I have a signal collected over time such that $X_t=\{X_1,X_2,...,X_N\}$. For the following system of stochastic differential equations,
$dX_t=-\lambda\left(X_t-V_t\right)dt+\sigma_1dW_t^1$
$dV_t=-\kappa V_t dt + \sigma_2 dW_t^2$
where $W^1_t$ and $W^2_t$ are two different Wiener process. How can I find $(\lambda,\kappa,\sigma_1,\sigma_2)$ by only observing $X_t=\{X_1,X_2,...,X_N\}$.
Besides that, I have found the solution of $X_t$ and $V_t$ as follows.
$X_t=e^{-\lambda t}X_0+\frac{\lambda}{\kappa-\lambda}(e^{-\lambda t}-e^{-\kappa t})V_0+\sigma_1 \int^{t}_0e^{-\lambda(t-s)}dW^1_s+\frac{\lambda}{\kappa-\lambda}\left[ \sigma_2 \int^{t}_0e^{-\lambda(t-s)}dW^2_s - \sigma_2 \int^{t}_0e^{-\kappa(t-s)}dW^2_s\right]$
$V_t=e^{-\kappa t}V_0+\sigma_2\int_{0}^{t}e^{-\kappa(t-s)}dW^2_s$
I really appreciate your help. Thanks for everyone.
$X_{t+\Delta t}=X_t-\lambda X_t \Delta t + \lambda (V_0 e^{-\kappa t} +\int_0^{t}e^{-\kappa (t-s)}dW^2_s)\Delta t + \sigma_1 (\Delta t)^{1/2}N^1(0,1)$
is reached after discretization and substitution of $V_t$. Yet it is still impossible to approximate $V_t=V_0 e^{-\kappa t} +\int_0^{t}e^{-\kappa (t-s)}dW^2_s$ since the process $V_t$ is not observed which is a similar problem to estimation of volatility process in Heston Model.
This issue solved in Atiya, A. F., & Wall, S. (2009). An analytic approximation of the likelihood function for the Heston model volatility estimation problem. Quantitative Finance, 9(3), 289–296. doi:10.1080/14697680802595601 through some assumptions.
After overcoming the issues about $V_t$, the likelihood function
$\prod_{i=1}^N p(X_{i+1}\mid X_i,\{\mu,\kappa,\sigma_1,\sigma_2\})$
can be maximized to determine the parameters.