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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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

Simeon Hester · Accepted Answer

Step 1It seems like you want us to do you homework, here I will explain you how to do it.The key is to do Ito&#039;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 2And 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  ,  .  .

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

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