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




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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














up vote
5
down vote

favorite
2












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




share|cite|improve this question




















  • 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












up vote
5
down vote

favorite
2









up vote
5
down vote

favorite
2






2





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




share|cite|improve this question












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





share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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
















  • 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










1 Answer
1






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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






share|cite|improve this answer






















  • Okay thanks! I'll go work on it.
    – Sharon Reed
    Dec 3 '11 at 15:07










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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






share|cite|improve this answer






















  • Okay thanks! I'll go work on it.
    – Sharon Reed
    Dec 3 '11 at 15:07














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






share|cite|improve this answer






















  • Okay thanks! I'll go work on it.
    – Sharon Reed
    Dec 3 '11 at 15:07












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






share|cite|improve this answer














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







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Aug 9 at 23:12









Community♦

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answered Dec 3 '11 at 15:02









Sasha

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  • 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




Okay thanks! I'll go work on it.
– Sharon Reed
Dec 3 '11 at 15:07












 

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