I have a stochastic difference equation of the form, $$ x_t = x_{t-1} + \frac{1}{t}(- x_{t-1} + \varepsilon_t) \,, $$ where $\varepsilon_t$ are iid random normal variables, which I would like to approximate with something like, $$ \frac{x_t-x_{t-1}}{\frac{1}{t}} \approx \dot x = -x + \varepsilon \,. $$

Now, I do not know how to go from the discrete stochastic process to a continuous one, i.e. what the limit continuous process would be. Some references would be greatly appreciated.



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