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, d S = μ S d t + σ S d Z, with μ and σ constants. Derive the stochastic differential equation satisfied by y = 2 S , 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 C 2 then
f ( S t ) = f ( S 0 ) + 0 t f ( S t ) d S t + 1 2 0 t f ( S t ) d [ S ] t
Step 2
And so considering the differential form:
d f ( S t ) = f ( S t ) d S t + 1 2 f ( S t ) d [ S ] t
Where of course [ S ] t denotes the quadratic variation of S. In your case: [ S ] t = 0 t ( σ S t ) 2 d t assuming Z t is a BM.
You can apply this formula to f ( x ) = 2 x , f ( x ) = x 2 , f ( x ) = e x , . .
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