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I have the following process $$dS_t = mu S_t dt + sigma S_t dz_t $$ and the function $$f(S) = S^2$$ where

$$fracpartial fpartial t = 0, fracpartial fpartial S = 2S, fracpartial^2 fpartial S^2 = 2, df = 2SdS + frac122 cdot dSdS$$

I have that using Ito's Lemma: $$df_t = 2S_t(mu S_td_t + sigma S_tdz_t) + frac122cdot sigma S_tdz_t cdot sigma S_tdz_t = (2 mu S_t^2 + sigma^2 S_t^2)dt + 2 sigma S_t^2dz_t$$ or $$df_t = (2 mu + sigma^2)f_t dt + 2sigma f_tdz_t. $$ Can someone explain how this works for this specific example for me? I realise that $$ df = left( fracpartial fpartial t + mu fracpartial fpartial s + frac12sigma^2 fracpartial^2fpartial S^2 right) dt + sigma fracpartial fpartial sdz_t $$ but i cant understand how $$ df = 2SdS + frac122 cdot dSdS$$ was obtained from that expression? Also, i don't understand how the larger first expression for $df_t$ was constructed? Any help would be great, thanks!

pde stochastic-processes stochastic-calculus stochastic-analysis stochastic-pde

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asked Mar 16 at 21:29

HallaoHallao

12

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Which version of Ito's lemma do you know? Without this information, it's not possible to give a reasoable answer to your question. Moreover, it would be good to know where exactly you are stuck when applying Itô's lemma. Currently you just wrote up the equations which you are supposed to end up with (... at least as far as I understand) but you do not give your own thoughts/computations.
  $endgroup$
  – saz
  Mar 17 at 7:05
  
  

  
  
  
  

add a comment | 

-1

1

$begingroup$

I have the following process $$dS_t = mu S_t dt + sigma S_t dz_t $$ and the function $$f(S) = S^2$$ where

$$fracpartial fpartial t = 0, fracpartial fpartial S = 2S, fracpartial^2 fpartial S^2 = 2, df = 2SdS + frac122 cdot dSdS$$

I have that using Ito's Lemma: $$df_t = 2S_t(mu S_td_t + sigma S_tdz_t) + frac122cdot sigma S_tdz_t cdot sigma S_tdz_t = (2 mu S_t^2 + sigma^2 S_t^2)dt + 2 sigma S_t^2dz_t$$ or $$df_t = (2 mu + sigma^2)f_t dt + 2sigma f_tdz_t. $$ Can someone explain how this works for this specific example for me? I realise that $$ df = left( fracpartial fpartial t + mu fracpartial fpartial s + frac12sigma^2 fracpartial^2fpartial S^2 right) dt + sigma fracpartial fpartial sdz_t $$ but i cant understand how $$ df = 2SdS + frac122 cdot dSdS$$ was obtained from that expression? Also, i don't understand how the larger first expression for $df_t$ was constructed? Any help would be great, thanks!

pde stochastic-processes stochastic-calculus stochastic-analysis stochastic-pde

share|cite|improve this question

asked Mar 16 at 21:29

HallaoHallao

12

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Which version of Ito's lemma do you know? Without this information, it's not possible to give a reasoable answer to your question. Moreover, it would be good to know where exactly you are stuck when applying Itô's lemma. Currently you just wrote up the equations which you are supposed to end up with (... at least as far as I understand) but you do not give your own thoughts/computations.
  $endgroup$
  – saz
  Mar 17 at 7:05
  
  

  
  
  
  

add a comment | 

$begingroup$

I have the following process $$dS_t = mu S_t dt + sigma S_t dz_t $$ and the function $$f(S) = S^2$$ where

$$fracpartial fpartial t = 0, fracpartial fpartial S = 2S, fracpartial^2 fpartial S^2 = 2, df = 2SdS + frac122 cdot dSdS$$

I have that using Ito's Lemma: $$df_t = 2S_t(mu S_td_t + sigma S_tdz_t) + frac122cdot sigma S_tdz_t cdot sigma S_tdz_t = (2 mu S_t^2 + sigma^2 S_t^2)dt + 2 sigma S_t^2dz_t$$ or $$df_t = (2 mu + sigma^2)f_t dt + 2sigma f_tdz_t. $$ Can someone explain how this works for this specific example for me? I realise that $$ df = left( fracpartial fpartial t + mu fracpartial fpartial s + frac12sigma^2 fracpartial^2fpartial S^2 right) dt + sigma fracpartial fpartial sdz_t $$ but i cant understand how $$ df = 2SdS + frac122 cdot dSdS$$ was obtained from that expression? Also, i don't understand how the larger first expression for $df_t$ was constructed? Any help would be great, thanks!

pde stochastic-processes stochastic-calculus stochastic-analysis stochastic-pde

share|cite|improve this question

asked Mar 16 at 21:29

HallaoHallao

12

$endgroup$

share|cite|improve this question

asked Mar 16 at 21:29

HallaoHallao

12

share|cite|improve this question

asked Mar 16 at 21:29

HallaoHallao

12

asked Mar 16 at 21:29

HallaoHallao

12

asked Mar 16 at 21:29

HallaoHallao

12