$lienar Filtering Problem
$$X_n^1 = X_{n-1}^1 + \epsilon_n *W_n $$ $$X_n^2 = (1-\alpha* \delta) X_{n-1}^2 + \beta*\delta X_n^1 $$ $$X_n^3 = X_{n-1}^3 + \delta*X_n^2$$
above is $$\approx$$
$$dX_n^1 = W_t * dN_t$$ $$dX_n^2/dt = -\alpha * X_t^2 + \beta *X_t^1 $$ $$dX_n^3/dt = X_t^2$$
Poisson $N_t$ and Wiener/Brownian $W_t$ $(\alpha = 1$, $\beta = 18$, $X_o^1$ dist. N$(30,\sigma_0)$ $X_0^2$ & $X_0^3$ dist $N(500,\sigma_o)$, $\sigma$ $\epsilon {[10,100]}$, $\epsilon_n$ iid Bernoulli with parameter $\delta = 10^{-2})$
Observation: $Y_n = X_n^3 + \delta* V_n$
Find $Law(X_0,....,X_n| (Y_0,......Y_{n-1(or)n} )$
and $p(y_o,....,y_n)$
Attempt at first question
$Prob((Y_n = X_n^3 + \delta* V_n))$ with $V_n$ i.i.d. ~ N(0,\sigma_v^2 = 100)$$ = 1/sqrt(2*100) *exp((y_n - x_n^3/(2*\delta^2\sigma^2))$ $$Potential function to select is $G_n(x^1,x^2,x^3) = exp((y_n - x_n^3/(2*\delta^2\sigma^2))$ $$
According to Del Moral's (Page 16) Result http://books.google.com.au/books?id=8LypfuG8ZLYC&pg=PA165&dq=feynman+kac+model&hl=en&sa=X&ei=igs0VLHAEoKzogS4nYLwBg&ved=0CB4Q6AEwAA#v=onepage&q=feynman%20kac%20model&f=false
$Law(X_0,....,X_n| (Y_0,......Y_n) = \prod_{1\leq k\leq n} 1/sqrt(2*100) *exp((y_k - x_k^3/(2*\delta^2\sigma^2))$
and $Law(X_0,....,X_n| (Y_0,......Y_n-1) = Q_n $(Feynman-Kac's path measure as denoted in Moral's book) $= 1/Z_n *\displaystyle\prod_{1\leq k\leq n} exp((y_k - x_k^3/(2*\delta^2\sigma^2)) * P(x_o,.....x_n)$
Don't know what P(x_o,.....x_n)
where Z_n is the normalising constant.
However, I haven't incrorporated the acceleration/speed X(2) or X(3) into the soln nor the stochastic equations. How do I make use of them to find the $Law(X_0,....,X_n| (Y_0,......Y_n)$ or $Law(X_0,....,X_n| (Y_0,......Y_n-1)$ in the first question?