# Deriving Ito process with a drift from geometric Brownian motion.

Let the Geometric Brownian motion be: $$\frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t}$$

$$\Delta S$$ = change in stock price (s)

$$\mu$$ = expected rate of return

$$\sigma$$ = volatility of shock

$$epsilon$$ has standard normal N(0,1) distribution

$$\sigma \epsilon \sqrt{\Delta t}$$ = stochastic companion

i) Derive the Ito process with a drift for the above ii) Given that the option price at time $$t$$ is $$f(s,t)$$, derive the process with Ito's lemma. Give an example.

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• This depends very much on what level of rigour you're requiring. A 'formal' answer may be out of the scope of the course/notes you're reading. In this case the It\^o process would be the one satisfying the SDE $$dS_t = \mu S_t dt + \sigma S_t dW_t,$$ i.e. $S_t = S_0 + \int_0^t \mu S_s ds + \int_0^t \sigma S_s dW_s$. – Panda Jun 16 at 9:38