# Deriving Stochastic differential equations. Assume S follows the geometric Brownian motion dynamics, dS = muSdt + sigma SdZ, with mu and sigma constants. Derive the stochastic differential equation satisfied by y = 2S, y = S^2, y=e^S

gobeurzb 2022-09-25 Answered
Deriving Stochastic differential equations. I am having a difficulty in deriving stochastic differential equations from geometric Brownian motion dynamics.
Assume S follows the geometric Brownian motion dynamics, $dS=\mu Sdt+\sigma SdZ$, with $\mu$ and $\sigma$ constants. Derive the stochastic differential equation satisfied by $y=2S,y={S}^{2},y={e}^{S}$
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## Answers (1)

Simeon Hester
Answered 2022-09-26 Author has 16 answers
Step 1
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
$f\left({S}_{t}\right)=f\left({S}_{0}\right)+{\int }_{0}^{t}{f}^{\prime }\left({S}_{t}\right)d{S}_{t}+\frac{1}{2}{\int }_{0}^{t}{f}^{″}\left({S}_{t}\right)d\left[S{\right]}_{t}$
Step 2
And so considering the differential form:
$df\left({S}_{t}\right)={f}^{\prime }\left({S}_{t}\right)d{S}_{t}+\frac{1}{2}{f}^{″}\left({S}_{t}\right)d\left[S{\right]}_{t}$
Where of course $\left[S{\right]}_{t}$ denotes the quadratic variation of S. In your case: $\left[S{\right]}_{t}={\int }_{0}^{t}\left(\sigma {S}_{t}{\right)}^{2}dt$ assuming ${Z}_{t}$ is a BM.
You can apply this formula to $f\left(x\right)=2x,f\left(x\right)=x2,f\left(x\right)=ex,..$
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