I am having a difficulty in deriving stochastic differential equations from geometric Brownian motion dynamics.
Assume S follows the geometric Brownian motion dynamics, dS = μSdt + σSdZ, with μ and σ constants. Derive the stochastic differential equation satisfied by y = 2S, y = S^2, y=e^S
Doing any of these examples will help me. Thanks in advance.
It seems like you want us to do you homework, here I will explain you how to do it.
The key is to do Ito's formula: If $ f \in \mathcal{C}^2$ then $$f(S_t) = f(S_0) + \int_0^t f'(S_t)dS_t + \frac{1}{2} \int_0^t f''(S_t) d[S]_t$$
And so considering the differential form: $$ df(S_t) = f'(S_t)dS_t + \frac{1}{2} f''(S_t) d[S]_t $$
Where of course $[S]_t$ denotes the quadratic variation of S. In your case: $[S]_t= \int_0^t (\sigma S_t)^2 dt $ assuming $Z_t$ is a BM.
You can apply this formula to $f(x)=2x, f(x)=x^2, f(x)=e^x,.. $
Can you finish the exercise?
