sampling from transition equation (state space model)

kasraa

Hi.

Consider the following state space equation
[tex]x_k = f(x_{k-1},v_k)[/tex]
where f(.) is a nonlinear function and v_k is a white Gaussian noise (independent of state).

I want to generate samples according to [tex] p(x_k | x_{k-1} = x_{k-1}^i) [/tex] for a particular [tex]x_{k-1}^i[/tex]. I've seen people generate samples [tex] v_k^j [/tex] from [tex] p(v) [/tex] (which is a Gaussian) and consider [tex] x_k^{(i,j)} = f(x_{k-1}^i,v_k^j) [/tex] as samples generated according to [tex] p(x_k | x_{k-1} = x_{k-1}^i) [/tex].
I can understand the intuition behind this approach, but can you suggest any mathematical proof?

Thanks,
K.