t\leq T\). If \({\mathcal Y}_ t=\sigma \{y(s):\) \(0\leq s\leq t\}\), the optimum filter for \(\phi\) (x(t)) is \(\pi_ t(\phi)=E[\phi (x(t))| {\mathcal Y}_ t]\), where \(\phi\) is any function satisfying \(E[| \phi (x(t))|]<\infty\). If \[ L(t)=\exp [\int^{t}_{0}h(x(s))dy(s)- \frac{1}{2}\int^{t}_{0}h^ 2(x(s))ds] \] and \(P_ 0\) is the probability measure defined by \(dP_ 0/dP=L^{-1}(T)\) the optimum filter can be expressed as \(\sigma_ t(\phi)/\sigma_ t(1)\) where \(\sigma_ t(\phi)=E_ 0[\phi (x(t))L(t)| {\mathcal Y}_ t]\). Here \(E_ 0\) denotes expectation with respect to \(P_ 0\). The Zakai equation for \(\sigma_ t(\phi)\) is: \(d\sigma_ t(\phi)=\sigma_ t(A\phi)dt+\sigma_ t(h\phi)dy(t)\), \(\sigma_ 0(\phi)=\pi_ 0(\phi)\). The usual derivation of the Zakai equation uses the identity \(\sigma_ t(\phi)=\pi_ t(\phi)\sigma_ t(1)\) and the Ito product rule. Under the ''finite energy'' condition: \(E\int^{T}_{0}h^ 2(x(s))ds<\infty\), the author proves that both L(T) and \(\sigma_ t(1)\) are \(H^ 1(P_ 0)\) martingales, i.e., that \(E_ 0[\sup_{0\leq t<T}L(t)<\infty]\), and using this property gives a direct derivation of the Zakai equation.