Suppose I have a discrete stochastic process, in the form of $$x_{t+1} = x_t + \varepsilon_t$$ where $\varepsilon_t$ is the random noise. The caveat is by examining the existing data, $\varepsilon_t$ does not follow a normal distribution. Suppose it follows a Cauchy distribution or more well-behavior distributions such as hyperbolic distribution. Is there any general procedure to convert this discrete stochastic process into a continuous stochastic process described by $$dx = \mu(x, t) \, dt + \sigma(x, t) \, dW$$ where $\mu$ is the drift, $\sigma$ is the square root of variance of the noise, and $dW$ is Wiener process?

## 1 Answer

You would first need to check whether Donsker's applies to your process https://math.stackexchange.com/questions/274685/joint-convergence-and-donskers-theorem. If not, it could be that the process has some different limiting process $L$ eg. some Levy process https://en.wikipedia.org/wiki/Cauchy_process and so you will need to study the stochastic integral wrt to that process.