Abstract: | Under some regularity assumptions and the following generalization of the well-known Bene
condition 1]: , where F(t,z) = g−2(t)∫f(t,z)dz, Ft, Fz, Fzz, are partial derivatives of F, we obtain explicit formulas for the unnormalized conditional density qt(z, x) α Pxt ε dz| ys, 0 st, where diffusion xt on R1 solves x0 = x, dxt = β(t) + α(t)xt + f(t, xt] dt + g(t) dw1, and observation yt = ∫oth(s)xs ds + ∫ot(s) dw2t, with w = (w1, w2) a two-dimensional Wiener process. |