PDE with terminal condition, Feynman-Kac
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I have the following PDE:
$$2u_t+9u_{xx}-2u_x=0$$
$$u(x,T)=e^x$$
and I get that
$$dX(s)=-dS+3dW(s)$$
$$X(t)=x$$
But how do I get the expected value of $e^x$?
I tried substituting $Y(s)=e^x$, but I get the wrong answer... The answer should be $u(x,t)=e^{(x+frac{7}{2}(T-t))}$.
pde stochastic-processes brownian-motion
asked Dec 18 '18 at 19:53
Natalie_94Natalie_94
316
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add a comment |
$begingroup$
I have the following PDE:
$$2u_t+9u_{xx}-2u_x=0$$
$$u(x,T)=e^x$$
and I get that
$$dX(s)=-dS+3dW(s)$$
$$X(t)=x$$
But how do I get the expected value of $e^x$?
I tried substituting $Y(s)=e^x$, but I get the wrong answer... The answer should be $u(x,t)=e^{(x+frac{7}{2}(T-t))}$.
pde stochastic-processes brownian-motion
asked Dec 18 '18 at 19:53
Natalie_94Natalie_94
316
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What is $X, S, W$ ?
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– Cesareo
Dec 18 '18 at 20:42
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Sorry, X is a stochastic process, W is a Wiener process and S is time
$endgroup$
– Natalie_94
Dec 19 '18 at 10:34
add a comment |
$begingroup$
I have the following PDE:
$$2u_t+9u_{xx}-2u_x=0$$
$$u(x,T)=e^x$$
and I get that
$$dX(s)=-dS+3dW(s)$$
$$X(t)=x$$
But how do I get the expected value of $e^x$?
I tried substituting $Y(s)=e^x$, but I get the wrong answer... The answer should be $u(x,t)=e^{(x+frac{7}{2}(T-t))}$.
pde stochastic-processes brownian-motion
asked Dec 18 '18 at 19:53
Natalie_94Natalie_94
316
$endgroup$
I have the following PDE:
$$2u_t+9u_{xx}-2u_x=0$$
$$u(x,T)=e^x$$
and I get that
$$dX(s)=-dS+3dW(s)$$
$$X(t)=x$$
But how do I get the expected value of $e^x$?
I tried substituting $Y(s)=e^x$, but I get the wrong answer... The answer should be $u(x,t)=e^{(x+frac{7}{2}(T-t))}$.
pde stochastic-processes brownian-motion
pde stochastic-processes brownian-motion
asked Dec 18 '18 at 19:53
Natalie_94Natalie_94
316
asked Dec 18 '18 at 19:53
Natalie_94Natalie_94
316
asked Dec 18 '18 at 19:53
Natalie_94Natalie_94
316
asked Dec 18 '18 at 19:53
Natalie_94Natalie_94
316
asked Dec 18 '18 at 19:53
Natalie_94Natalie_94
316
316
$begingroup$
What is $X, S, W$ ?
$endgroup$
– Cesareo
Dec 18 '18 at 20:42
$begingroup$
Sorry, X is a stochastic process, W is a Wiener process and S is time
$endgroup$
– Natalie_94
Dec 19 '18 at 10:34
add a comment |
$begingroup$
What is $X, S, W$ ?
$endgroup$
– Cesareo
Dec 18 '18 at 20:42
$begingroup$
Sorry, X is a stochastic process, W is a Wiener process and S is time
$endgroup$
– Natalie_94
Dec 19 '18 at 10:34
$begingroup$
What is $X, S, W$ ?
$endgroup$
– Cesareo
Dec 18 '18 at 20:42
$begingroup$
What is $X, S, W$ ?
$endgroup$
– Cesareo
Dec 18 '18 at 20:42
$begingroup$
Sorry, X is a stochastic process, W is a Wiener process and S is time
$endgroup$
– Natalie_94
Dec 19 '18 at 10:34
$begingroup$
Sorry, X is a stochastic process, W is a Wiener process and S is time
$endgroup$
– Natalie_94
Dec 19 '18 at 10:34
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint:
Your SDE results in the following expression:
$X(T) = x - (T - t) + 3(W(T) - W(t))$
Now you just need to find $E[e^{X(T)}]$
Can you do that?
Hint2: MGF
answered Dec 19 '18 at 15:43
MakinaMakina
1,1901316
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add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint:
Your SDE results in the following expression:
$X(T) = x - (T - t) + 3(W(T) - W(t))$
Now you just need to find $E[e^{X(T)}]$
Can you do that?
Hint2: MGF
answered Dec 19 '18 at 15:43
MakinaMakina
1,1901316
$endgroup$
add a comment |
$begingroup$
Hint:
Your SDE results in the following expression:
$X(T) = x - (T - t) + 3(W(T) - W(t))$
Now you just need to find $E[e^{X(T)}]$
Can you do that?
Hint2: MGF
answered Dec 19 '18 at 15:43
MakinaMakina
1,1901316
$endgroup$
add a comment |
$begingroup$
Hint:
Your SDE results in the following expression:
$X(T) = x - (T - t) + 3(W(T) - W(t))$
Now you just need to find $E[e^{X(T)}]$
Can you do that?
Hint2: MGF
answered Dec 19 '18 at 15:43
MakinaMakina
1,1901316
$endgroup$
Hint:
Your SDE results in the following expression:
$X(T) = x - (T - t) + 3(W(T) - W(t))$
Now you just need to find $E[e^{X(T)}]$
Can you do that?
Hint2: MGF
answered Dec 19 '18 at 15:43
MakinaMakina
1,1901316
answered Dec 19 '18 at 15:43
MakinaMakina
1,1901316
answered Dec 19 '18 at 15:43
MakinaMakina
1,1901316
answered Dec 19 '18 at 15:43
MakinaMakina
1,1901316
1,1901316
add a comment |
add a comment |
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$begingroup$
What is $X, S, W$ ?
$endgroup$
– Cesareo
Dec 18 '18 at 20:42
$begingroup$
Sorry, X is a stochastic process, W is a Wiener process and S is time
$endgroup$
– Natalie_94
Dec 19 '18 at 10:34