Mixing
Hypotheses of Itô's Lemma
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Let $X_t$ be an Itô process satisfying
$$ dX_t=mu(t,X_t),dt+sigma(t,X_t),dB_t, $$
$tin I$, being $I$ an interval in $(0,infty)$ and $B_t$ a standard Brownian motion. I studied that Itô's formula can be applied to $f(t,X_t)$ when $fin C^1,2(ItimesmathbbR)$. Let $mathcalD$ be an open set of $mathbbR$ such that $X(t)in mathcalD$ for all $tin I$. My question is whether the assumption $fin C^1,2(ItimesmathcalD)$ is sufficient to apply Itô's formula.
Motivation: Consider the lognormal model $dS_t=mu S_t,dt+sigma S_t,dB_t$ for the asset price $S_t>0$. To find the solution, Itô's formula is applied to $f(x)=log x$. But $f$ is $C^2$ on $(0,infty)$, not on the whole $mathbbR$.
probability-theory stochastic-processes stochastic-calculus brownian-motion
edited Aug 1 at 13:39
saz
72.7k552111
asked Jul 30 at 21:25
user39756
181112
add a comment |Â
up vote
2
down vote
favorite
Let $X_t$ be an Itô process satisfying
$$ dX_t=mu(t,X_t),dt+sigma(t,X_t),dB_t, $$
$tin I$, being $I$ an interval in $(0,infty)$ and $B_t$ a standard Brownian motion. I studied that Itô's formula can be applied to $f(t,X_t)$ when $fin C^1,2(ItimesmathbbR)$. Let $mathcalD$ be an open set of $mathbbR$ such that $X(t)in mathcalD$ for all $tin I$. My question is whether the assumption $fin C^1,2(ItimesmathcalD)$ is sufficient to apply Itô's formula.
Motivation: Consider the lognormal model $dS_t=mu S_t,dt+sigma S_t,dB_t$ for the asset price $S_t>0$. To find the solution, Itô's formula is applied to $f(x)=log x$. But $f$ is $C^2$ on $(0,infty)$, not on the whole $mathbbR$.
probability-theory stochastic-processes stochastic-calculus brownian-motion
edited Aug 1 at 13:39
saz
72.7k552111
asked Jul 30 at 21:25
user39756
181112
1
Typically, the reasoning for SDEs is as follows: First Itô's formula is applied (without caring too much whether the assumptions are satisfied) to obtain a candidate for the solution. Once we have a candidate for the solution, we can apply Itô's formula (properly, this time) to prove that the candidate is indeed a solution.
– saz
Aug 1 at 13:10
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $X_t$ be an Itô process satisfying
$$ dX_t=mu(t,X_t),dt+sigma(t,X_t),dB_t, $$
$tin I$, being $I$ an interval in $(0,infty)$ and $B_t$ a standard Brownian motion. I studied that Itô's formula can be applied to $f(t,X_t)$ when $fin C^1,2(ItimesmathbbR)$. Let $mathcalD$ be an open set of $mathbbR$ such that $X(t)in mathcalD$ for all $tin I$. My question is whether the assumption $fin C^1,2(ItimesmathcalD)$ is sufficient to apply Itô's formula.
Motivation: Consider the lognormal model $dS_t=mu S_t,dt+sigma S_t,dB_t$ for the asset price $S_t>0$. To find the solution, Itô's formula is applied to $f(x)=log x$. But $f$ is $C^2$ on $(0,infty)$, not on the whole $mathbbR$.
probability-theory stochastic-processes stochastic-calculus brownian-motion
edited Aug 1 at 13:39
saz
72.7k552111
asked Jul 30 at 21:25
user39756
181112
Let $X_t$ be an Itô process satisfying
$$ dX_t=mu(t,X_t),dt+sigma(t,X_t),dB_t, $$
$tin I$, being $I$ an interval in $(0,infty)$ and $B_t$ a standard Brownian motion. I studied that Itô's formula can be applied to $f(t,X_t)$ when $fin C^1,2(ItimesmathbbR)$. Let $mathcalD$ be an open set of $mathbbR$ such that $X(t)in mathcalD$ for all $tin I$. My question is whether the assumption $fin C^1,2(ItimesmathcalD)$ is sufficient to apply Itô's formula.
Motivation: Consider the lognormal model $dS_t=mu S_t,dt+sigma S_t,dB_t$ for the asset price $S_t>0$. To find the solution, Itô's formula is applied to $f(x)=log x$. But $f$ is $C^2$ on $(0,infty)$, not on the whole $mathbbR$.
probability-theory stochastic-processes stochastic-calculus brownian-motion
edited Aug 1 at 13:39
saz
72.7k552111
asked Jul 30 at 21:25
user39756
181112
edited Aug 1 at 13:39
saz
72.7k552111
edited Aug 1 at 13:39
saz
72.7k552111
edited Aug 1 at 13:39
saz
72.7k552111
72.7k552111
asked Jul 30 at 21:25
user39756
181112
asked Jul 30 at 21:25
user39756
181112
asked Jul 30 at 21:25
user39756
181112
181112
1
Typically, the reasoning for SDEs is as follows: First Itô's formula is applied (without caring too much whether the assumptions are satisfied) to obtain a candidate for the solution. Once we have a candidate for the solution, we can apply Itô's formula (properly, this time) to prove that the candidate is indeed a solution.
– saz
Aug 1 at 13:10
add a comment |Â
1
Typically, the reasoning for SDEs is as follows: First Itô's formula is applied (without caring too much whether the assumptions are satisfied) to obtain a candidate for the solution. Once we have a candidate for the solution, we can apply Itô's formula (properly, this time) to prove that the candidate is indeed a solution.
– saz
Aug 1 at 13:10
1
1
Typically, the reasoning for SDEs is as follows: First Itô's formula is applied (without caring too much whether the assumptions are satisfied) to obtain a candidate for the solution. Once we have a candidate for the solution, we can apply Itô's formula (properly, this time) to prove that the candidate is indeed a solution.
– saz
Aug 1 at 13:10
Typically, the reasoning for SDEs is as follows: First Itô's formula is applied (without caring too much whether the assumptions are satisfied) to obtain a candidate for the solution. Once we have a candidate for the solution, we can apply Itô's formula (properly, this time) to prove that the candidate is indeed a solution.
– saz
Aug 1 at 13:10
add a comment |Â
1 Answer
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2
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Yes, it's enough to assume that $f in C^1,2(I times D)$.
The reason is, essentially, that Itô's formula can be applied to stopped processes. If we define
$$tau_D := inft>0; X_t notin D$$
then $tau_D$ is a stopping time (with respect to the filtration $(mathcalF_t+)_t geq 0)$, and therefore Itô's formula gives
$$beginalign* f(t,X_t wedge tau_D)-f(0,X_0) &= int_0^t wedge tau_D partial_x f(s,X_s) , dX_s \ &quad + int_0^t wedge tau_D left( fracsigma^2(s,X_s)2 partial_x^2 f(s,X_s) + partial_t f(s,X_s) right) , ds.endalign*$$
Since $(X_t)_t geq 0$ takes only values in $D$, we have $tau_D = infty$ almost surely, and this gives the desired identity.
answered Aug 1 at 13:39
saz
72.7k552111
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Yes, it's enough to assume that $f in C^1,2(I times D)$.
The reason is, essentially, that Itô's formula can be applied to stopped processes. If we define
$$tau_D := inft>0; X_t notin D$$
then $tau_D$ is a stopping time (with respect to the filtration $(mathcalF_t+)_t geq 0)$, and therefore Itô's formula gives
$$beginalign* f(t,X_t wedge tau_D)-f(0,X_0) &= int_0^t wedge tau_D partial_x f(s,X_s) , dX_s \ &quad + int_0^t wedge tau_D left( fracsigma^2(s,X_s)2 partial_x^2 f(s,X_s) + partial_t f(s,X_s) right) , ds.endalign*$$
Since $(X_t)_t geq 0$ takes only values in $D$, we have $tau_D = infty$ almost surely, and this gives the desired identity.
answered Aug 1 at 13:39
saz
72.7k552111
add a comment |Â
up vote
2
down vote
accepted
Yes, it's enough to assume that $f in C^1,2(I times D)$.
The reason is, essentially, that Itô's formula can be applied to stopped processes. If we define
$$tau_D := inft>0; X_t notin D$$
then $tau_D$ is a stopping time (with respect to the filtration $(mathcalF_t+)_t geq 0)$, and therefore Itô's formula gives
$$beginalign* f(t,X_t wedge tau_D)-f(0,X_0) &= int_0^t wedge tau_D partial_x f(s,X_s) , dX_s \ &quad + int_0^t wedge tau_D left( fracsigma^2(s,X_s)2 partial_x^2 f(s,X_s) + partial_t f(s,X_s) right) , ds.endalign*$$
Since $(X_t)_t geq 0$ takes only values in $D$, we have $tau_D = infty$ almost surely, and this gives the desired identity.
answered Aug 1 at 13:39
saz
72.7k552111
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Yes, it's enough to assume that $f in C^1,2(I times D)$.
The reason is, essentially, that Itô's formula can be applied to stopped processes. If we define
$$tau_D := inft>0; X_t notin D$$
then $tau_D$ is a stopping time (with respect to the filtration $(mathcalF_t+)_t geq 0)$, and therefore Itô's formula gives
$$beginalign* f(t,X_t wedge tau_D)-f(0,X_0) &= int_0^t wedge tau_D partial_x f(s,X_s) , dX_s \ &quad + int_0^t wedge tau_D left( fracsigma^2(s,X_s)2 partial_x^2 f(s,X_s) + partial_t f(s,X_s) right) , ds.endalign*$$
Since $(X_t)_t geq 0$ takes only values in $D$, we have $tau_D = infty$ almost surely, and this gives the desired identity.
answered Aug 1 at 13:39
saz
72.7k552111
Yes, it's enough to assume that $f in C^1,2(I times D)$.
The reason is, essentially, that Itô's formula can be applied to stopped processes. If we define
$$tau_D := inft>0; X_t notin D$$
then $tau_D$ is a stopping time (with respect to the filtration $(mathcalF_t+)_t geq 0)$, and therefore Itô's formula gives
$$beginalign* f(t,X_t wedge tau_D)-f(0,X_0) &= int_0^t wedge tau_D partial_x f(s,X_s) , dX_s \ &quad + int_0^t wedge tau_D left( fracsigma^2(s,X_s)2 partial_x^2 f(s,X_s) + partial_t f(s,X_s) right) , ds.endalign*$$
Since $(X_t)_t geq 0$ takes only values in $D$, we have $tau_D = infty$ almost surely, and this gives the desired identity.
answered Aug 1 at 13:39
saz
72.7k552111
answered Aug 1 at 13:39
saz
72.7k552111
answered Aug 1 at 13:39
saz
72.7k552111
answered Aug 1 at 13:39
saz
72.7k552111
72.7k552111
add a comment |Â
add a comment |Â
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1
Typically, the reasoning for SDEs is as follows: First Itô's formula is applied (without caring too much whether the assumptions are satisfied) to obtain a candidate for the solution. Once we have a candidate for the solution, we can apply Itô's formula (properly, this time) to prove that the candidate is indeed a solution.
– saz
Aug 1 at 13:10