How would one solve for a process where a stochastic random variable is divided by a deterministic random...












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I'm currently working on a problem where I am trying to model a stochastic process where a stochastic random variable is divided by a deterministic random variable following a normal distribution. Its been a while wince I had a look at stochastic calc so apologies in advance if my math is a bit rusty!



Consider a simple stochastic process with no drift that satisfies the following SDE



Y(t) = X(t)



dX(t) = $sigma$dB(t) where B(t) is Brownian motion and B(t) ~ N($0$,t).



Solving such a process would give me Y(t) = Y$_0$ + $sigma$B(t)



I'm currently trying to find out what the resultant stochastic process would be if Y(t) = $frac{X(t)}{Z}$ where Z is a Normal distribution with mean $mu_z$ and variance $sigma_z^2$ i.e Z ~ N($mu_z$,$sigma_z^2$).



How would one apply Ito's lemma in such a case?










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  • $begingroup$
    You already know that $X(t) = X(0) + sigma B(t)$ so $Y(t) = frac{X(0)}{Z} + frac{sigma}{Z} B(t)$.
    $endgroup$
    – Rhys Steele
    Jan 9 at 15:04












  • $begingroup$
    Perfect! That makes total sense! Cheers :)
    $endgroup$
    – Hamza Juzer
    Jan 10 at 22:07
















1












$begingroup$


I'm currently working on a problem where I am trying to model a stochastic process where a stochastic random variable is divided by a deterministic random variable following a normal distribution. Its been a while wince I had a look at stochastic calc so apologies in advance if my math is a bit rusty!



Consider a simple stochastic process with no drift that satisfies the following SDE



Y(t) = X(t)



dX(t) = $sigma$dB(t) where B(t) is Brownian motion and B(t) ~ N($0$,t).



Solving such a process would give me Y(t) = Y$_0$ + $sigma$B(t)



I'm currently trying to find out what the resultant stochastic process would be if Y(t) = $frac{X(t)}{Z}$ where Z is a Normal distribution with mean $mu_z$ and variance $sigma_z^2$ i.e Z ~ N($mu_z$,$sigma_z^2$).



How would one apply Ito's lemma in such a case?










share|cite|improve this question









$endgroup$












  • $begingroup$
    You already know that $X(t) = X(0) + sigma B(t)$ so $Y(t) = frac{X(0)}{Z} + frac{sigma}{Z} B(t)$.
    $endgroup$
    – Rhys Steele
    Jan 9 at 15:04












  • $begingroup$
    Perfect! That makes total sense! Cheers :)
    $endgroup$
    – Hamza Juzer
    Jan 10 at 22:07














1












1








1


1



$begingroup$


I'm currently working on a problem where I am trying to model a stochastic process where a stochastic random variable is divided by a deterministic random variable following a normal distribution. Its been a while wince I had a look at stochastic calc so apologies in advance if my math is a bit rusty!



Consider a simple stochastic process with no drift that satisfies the following SDE



Y(t) = X(t)



dX(t) = $sigma$dB(t) where B(t) is Brownian motion and B(t) ~ N($0$,t).



Solving such a process would give me Y(t) = Y$_0$ + $sigma$B(t)



I'm currently trying to find out what the resultant stochastic process would be if Y(t) = $frac{X(t)}{Z}$ where Z is a Normal distribution with mean $mu_z$ and variance $sigma_z^2$ i.e Z ~ N($mu_z$,$sigma_z^2$).



How would one apply Ito's lemma in such a case?










share|cite|improve this question









$endgroup$




I'm currently working on a problem where I am trying to model a stochastic process where a stochastic random variable is divided by a deterministic random variable following a normal distribution. Its been a while wince I had a look at stochastic calc so apologies in advance if my math is a bit rusty!



Consider a simple stochastic process with no drift that satisfies the following SDE



Y(t) = X(t)



dX(t) = $sigma$dB(t) where B(t) is Brownian motion and B(t) ~ N($0$,t).



Solving such a process would give me Y(t) = Y$_0$ + $sigma$B(t)



I'm currently trying to find out what the resultant stochastic process would be if Y(t) = $frac{X(t)}{Z}$ where Z is a Normal distribution with mean $mu_z$ and variance $sigma_z^2$ i.e Z ~ N($mu_z$,$sigma_z^2$).



How would one apply Ito's lemma in such a case?







stochastic-processes random-variables normal-distribution stochastic-calculus stochastic-integrals






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 9 at 12:11









Hamza JuzerHamza Juzer

61




61












  • $begingroup$
    You already know that $X(t) = X(0) + sigma B(t)$ so $Y(t) = frac{X(0)}{Z} + frac{sigma}{Z} B(t)$.
    $endgroup$
    – Rhys Steele
    Jan 9 at 15:04












  • $begingroup$
    Perfect! That makes total sense! Cheers :)
    $endgroup$
    – Hamza Juzer
    Jan 10 at 22:07


















  • $begingroup$
    You already know that $X(t) = X(0) + sigma B(t)$ so $Y(t) = frac{X(0)}{Z} + frac{sigma}{Z} B(t)$.
    $endgroup$
    – Rhys Steele
    Jan 9 at 15:04












  • $begingroup$
    Perfect! That makes total sense! Cheers :)
    $endgroup$
    – Hamza Juzer
    Jan 10 at 22:07
















$begingroup$
You already know that $X(t) = X(0) + sigma B(t)$ so $Y(t) = frac{X(0)}{Z} + frac{sigma}{Z} B(t)$.
$endgroup$
– Rhys Steele
Jan 9 at 15:04






$begingroup$
You already know that $X(t) = X(0) + sigma B(t)$ so $Y(t) = frac{X(0)}{Z} + frac{sigma}{Z} B(t)$.
$endgroup$
– Rhys Steele
Jan 9 at 15:04














$begingroup$
Perfect! That makes total sense! Cheers :)
$endgroup$
– Hamza Juzer
Jan 10 at 22:07




$begingroup$
Perfect! That makes total sense! Cheers :)
$endgroup$
– Hamza Juzer
Jan 10 at 22:07










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