Integral representation of a stochastic differential equation
$begingroup$
If I have a stochastic differential equation gives as
$$
dX_t = aW_t^2dt
+bW_t^3dW_t$$
where $w_t$ is a wiener process and $a,b$ are real numbers. How can I reach the integral representation of $X_t$?
In other words how do I compute $f,g$ such that:
$$X_t=X_s+int_s^tf(t,W_t)dt+int_s^tg(t,W_t)dW_t
$$
when $t>s$
stochastic-processes stochastic-calculus stochastic-integrals
asked Dec 23 '18 at 1:30
k.dkhkk.dkhk
16410
$endgroup$
add a comment |
$begingroup$
If I have a stochastic differential equation gives as
$$
dX_t = aW_t^2dt
+bW_t^3dW_t$$
where $w_t$ is a wiener process and $a,b$ are real numbers. How can I reach the integral representation of $X_t$?
In other words how do I compute $f,g$ such that:
$$X_t=X_s+int_s^tf(t,W_t)dt+int_s^tg(t,W_t)dW_t
$$
when $t>s$
stochastic-processes stochastic-calculus stochastic-integrals
asked Dec 23 '18 at 1:30
k.dkhkk.dkhk
16410
$endgroup$
1
$begingroup$
You just ... integrate, from $s$ to $t$...? I mean, its not like you will solve those integrals, but the representation is pretty much this. As a result, your $f(t, W_t) = aW_t^2$ and $g(t, W_t) = bW_t^3$ Also, keep in mind, the $t$ should actually be something else, like $s$, since $t$ is in your limits of integration, so replace all $t$ with $s$
$endgroup$
– Makina
Dec 23 '18 at 1:36
add a comment |
$begingroup$
If I have a stochastic differential equation gives as
$$
dX_t = aW_t^2dt
+bW_t^3dW_t$$
where $w_t$ is a wiener process and $a,b$ are real numbers. How can I reach the integral representation of $X_t$?
In other words how do I compute $f,g$ such that:
$$X_t=X_s+int_s^tf(t,W_t)dt+int_s^tg(t,W_t)dW_t
$$
when $t>s$
stochastic-processes stochastic-calculus stochastic-integrals
asked Dec 23 '18 at 1:30
k.dkhkk.dkhk
16410
$endgroup$
If I have a stochastic differential equation gives as
$$
dX_t = aW_t^2dt
+bW_t^3dW_t$$
where $w_t$ is a wiener process and $a,b$ are real numbers. How can I reach the integral representation of $X_t$?
In other words how do I compute $f,g$ such that:
$$X_t=X_s+int_s^tf(t,W_t)dt+int_s^tg(t,W_t)dW_t
$$
when $t>s$
stochastic-processes stochastic-calculus stochastic-integrals
stochastic-processes stochastic-calculus stochastic-integrals
asked Dec 23 '18 at 1:30
k.dkhkk.dkhk
16410
asked Dec 23 '18 at 1:30
k.dkhkk.dkhk
16410
asked Dec 23 '18 at 1:30
k.dkhkk.dkhk
16410
asked Dec 23 '18 at 1:30
k.dkhkk.dkhk
16410
asked Dec 23 '18 at 1:30
k.dkhkk.dkhk
16410
16410
1
$begingroup$
You just ... integrate, from $s$ to $t$...? I mean, its not like you will solve those integrals, but the representation is pretty much this. As a result, your $f(t, W_t) = aW_t^2$ and $g(t, W_t) = bW_t^3$ Also, keep in mind, the $t$ should actually be something else, like $s$, since $t$ is in your limits of integration, so replace all $t$ with $s$
$endgroup$
– Makina
Dec 23 '18 at 1:36
add a comment |
1
$begingroup$
You just ... integrate, from $s$ to $t$...? I mean, its not like you will solve those integrals, but the representation is pretty much this. As a result, your $f(t, W_t) = aW_t^2$ and $g(t, W_t) = bW_t^3$ Also, keep in mind, the $t$ should actually be something else, like $s$, since $t$ is in your limits of integration, so replace all $t$ with $s$
$endgroup$
– Makina
Dec 23 '18 at 1:36
1
1
$begingroup$
You just ... integrate, from $s$ to $t$...? I mean, its not like you will solve those integrals, but the representation is pretty much this. As a result, your $f(t, W_t) = aW_t^2$ and $g(t, W_t) = bW_t^3$ Also, keep in mind, the $t$ should actually be something else, like $s$, since $t$ is in your limits of integration, so replace all $t$ with $s$
$endgroup$
– Makina
Dec 23 '18 at 1:36
$begingroup$
You just ... integrate, from $s$ to $t$...? I mean, its not like you will solve those integrals, but the representation is pretty much this. As a result, your $f(t, W_t) = aW_t^2$ and $g(t, W_t) = bW_t^3$ Also, keep in mind, the $t$ should actually be something else, like $s$, since $t$ is in your limits of integration, so replace all $t$ with $s$
$endgroup$
– Makina
Dec 23 '18 at 1:36
add a comment |
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$begingroup$
You just ... integrate, from $s$ to $t$...? I mean, its not like you will solve those integrals, but the representation is pretty much this. As a result, your $f(t, W_t) = aW_t^2$ and $g(t, W_t) = bW_t^3$ Also, keep in mind, the $t$ should actually be something else, like $s$, since $t$ is in your limits of integration, so replace all $t$ with $s$
$endgroup$
– Makina
Dec 23 '18 at 1:36