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Suppose I have signal process $\lambda_t$ following the dynamics \begin{equation} \begin{aligned} \zeta_t&=\mu^{\zeta}(t,{\zeta}_t)dt+\sigma^{\zeta}(t,{\zeta}_t)dW^{\zeta}_t\\ \xi_t&=\mu^{\xi}(t,\xi_t)dt+\sigma^{\xi}(t,{\xi}_t)dW^{\xi}_t\\ \frac{\partial p_t}{\partial t}&= A p_t\\ \end{aligned} \end{equation} where $p_t^1\triangleq \mathbb{P}\left(\lambda_t={\zeta}\right)$, and $p_t^2\triangleq \mathbb{P}\left(\lambda_t={\xi}\right)$, are the states of the continuous-time finite state Markov process $\lambda_t$ (with state space $\mathbb{S}\triangleq \{\zeta_t,\xi_t\}$).

Moreover, my observation process follows the dynamics $$ dY_t = a(t,\lambda_t,Y_t)dt + b(t,\lambda_t,Y_t)d\tilde{W}_t, $$ where $\tilde{W}_t$ and $W_t$ are correlated Brownian motions. I know there are explicit solutions to the "The Kushner–Stratonovich Equation" for the general semi-martingale case (see citation below) as well as the particular KS Equation for finite-state continous-time Markov process. My question is how can I (is it known) how to combine these to obtain the explicit KS equation for the filtering problem I have stated. (Can I just combine the general KS solutions for both with the KS equations for the Markov process goverened by $A$?

Cohen, Samuel N.; Elliott, Robert J., Stochastic calculus and applications, Probability and Its Applications. New York, NY: Birkhäuser/Springer (ISBN 978-1-4939-2866-8/hbk; 978-1-4939-2867-5/ebook). xxiii, 666 p. (2015). ZBL1338.60001.)

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I'm only going to answer the special case $b\equiv 1$ and $\bar W$ independent from the signal noise because I'm not familiar enough with the case of multiplicative/correlated observation noise. However, as far as I have been able to glean from the literature, adding those generalizations is pretty straightforward, if a little technical.

With this restriction, quite generally, suppose that our Markov signal process is called $X_t$ with an infinitesimal generator $\mathcal{A}$, and our observations are given by

$$dY_t=a(t,X_t,Y_t)dt+d\bar W_t, \quad Y_0=0$$

We can find a reference measure $\tilde P$ that is absolutely continuous to the original measure $P$ such that

$$L_t\doteq\frac{dP}{d\tilde P}\Bigg\vert_{\mathcal{F}_t}=\exp\left[\int_0^ta(s,X_s,Y_s)dY_s-\frac{1}{2}\int_0^ta(s,X_s,Y_s)^2ds\right]$$

and $Y_t$ is a Wiener process under $\tilde P$. We may then write down a Zakai equation for the conditional expectation $\rho_t[\varphi]=\mathbb{\tilde E}[L_t\varphi(X_t)|\mathcal{F}^Y_t]$:

$$d\rho_t[\varphi]=\rho_t[\mathcal{A}\varphi]dt+\rho_t[a\varphi]dY_t,$$

where $\varphi$ is a measurable function with $\mathbb{E}[\varphi(X_t)]<\infty$ that is in the domain of $\mathcal{A}$.

To get back to the conditional expectations that you need for filtering, you use the Kallianpur-Striebel formula

$$\mathbb{E}[\varphi(X_t)|\mathcal{F}^Y_t]=\frac{\rho_t[\varphi]}{\rho_t[1]}.$$

The expectation $\mathbb{E}[\varphi(X_t)|\mathcal{F}^Y_t]$ satisfies a Kushner-Stratonovich equation which you can obtain from the Zakai equation (if required) by means of stochastic calculus.


Let's go back to the specific example that you provide. I have to modify the notation a little because $\lambda_t$ as you defined it is not Markov. Let's call the signal process $X_t$, given by

$$X_t=\begin{pmatrix}\zeta_t\\\xi_t\\\alpha_t\end{pmatrix}\in \mathbb{R}\times\mathbb{R}\times\{1,2\},$$

so the process that you want to filter is now given by $\lambda_t=\psi(X_t)$, where

$$\psi(\zeta,\xi,1)=\zeta, \quad \psi(\zeta,\xi,2)=\xi.$$

The infinitesimal generator of $X_t$ (which is now Markov) is given by (please verify)

\begin{multline} \mathcal{A}\varphi(\zeta,\xi,\alpha)=\mu^{\zeta}(t,\zeta)\partial_{\zeta}\varphi(\zeta,\xi,\alpha)+\mu^{\xi}(t,\xi)\partial_{\xi}\varphi(\zeta,\xi,\alpha)\\ +\frac{1}{2}\sigma^{\zeta}(t,\zeta)^2\partial^2_{\zeta}\varphi(\zeta,\xi,\alpha) +\frac{1}{2}\sigma^{\xi}(t,\xi)^2\partial^2_{\xi}\varphi(\zeta,\xi,\alpha)\\ +A^{\ast}_{\alpha,1}\varphi(\zeta,\xi,1)+A^{\ast}_{\alpha,2}\varphi(\zeta,\xi,2), \end{multline}

where $A^{\ast}$ is the transpose of the transition rate matrix $A$. From here on, you should be able to use the standard machinery in order to try to find approximate solutions to the Zakai equation.

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