i have a rather complicated problem for a process which is connected to a kalman-bucy filter with a riccati equation. More precisely, let $dX_t =A(X_t)dt + R_1^{1/2} dWt, dY_=BX_tdt + R_2^{1/2}dV_t $ be the signal eqations,i.e. the filtering problem and $d\hat{X}_t = A(\hat{X}_t) dt + P_t B^T R_2^{-1} \left[dY_t -B\hat{X}_t dt\right] \textrm{where } \hat{X}_0=\mathbb{E}[X_0] \\ \partial P_t = \partial A(\hat{X}_t)P_t+P_t\partial A(\hat{X}_t)+ R_1 - P_tSP_t \textrm{where }S:=B^TR_2^{-1}B$ be the Kalman Bucy filter. the second equation is a riccati equation. then there is the estimated distance between the filter $\hat{X}_t $ and the signal $X_t$ which by a taylor expansion becomes
$ d\tilde{X}_t:= [\partial A (\hat{X}_t)-P_t S]\hat{X}_tdt + R_1^{1/2}dW_t -P_tB^T R_2^{-1/2} dV_t. $ Now, $P_t=\mathbb{E}[\tilde{X}_t \tilde{X}_t^T| \mathcal{F}_t].$ solves the riccati equation. In the above situation, $V_t, W_t$ are independant (m+n)(finite)- dimensional brownian motions, $A$ is the vectorvalued, differentiable drift of the signal, $\partial A$ its jacobian. R_1,R_2 are positive definite and symmetric dispersion matrices withe dimensions $n\times n, m\times m$ respectively, $B$ is $m \times n$.
And now the process in question can be defined via $dM_t = (P_t-\check{P}_t)B^T R_2^{-1/2} dV_t. $. I have tried to show that this is a locale martingale, but the variables are so entangled and codependant, that i am not able to find any conlusive representation. This is a problem from a course i am taking and the literature claims that this process is a martingale without any other explanation. Can anybody help?