Abstract nonlinear filtering theory in the presence of fractional Brownian motion (Q1578603)

The authors consider stochastic differential equations of the form \[ X^l_t = x^l_0 + \sum_{i=1}^M \int_0^t K_{H_i} (t,s) b^{l,i}(X_s) ds + \int_0^t K_{H_i} (t,s) a^{l,i} (X_s) dB^i_s, \] \[ Y^k_t = \int_0^t K_H(t,s)h^k(X_s) ds + \sum_{j=M+1}^d \int_0^t K_H (t,s)\tau^{k,j}(Y_s) dB_s^j, \] where \((B_s^j)_{j=1,\dots ,d}\) is a standard \(d\)-dimensional Brownian motion and \(K_{H_j}(t,s)\) is the kernel that makes \(t\mapsto \int_0^t K_{H_j}(t,s)dB_s^j\) a fractional Brownian motion of Hurst parameter \(H_j\geq 1/2\). The associated filtering problem, i.e. the computation of the normalized filter \(\pi_t = E[f(X_t)\mid Y_s, \;s\leq t]\) is considered. Using a Girsanov transform and an Itô type change of variable formula for processes such as \((X_t)_{t\in R}\), a formula which is analogous to the Zakai equation is derived for the unnormalized filter \(\sigma_t\). However this formula is not a closed equation. In order to tackle this problem the authors suggest an infinite-dimensional approach, by considering \[ X^l_r(\cdot) = x^l_0 + \sum_{i=1}^M \int_0^r K_H (\cdot ,s) b^{l,i}(X_s) ds + \int_0^r K_H (\cdot ,s) a^{l,i} (X_s) dB^i_s \] as an infinite-dimensional process indexed by \(r\). Existence and uniqueness results for the solution process \((X_t)\), as well as the change of variable formula, are proved in an appendix.