Filtering of some nonlinear diffusions satisfying the general Beneš condition (Q1117891)

Under some regularity assumptions and the following generalization of the well-known condition of \textit{V. E. Beneš} [Stochastics 5, 65-92 (1981; Zbl 0458.60030)]: \(F_ t+g^ 2(t)[F_{zz}+F^ 2_ z]+[\beta (t)+\alpha (t)z]F_ z=A(t)z^ 2+B(t)z+C(t)\), where \(F(t,z)=g^{- 2}(t)\int f(t,z)dz\), \(F_ t\), \(F_ z\), \(F_{zz}\) are partial derivatives of F, we obtain explicit formulas for the unnormalized conditional density \(q_ t(z,x)\infty P\{x_ t\in dz| y_ s\), \(0\leq s\leq t\}\), where diffusion \(x_ t\) on \(R^ 1\) solves \(x_ 0=x\), \(dx_ t=[\beta (t)+\alpha (t)x_ t+f(t,s_ t)]dt+g(t)dw_{1t}\), and observation \(y_ t=\int^{t}_{0}h(s)x_ sds+\int^{t}_{0}e(s)dw_{2t}\), with \(w=(w_ 1,w_ 2)\) a two- dimensional Wiener process.