If I am modelling my returns as $\sim N(0, \sigma^2)$, then I can evolve my spot distribution as: $$S_{t} = S_{0}e^{(\mu - \frac{1}{2}\sigma^{2})t + \sigma dW_{t}}$$ where $S_{0}$ is the spot, $\mu$ is the mean , and $\sigma$ is the returns volatility and $dW_{t}$ is the gaussian noise.
How should I amend my Spot (lognormal) distribution if I assuming my returns follow a student-t distribution $\sim t-dist(\nu)$
Thanks