Sampling from transition equation (state space model)

  1. 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.
     
  2. jcsd
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