Please check this book... I hope will help you. Good Luck

https://www.dropbox.com/s/0y4pklfhdi43hhx/Steven%20M.%20Kay-Fundamentals%20of%20Statistical%20Signal%20Processing%2C%20Volume%20I_%20Estimation%20Theory-Prentice%20Hall%20%281993%29_2.pdf?dl=0

Dear Amine,

Please have a look on the following paper. Hope it could be helpful!!

Regards,

Mohamed

Some references about using the Maximum Likelihood method for the state estimation of a system modelled by stochastic differential equations:

J. Durbin and S.J. Koopman. Monte Carlo maximum likelihood estimation for nonGaussian state space models. Biometrika, 84,3:669–684, 1997.

A.R Pedersen. A New Approach to Maximum Likelihood Estimation for Stochastic Differential Equations Based on Discrete Observations. Scandinavian Journal of Statistics, 22:55–71, 1995.

H. Singer. Simulated Maximum Likelihood in Nonlinear Continuous-Discrete State Space Models: Importance Sampling by Approximate Smoothing. Computational Statistics, 18,1:79–106, 2003.

Good Luck

The Maximum Likelihood Estimation (MLE) method in its basic form is applicable to linear stochastic model  of the following matrix form  Y= XA+ W.   

Where Y is the  system output vectoror observation, X is the system state Matrix, A is the system parameters  vector and W is the noise vector representing all the disturbances affecting the considered system.

In this type of model all variables are assumed to be Gaussian.

The MLE  for parameters estimation is  A= (XTR-1X)-1XTR-1Y.

In this relation R represent the varianace of the parameter error estimation assuming that the measuments of the state X is  available.

In the case of state estimation we assume the measurement of the parameters are available and we use the following similar model: Y=AX+W

Note that we have just inverted X and A.

I am not sure of the formulation of your problem. So, I propose 3 interpretations in the following. It will be easier to explain my point with some notations.

Let X be your state vector, Y your observation, theta your vector of parameters if there are any.

Your stochastic differential equation:

d X_t = F(t, X_t, theta) dt + G(t, X_t, theta) dBt  (Eq. 1) .

Your observation equation:

Y_t = H(t, X_t, epsilon, theta),  (Eq. 2)

with epsilon being random.

1) If you want to maximize the likelihood for a "state estimation" i.e. to estimate the state X_t at several times t knowing Y_t' at several times t'

However, this problem is trivial and I do not think this is what you want to do.

Using the Markov property of your system, in a discrete-time point of view, it is easy to show that the function to be maximized reads:

p( Y_1 , ... , Y_n | X_1, ... , X_n) = p( Y_1 | X_1) ... p( Y_n | X_n) .

It only relies on Eq. 2 (your model of observation) and not on Eq.1 (your hidden dynamics).

2) If you want to maximize the likelihood for a "parameter estimation" i.e. to estimate the vector of parameter, theta, knowing Y_t' at several times t'. 

I think Sergiy Prykhodko understood your problem that way. By the way, the papers he suggested seem interesting.

For this problem, a very powerful tools exist in stochastic calculus, even if X_t is not Gaussian. Based on Girsanov theorem, an explicit expression of the log-likelihood can be derived in a general case. You can find it for instance in:

P. Rao. Statistical inference for diffusion type processes. Arnold, 1999

3) If you want to focus on "state estimation" i.e. to estimate the state X_t at several times t knowing Y_t' at several times t' using the likelihood p(Y_t | X_t) (without dealing with parameters).

This problem, so-called filtering problem or smoothing problem, is well documented and widely used. I advice the following papers:

 A. Doucet, N. De Freitas, and N. Gordon. Sequential Monte Carlo methods in practice . Springer, 2001.

A. Doucet and A. Johansen. A tutorial on particle filtering and smoothing: Fifteen years later. Handbook of Nonlinear Filtering , 12:656–704, 2009.

Thank you all dear colleagues for your answers and advises