How to derive the Ornstein-Uhlenbeck Stochastic Integral Equation?
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I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-aint^t_0 X_s,ds +B_t$. B is the Brownian motion.
And its analytic solution is $X_t=e^-atint^t_0 e^as,dB_s$.
How do I prove that this is the case? I know that $e^-at=-aint^t_0 e^-as,ds$ and that I'm somehow supposed to make use of this result but I am still unable to get from the analytic solution to the integral equation.
Thanks!
stochastic-processes integration stochastic-integrals brownian-motion
asked Dec 3 '11 at 11:57
Sharon Reed
2813
add a comment |Â
up vote
5
down vote
favorite
I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-aint^t_0 X_s,ds +B_t$. B is the Brownian motion.
And its analytic solution is $X_t=e^-atint^t_0 e^as,dB_s$.
How do I prove that this is the case? I know that $e^-at=-aint^t_0 e^-as,ds$ and that I'm somehow supposed to make use of this result but I am still unable to get from the analytic solution to the integral equation.
Thanks!
stochastic-processes integration stochastic-integrals brownian-motion
asked Dec 3 '11 at 11:57
Sharon Reed
2813
If you want to check if solution is correct, why not apply Ito formula? In the case you wonder how the analytic solution was obtained from the integral equation - that's a bit more elaborate.
– Ilya
Dec 3 '11 at 13:22
Could you elaborate on how to apply Ito's formula? Sorry I'm not very good with this subject. I know that Ito formula is $f(B_t)-f(B_0)=int^t_0 f'(B_u),dB_u+1/2int^t_0 f''(B_u),du$ but I don't know what function should f(x) be.
– Sharon Reed
Dec 3 '11 at 14:42
No, its in my lecture notes. I just can't understand that bit.
– Sharon Reed
Dec 3 '11 at 15:02
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-aint^t_0 X_s,ds +B_t$. B is the Brownian motion.
And its analytic solution is $X_t=e^-atint^t_0 e^as,dB_s$.
How do I prove that this is the case? I know that $e^-at=-aint^t_0 e^-as,ds$ and that I'm somehow supposed to make use of this result but I am still unable to get from the analytic solution to the integral equation.
Thanks!
stochastic-processes integration stochastic-integrals brownian-motion
asked Dec 3 '11 at 11:57
Sharon Reed
2813
I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-aint^t_0 X_s,ds +B_t$. B is the Brownian motion.
And its analytic solution is $X_t=e^-atint^t_0 e^as,dB_s$.
How do I prove that this is the case? I know that $e^-at=-aint^t_0 e^-as,ds$ and that I'm somehow supposed to make use of this result but I am still unable to get from the analytic solution to the integral equation.
Thanks!
stochastic-processes integration stochastic-integrals brownian-motion
asked Dec 3 '11 at 11:57
Sharon Reed
2813
asked Dec 3 '11 at 11:57
Sharon Reed
2813
asked Dec 3 '11 at 11:57
Sharon Reed
2813
asked Dec 3 '11 at 11:57
Sharon Reed
2813
2813
If you want to check if solution is correct, why not apply Ito formula? In the case you wonder how the analytic solution was obtained from the integral equation - that's a bit more elaborate.
– Ilya
Dec 3 '11 at 13:22
Could you elaborate on how to apply Ito's formula? Sorry I'm not very good with this subject. I know that Ito formula is $f(B_t)-f(B_0)=int^t_0 f'(B_u),dB_u+1/2int^t_0 f''(B_u),du$ but I don't know what function should f(x) be.
– Sharon Reed
Dec 3 '11 at 14:42
No, its in my lecture notes. I just can't understand that bit.
– Sharon Reed
Dec 3 '11 at 15:02
add a comment |Â
If you want to check if solution is correct, why not apply Ito formula? In the case you wonder how the analytic solution was obtained from the integral equation - that's a bit more elaborate.
– Ilya
Dec 3 '11 at 13:22
Could you elaborate on how to apply Ito's formula? Sorry I'm not very good with this subject. I know that Ito formula is $f(B_t)-f(B_0)=int^t_0 f'(B_u),dB_u+1/2int^t_0 f''(B_u),du$ but I don't know what function should f(x) be.
– Sharon Reed
Dec 3 '11 at 14:42
No, its in my lecture notes. I just can't understand that bit.
– Sharon Reed
Dec 3 '11 at 15:02
If you want to check if solution is correct, why not apply Ito formula? In the case you wonder how the analytic solution was obtained from the integral equation - that's a bit more elaborate.
– Ilya
Dec 3 '11 at 13:22
If you want to check if solution is correct, why not apply Ito formula? In the case you wonder how the analytic solution was obtained from the integral equation - that's a bit more elaborate.
– Ilya
Dec 3 '11 at 13:22
Could you elaborate on how to apply Ito's formula? Sorry I'm not very good with this subject. I know that Ito formula is $f(B_t)-f(B_0)=int^t_0 f'(B_u),dB_u+1/2int^t_0 f''(B_u),du$ but I don't know what function should f(x) be.
– Sharon Reed
Dec 3 '11 at 14:42
Could you elaborate on how to apply Ito's formula? Sorry I'm not very good with this subject. I know that Ito formula is $f(B_t)-f(B_0)=int^t_0 f'(B_u),dB_u+1/2int^t_0 f''(B_u),du$ but I don't know what function should f(x) be.
– Sharon Reed
Dec 3 '11 at 14:42
No, its in my lecture notes. I just can't understand that bit.
– Sharon Reed
Dec 3 '11 at 15:02
No, its in my lecture notes. I just can't understand that bit.
– Sharon Reed
Dec 3 '11 at 15:02
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
11
down vote
accepted
The Ito formula, for $mathrmd X_t = - a X_t mathrmd t + mathrmd B_t$, you need is:
$$
mathrmdleft( f(t, X_t) right) =
left( partial_t f(t,X_t) - a X_t partial_x f(t,X_t) + frac12 partial_xx f(t,X_t) right) mathrmd t +
left( partial_x f(t,X_t) right) mathrmd B_t
$$
HINT: use $f(t,X_t) = mathrme^a t X_t$.
Edit 9/8/2018: Added $X_t$ to formula
edited Aug 9 at 23:12
Community♦
1
answered Dec 3 '11 at 15:02
Sasha
59.7k5106178
Okay thanks! I'll go work on it.
– Sharon Reed
Dec 3 '11 at 15:07
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
11
down vote
accepted
The Ito formula, for $mathrmd X_t = - a X_t mathrmd t + mathrmd B_t$, you need is:
$$
mathrmdleft( f(t, X_t) right) =
left( partial_t f(t,X_t) - a X_t partial_x f(t,X_t) + frac12 partial_xx f(t,X_t) right) mathrmd t +
left( partial_x f(t,X_t) right) mathrmd B_t
$$
HINT: use $f(t,X_t) = mathrme^a t X_t$.
Edit 9/8/2018: Added $X_t$ to formula
edited Aug 9 at 23:12
Community♦
1
answered Dec 3 '11 at 15:02
Sasha
59.7k5106178
Okay thanks! I'll go work on it.
– Sharon Reed
Dec 3 '11 at 15:07
add a comment |Â
up vote
11
down vote
accepted
The Ito formula, for $mathrmd X_t = - a X_t mathrmd t + mathrmd B_t$, you need is:
$$
mathrmdleft( f(t, X_t) right) =
left( partial_t f(t,X_t) - a X_t partial_x f(t,X_t) + frac12 partial_xx f(t,X_t) right) mathrmd t +
left( partial_x f(t,X_t) right) mathrmd B_t
$$
HINT: use $f(t,X_t) = mathrme^a t X_t$.
Edit 9/8/2018: Added $X_t$ to formula
edited Aug 9 at 23:12
Community♦
1
answered Dec 3 '11 at 15:02
Sasha
59.7k5106178
Okay thanks! I'll go work on it.
– Sharon Reed
Dec 3 '11 at 15:07
add a comment |Â
up vote
11
down vote
accepted
up vote
11
down vote
accepted
The Ito formula, for $mathrmd X_t = - a X_t mathrmd t + mathrmd B_t$, you need is:
$$
mathrmdleft( f(t, X_t) right) =
left( partial_t f(t,X_t) - a X_t partial_x f(t,X_t) + frac12 partial_xx f(t,X_t) right) mathrmd t +
left( partial_x f(t,X_t) right) mathrmd B_t
$$
HINT: use $f(t,X_t) = mathrme^a t X_t$.
Edit 9/8/2018: Added $X_t$ to formula
edited Aug 9 at 23:12
Community♦
1
answered Dec 3 '11 at 15:02
Sasha
59.7k5106178
The Ito formula, for $mathrmd X_t = - a X_t mathrmd t + mathrmd B_t$, you need is:
$$
mathrmdleft( f(t, X_t) right) =
left( partial_t f(t,X_t) - a X_t partial_x f(t,X_t) + frac12 partial_xx f(t,X_t) right) mathrmd t +
left( partial_x f(t,X_t) right) mathrmd B_t
$$
HINT: use $f(t,X_t) = mathrme^a t X_t$.
Edit 9/8/2018: Added $X_t$ to formula
edited Aug 9 at 23:12
Community♦
1
answered Dec 3 '11 at 15:02
Sasha
59.7k5106178
edited Aug 9 at 23:12
Community♦
1
edited Aug 9 at 23:12
Community♦
1
edited Aug 9 at 23:12
Community♦
1
1
answered Dec 3 '11 at 15:02
Sasha
59.7k5106178
answered Dec 3 '11 at 15:02
Sasha
59.7k5106178
answered Dec 3 '11 at 15:02
Sasha
59.7k5106178
59.7k5106178
Okay thanks! I'll go work on it.
– Sharon Reed
Dec 3 '11 at 15:07
add a comment |Â
Okay thanks! I'll go work on it.
– Sharon Reed
Dec 3 '11 at 15:07
Okay thanks! I'll go work on it.
– Sharon Reed
Dec 3 '11 at 15:07
Okay thanks! I'll go work on it.
– Sharon Reed
Dec 3 '11 at 15:07
add a comment |Â
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If you want to check if solution is correct, why not apply Ito formula? In the case you wonder how the analytic solution was obtained from the integral equation - that's a bit more elaborate.
– Ilya
Dec 3 '11 at 13:22
Could you elaborate on how to apply Ito's formula? Sorry I'm not very good with this subject. I know that Ito formula is $f(B_t)-f(B_0)=int^t_0 f'(B_u),dB_u+1/2int^t_0 f''(B_u),du$ but I don't know what function should f(x) be.
– Sharon Reed
Dec 3 '11 at 14:42
No, its in my lecture notes. I just can't understand that bit.
– Sharon Reed
Dec 3 '11 at 15:02