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Given a progressively measurable processes $Delta_s(omega),Delta'_s(omega)$ and real numbers $z',z$, there was a claim that if $$int_0^T(Delta_s-Delta's)mathrm{d}X_t=z-z'$$ for a non-trivial local martingale $X$, then $Delta_s=Delta'_s$. (All equal is meant to be in almost surely sense). I don't see why it is follows. Any hint are appreciated.

probability stochastic-processes stochastic-calculus

share|cite|improve this question

asked Mar 14 at 20:10

quallenjägerquallenjäger

444519

$endgroup$

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  $begingroup$
  Do you assume that the equation holds for all $T>0$ or just for some fixed $T>0$...?The counterexamples here seem to suggest that assertion is, in general, wrong for fixed $T$ (... but perhaps I'm missing something).
  $endgroup$
  – saz
  Mar 14 at 20:49
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @saz Would it work for all t? Or with some more additional assumptions?
  $endgroup$
  – quallenjäger
  Mar 14 at 21:11
  

  
  
  
  


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  $begingroup$
  Well, under some additional assumptions it holds true, yes... but it would be good to know which statement exactly you are after. It seems that you read this claim somewhere, so perhaps you can look up the precise statement?
  $endgroup$
  – saz
  Mar 15 at 15:02
  

  
  
  
  

add a comment | 

0

$begingroup$

Given a progressively measurable processes $Delta_s(omega),Delta'_s(omega)$ and real numbers $z',z$, there was a claim that if $$int_0^T(Delta_s-Delta's)mathrm{d}X_t=z-z'$$ for a non-trivial local martingale $X$, then $Delta_s=Delta'_s$. (All equal is meant to be in almost surely sense). I don't see why it is follows. Any hint are appreciated.

probability stochastic-processes stochastic-calculus

share|cite|improve this question

asked Mar 14 at 20:10

quallenjägerquallenjäger

444519

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you assume that the equation holds for all $T>0$ or just for some fixed $T>0$...?The counterexamples here seem to suggest that assertion is, in general, wrong for fixed $T$ (... but perhaps I'm missing something).
  $endgroup$
  – saz
  Mar 14 at 20:49
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @saz Would it work for all t? Or with some more additional assumptions?
  $endgroup$
  – quallenjäger
  Mar 14 at 21:11
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Well, under some additional assumptions it holds true, yes... but it would be good to know which statement exactly you are after. It seems that you read this claim somewhere, so perhaps you can look up the precise statement?
  $endgroup$
  – saz
  Mar 15 at 15:02
  

  
  
  
  

add a comment | 

$begingroup$

Given a progressively measurable processes $Delta_s(omega),Delta'_s(omega)$ and real numbers $z',z$, there was a claim that if $$int_0^T(Delta_s-Delta's)mathrm{d}X_t=z-z'$$ for a non-trivial local martingale $X$, then $Delta_s=Delta'_s$. (All equal is meant to be in almost surely sense). I don't see why it is follows. Any hint are appreciated.

probability stochastic-processes stochastic-calculus

share|cite|improve this question

asked Mar 14 at 20:10

quallenjägerquallenjäger

444519

$endgroup$

share|cite|improve this question

asked Mar 14 at 20:10

quallenjägerquallenjäger

444519

share|cite|improve this question

asked Mar 14 at 20:10

quallenjägerquallenjäger

444519

asked Mar 14 at 20:10

quallenjägerquallenjäger

444519

asked Mar 14 at 20:10

quallenjägerquallenjäger

444519