Ornstein-Uhlenbeck equation

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September 30th 2009, 12:40 PM
elenaas

Ornstein-Uhlenbeck equation

I find out the solution of Ornstein-Uhlenbeck equation http://www.mymathforum.com/cgi-bin/m...0bB_%7Bt%7D%5C is
http://www.mymathforum.com/cgi-bin/m...DdB_%7Bs%7D%5C

My problem is how to find http://www.mymathforum.com/cgi-bin/m...5Cright%20]%5C and http://www.mymathforum.com/cgi-bin/m...5Cright%20]%5C

It seems like it is easy to find the mean and variance.But I was trying and got no solution http://www.mymathforum.com/images/sm...on_redface.gif
September 30th 2009, 11:19 PM
Laurent

Quote:

Originally Posted by elenaas

I find out the solution of Ornstein-Uhlenbeck equation http://www.mymathforum.com/cgi-bin/m...0bB_%7Bt%7D%5C is
http://www.mymathforum.com/cgi-bin/m...DdB_%7Bs%7D%5C

My problem is how to find http://www.mymathforum.com/cgi-bin/m...5Cright%20]%5C and http://www.mymathforum.com/cgi-bin/m...5Cright%20]%5C

It seems like it is easy to find the mean and variance.But I was trying and got no solution http://www.mymathforum.com/images/sm...on_redface.gif

All you need to know is $E[\int_0^t f(s) dB_s]=0$ , because this integral is a martingale (as a function of $t$ ), and $E[\left(\int_0^t f(s) dB_s\right)^2]=\int_0^t f(s)^2 ds$ , as follows from the definition of the stochastic integral.
October 1st 2009, 10:47 AM
elenaas

I suppose that I got the correct solution, but I would like you to confirm me.
I started with an explicit solution of the SDE
http://alt1.mathlinks.ro/latexrender...a974e70b86.gif

http://alt1.mathlinks.ro/latexrender...05a04dad97.gif

http://alt2.mathlinks.ro/latexrender...a35b4b6510.gif because this integral is an Ito integral and it is martingale
So the mean is: http://alt1.mathlinks.ro/latexrender...32f5c0ff36.gif

and for the variance I got:
http://alt1.mathlinks.ro/latexrender...fd60186e63.gif

I used that http://alt2.mathlinks.ro/latexrender...97b25e362d.gif
October 4th 2009, 02:09 AM
Laurent

Quote:

Originally Posted by elenaas

I suppose that I got the correct solution, but I would like you to confirm me.
I started with an explicit solution of the SDE
http://alt1.mathlinks.ro/latexrender...a974e70b86.gif

http://alt1.mathlinks.ro/latexrender...05a04dad97.gif

http://alt2.mathlinks.ro/latexrender...a35b4b6510.gif because this integral is an Ito integral and it is martingale
So the mean is: http://alt1.mathlinks.ro/latexrender...32f5c0ff36.gif

and for the variance I got:
http://alt1.mathlinks.ro/latexrender...fd60186e63.gif

I used that http://alt2.mathlinks.ro/latexrender...97b25e362d.gif

It depends on what is $X_0$ . If it is constant, then you are correct (and $E[X_t]=X_0e^{\mu t}$ ). If it is a random variable, then it must appear in the variance: $X_t-E[X_t]=(X_0-E[X_0])e^{\mu t} +\int_0^t (\cdots)$ and both terms are independent and centered hence $E[(X_t-E[X_t])^2]={\rm Var}(X_0)e^{2\mu t}+E[(\int_0^t(\cdots))^2]$ = ${\rm Var}(X_0)e^{2\mu t}+\frac{\sigma^2}{2\mu}(e^{2\mu t}-1)$ . I think.