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integrate over the Ornstein-Uhlenbeck process
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I have a stochastic process $dX(t)=k[A-Y(t)]f(t)dt$, where $Y(t)$ is the Ornstein-Uhlenbeck process, and satisfies: $dY(t)=k[A-Y(t)]dt+sigma dW(t)$ and $Y(0)=0$. $f(t)=0$ for $t<tau$ and $f(t)=1$ for $tgeqtau$. Now I want to solve for $X(t)$ and $mathbbE(X(t))$, $mathbbV(X(t))$.
I could plug in $Y(t)=A(1-e^-kt)+sigmaint_0^t e^-k(t-s)dW_s$, but how shall I proceed? I am new to SDE, and any help is highly appreciated!
(I find this question related to How to solve system of stochastic differential equations?)
EDIT:
I found this question: https://mathoverflow.net/questions/143245/distribution-of-integral-of-exponential-of-wiener-process. Why does this hold? $sigma e^asint_0^s e^-asdW_s=beta W_s$ where $beta^2=fracsigma^22a(1-e^-2as)$? Actually I cannot obtain the right formula of variance for the OU process based on this equation.
Assuming this is correct, my attempt for $tau=0$: (Let me know if anything is wrong)
$Y(t)=A(1-e^-kt)+beta(t)W(t)$ where $(beta(t))^2=fracsigma^22k(1-e^-2kt)$. So $dX(t)=(kAe^-kt-kbeta(t) W_t)dt$. Thus $mathbbE(X(T))=int_0^T kAe^-ktdt$.
I am still confused in obtaining $mathbbV(X(t))$, and thanks for any help in advance!
differential-equations stochastic-processes stochastic-calculus
asked Aug 1 at 22:29
huighlh
465
add a comment |Â
up vote
1
down vote
favorite
I have a stochastic process $dX(t)=k[A-Y(t)]f(t)dt$, where $Y(t)$ is the Ornstein-Uhlenbeck process, and satisfies: $dY(t)=k[A-Y(t)]dt+sigma dW(t)$ and $Y(0)=0$. $f(t)=0$ for $t<tau$ and $f(t)=1$ for $tgeqtau$. Now I want to solve for $X(t)$ and $mathbbE(X(t))$, $mathbbV(X(t))$.
I could plug in $Y(t)=A(1-e^-kt)+sigmaint_0^t e^-k(t-s)dW_s$, but how shall I proceed? I am new to SDE, and any help is highly appreciated!
(I find this question related to How to solve system of stochastic differential equations?)
EDIT:
I found this question: https://mathoverflow.net/questions/143245/distribution-of-integral-of-exponential-of-wiener-process. Why does this hold? $sigma e^asint_0^s e^-asdW_s=beta W_s$ where $beta^2=fracsigma^22a(1-e^-2as)$? Actually I cannot obtain the right formula of variance for the OU process based on this equation.
Assuming this is correct, my attempt for $tau=0$: (Let me know if anything is wrong)
$Y(t)=A(1-e^-kt)+beta(t)W(t)$ where $(beta(t))^2=fracsigma^22k(1-e^-2kt)$. So $dX(t)=(kAe^-kt-kbeta(t) W_t)dt$. Thus $mathbbE(X(T))=int_0^T kAe^-ktdt$.
I am still confused in obtaining $mathbbV(X(t))$, and thanks for any help in advance!
differential-equations stochastic-processes stochastic-calculus
asked Aug 1 at 22:29
huighlh
465
Is $tau$ a random variable or a deterministic constant? What about $A$...?
– saz
Aug 2 at 5:32
$tau$ and $A$ are constants. Thanks! @saz
– huighlh
Aug 2 at 14:04
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have a stochastic process $dX(t)=k[A-Y(t)]f(t)dt$, where $Y(t)$ is the Ornstein-Uhlenbeck process, and satisfies: $dY(t)=k[A-Y(t)]dt+sigma dW(t)$ and $Y(0)=0$. $f(t)=0$ for $t<tau$ and $f(t)=1$ for $tgeqtau$. Now I want to solve for $X(t)$ and $mathbbE(X(t))$, $mathbbV(X(t))$.
I could plug in $Y(t)=A(1-e^-kt)+sigmaint_0^t e^-k(t-s)dW_s$, but how shall I proceed? I am new to SDE, and any help is highly appreciated!
(I find this question related to How to solve system of stochastic differential equations?)
EDIT:
I found this question: https://mathoverflow.net/questions/143245/distribution-of-integral-of-exponential-of-wiener-process. Why does this hold? $sigma e^asint_0^s e^-asdW_s=beta W_s$ where $beta^2=fracsigma^22a(1-e^-2as)$? Actually I cannot obtain the right formula of variance for the OU process based on this equation.
Assuming this is correct, my attempt for $tau=0$: (Let me know if anything is wrong)
$Y(t)=A(1-e^-kt)+beta(t)W(t)$ where $(beta(t))^2=fracsigma^22k(1-e^-2kt)$. So $dX(t)=(kAe^-kt-kbeta(t) W_t)dt$. Thus $mathbbE(X(T))=int_0^T kAe^-ktdt$.
I am still confused in obtaining $mathbbV(X(t))$, and thanks for any help in advance!
differential-equations stochastic-processes stochastic-calculus
asked Aug 1 at 22:29
huighlh
465
I have a stochastic process $dX(t)=k[A-Y(t)]f(t)dt$, where $Y(t)$ is the Ornstein-Uhlenbeck process, and satisfies: $dY(t)=k[A-Y(t)]dt+sigma dW(t)$ and $Y(0)=0$. $f(t)=0$ for $t<tau$ and $f(t)=1$ for $tgeqtau$. Now I want to solve for $X(t)$ and $mathbbE(X(t))$, $mathbbV(X(t))$.
I could plug in $Y(t)=A(1-e^-kt)+sigmaint_0^t e^-k(t-s)dW_s$, but how shall I proceed? I am new to SDE, and any help is highly appreciated!
(I find this question related to How to solve system of stochastic differential equations?)
EDIT:
I found this question: https://mathoverflow.net/questions/143245/distribution-of-integral-of-exponential-of-wiener-process. Why does this hold? $sigma e^asint_0^s e^-asdW_s=beta W_s$ where $beta^2=fracsigma^22a(1-e^-2as)$? Actually I cannot obtain the right formula of variance for the OU process based on this equation.
Assuming this is correct, my attempt for $tau=0$: (Let me know if anything is wrong)
$Y(t)=A(1-e^-kt)+beta(t)W(t)$ where $(beta(t))^2=fracsigma^22k(1-e^-2kt)$. So $dX(t)=(kAe^-kt-kbeta(t) W_t)dt$. Thus $mathbbE(X(T))=int_0^T kAe^-ktdt$.
I am still confused in obtaining $mathbbV(X(t))$, and thanks for any help in advance!
differential-equations stochastic-processes stochastic-calculus
asked Aug 1 at 22:29
huighlh
465
edited Aug 2 at 15:22
asked Aug 1 at 22:29
huighlh
465
asked Aug 1 at 22:29
huighlh
465
asked Aug 1 at 22:29
huighlh
465
465
Is $tau$ a random variable or a deterministic constant? What about $A$...?
– saz
Aug 2 at 5:32
$tau$ and $A$ are constants. Thanks! @saz
– huighlh
Aug 2 at 14:04
add a comment |Â
Is $tau$ a random variable or a deterministic constant? What about $A$...?
– saz
Aug 2 at 5:32
$tau$ and $A$ are constants. Thanks! @saz
– huighlh
Aug 2 at 14:04
Is $tau$ a random variable or a deterministic constant? What about $A$...?
– saz
Aug 2 at 5:32
Is $tau$ a random variable or a deterministic constant? What about $A$...?
– saz
Aug 2 at 5:32
$tau$ and $A$ are constants. Thanks! @saz
– huighlh
Aug 2 at 14:04
$tau$ and $A$ are constants. Thanks! @saz
– huighlh
Aug 2 at 14:04
add a comment |Â
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Is $tau$ a random variable or a deterministic constant? What about $A$...?
– saz
Aug 2 at 5:32
$tau$ and $A$ are constants. Thanks! @saz
– huighlh
Aug 2 at 14:04