Mixing
When is the local martingale in the Itō formula a (strict) martingale?
Let
$(Omega,mathcal A,operatorname P)$ be a complete probability space
$(mathcal F_t)_{tge0}$ be a filtration on $(Omega,mathcal A)$
$W$ be an $mathcal F$-Brownian motion on $(Omega,mathcal A,operatorname P)$
$X$ be a continuous $mathcal F$-adapted process on $(Omega,mathcal A,operatorname P)$ with $$X_t=X_0+int_0^tmu_s:{rm d}s+int_0^tsigma_s:{rm d}W_s;;;text{for all }tge0text{ almost surely}tag1$$ for some $mathcal F$-progressive processes $mu,sigma$ with $$int_0^t|mu_s|+|sigma_s|^2:{rm d}s<infty;;;text{almost surely for all }tge0tag2$$
Now, let $fin C^2(mathbb R)$. By the Itō formula, $$f(X_t)=f(X_0)+int_0^tf'(X_s):{rm d}X_s+frac12int_0^tf''(X_s):{rm d}[X]_s;;;text{for all }tge0text{ almost surely}tag3.$$
Which assumption do we need to impose, if we want that $$left(int_0^tsigma_sf'(X_s):{rm d}W_sright)_{tge0}$$ is an $mathcal F$-martingale?
One possible assumption would be that $f$ has compact support and that $mu$ and $sigma$ are bounded on $left{(omega,t)inOmegatimes[0,infty):X_t(omega)inoperatorname{supp}fright}$.
I'm particularly interested in the case $mu_t=tildemu(t,X_t)$ and $sigma_t=tildesigma(t,X_t)$ for all $tge0$ for some Borel measurable $tildemu,tildesigma:[0,infty)timesmathbb Rtomathbb R$.
probability-theory stochastic-processes stochastic-integrals stochastic-analysis sde
asked Nov 21 '18 at 18:41
0xbadf00d0xbadf00d
1,75941430
add a comment |
Let
$(Omega,mathcal A,operatorname P)$ be a complete probability space
$(mathcal F_t)_{tge0}$ be a filtration on $(Omega,mathcal A)$
$W$ be an $mathcal F$-Brownian motion on $(Omega,mathcal A,operatorname P)$
$X$ be a continuous $mathcal F$-adapted process on $(Omega,mathcal A,operatorname P)$ with $$X_t=X_0+int_0^tmu_s:{rm d}s+int_0^tsigma_s:{rm d}W_s;;;text{for all }tge0text{ almost surely}tag1$$ for some $mathcal F$-progressive processes $mu,sigma$ with $$int_0^t|mu_s|+|sigma_s|^2:{rm d}s<infty;;;text{almost surely for all }tge0tag2$$
Now, let $fin C^2(mathbb R)$. By the Itō formula, $$f(X_t)=f(X_0)+int_0^tf'(X_s):{rm d}X_s+frac12int_0^tf''(X_s):{rm d}[X]_s;;;text{for all }tge0text{ almost surely}tag3.$$
Which assumption do we need to impose, if we want that $$left(int_0^tsigma_sf'(X_s):{rm d}W_sright)_{tge0}$$ is an $mathcal F$-martingale?
One possible assumption would be that $f$ has compact support and that $mu$ and $sigma$ are bounded on $left{(omega,t)inOmegatimes[0,infty):X_t(omega)inoperatorname{supp}fright}$.
I'm particularly interested in the case $mu_t=tildemu(t,X_t)$ and $sigma_t=tildesigma(t,X_t)$ for all $tge0$ for some Borel measurable $tildemu,tildesigma:[0,infty)timesmathbb Rtomathbb R$.
probability-theory stochastic-processes stochastic-integrals stochastic-analysis sde
asked Nov 21 '18 at 18:41
0xbadf00d0xbadf00d
1,75941430
have a look here math.stackexchange.com/questions/38908/…
– TheBridge
Nov 21 '18 at 22:05
add a comment |
Let
$(Omega,mathcal A,operatorname P)$ be a complete probability space
$(mathcal F_t)_{tge0}$ be a filtration on $(Omega,mathcal A)$
$W$ be an $mathcal F$-Brownian motion on $(Omega,mathcal A,operatorname P)$
$X$ be a continuous $mathcal F$-adapted process on $(Omega,mathcal A,operatorname P)$ with $$X_t=X_0+int_0^tmu_s:{rm d}s+int_0^tsigma_s:{rm d}W_s;;;text{for all }tge0text{ almost surely}tag1$$ for some $mathcal F$-progressive processes $mu,sigma$ with $$int_0^t|mu_s|+|sigma_s|^2:{rm d}s<infty;;;text{almost surely for all }tge0tag2$$
Now, let $fin C^2(mathbb R)$. By the Itō formula, $$f(X_t)=f(X_0)+int_0^tf'(X_s):{rm d}X_s+frac12int_0^tf''(X_s):{rm d}[X]_s;;;text{for all }tge0text{ almost surely}tag3.$$
Which assumption do we need to impose, if we want that $$left(int_0^tsigma_sf'(X_s):{rm d}W_sright)_{tge0}$$ is an $mathcal F$-martingale?
One possible assumption would be that $f$ has compact support and that $mu$ and $sigma$ are bounded on $left{(omega,t)inOmegatimes[0,infty):X_t(omega)inoperatorname{supp}fright}$.
I'm particularly interested in the case $mu_t=tildemu(t,X_t)$ and $sigma_t=tildesigma(t,X_t)$ for all $tge0$ for some Borel measurable $tildemu,tildesigma:[0,infty)timesmathbb Rtomathbb R$.
probability-theory stochastic-processes stochastic-integrals stochastic-analysis sde
asked Nov 21 '18 at 18:41
0xbadf00d0xbadf00d
1,75941430
Let
$(Omega,mathcal A,operatorname P)$ be a complete probability space
$(mathcal F_t)_{tge0}$ be a filtration on $(Omega,mathcal A)$
$W$ be an $mathcal F$-Brownian motion on $(Omega,mathcal A,operatorname P)$
$X$ be a continuous $mathcal F$-adapted process on $(Omega,mathcal A,operatorname P)$ with $$X_t=X_0+int_0^tmu_s:{rm d}s+int_0^tsigma_s:{rm d}W_s;;;text{for all }tge0text{ almost surely}tag1$$ for some $mathcal F$-progressive processes $mu,sigma$ with $$int_0^t|mu_s|+|sigma_s|^2:{rm d}s<infty;;;text{almost surely for all }tge0tag2$$
Now, let $fin C^2(mathbb R)$. By the Itō formula, $$f(X_t)=f(X_0)+int_0^tf'(X_s):{rm d}X_s+frac12int_0^tf''(X_s):{rm d}[X]_s;;;text{for all }tge0text{ almost surely}tag3.$$
Which assumption do we need to impose, if we want that $$left(int_0^tsigma_sf'(X_s):{rm d}W_sright)_{tge0}$$ is an $mathcal F$-martingale?
One possible assumption would be that $f$ has compact support and that $mu$ and $sigma$ are bounded on $left{(omega,t)inOmegatimes[0,infty):X_t(omega)inoperatorname{supp}fright}$.
I'm particularly interested in the case $mu_t=tildemu(t,X_t)$ and $sigma_t=tildesigma(t,X_t)$ for all $tge0$ for some Borel measurable $tildemu,tildesigma:[0,infty)timesmathbb Rtomathbb R$.
probability-theory stochastic-processes stochastic-integrals stochastic-analysis sde
probability-theory stochastic-processes stochastic-integrals stochastic-analysis sde
asked Nov 21 '18 at 18:41
0xbadf00d0xbadf00d
1,75941430
asked Nov 21 '18 at 18:41
0xbadf00d0xbadf00d
1,75941430
asked Nov 21 '18 at 18:41
0xbadf00d0xbadf00d
1,75941430
asked Nov 21 '18 at 18:41
0xbadf00d0xbadf00d
1,75941430
asked Nov 21 '18 at 18:41
0xbadf00d0xbadf00d
1,75941430
1,75941430
have a look here math.stackexchange.com/questions/38908/…
– TheBridge
Nov 21 '18 at 22:05
add a comment |
have a look here math.stackexchange.com/questions/38908/…
– TheBridge
Nov 21 '18 at 22:05
have a look here math.stackexchange.com/questions/38908/…
– TheBridge
Nov 21 '18 at 22:05
have a look here math.stackexchange.com/questions/38908/…
– TheBridge
Nov 21 '18 at 22:05
add a comment |
1 Answer
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It is a martingale if the integrand is adapted and the expectation of the square of this expression (using ito's isometry) is finite. That is the general approach to such question.
answered Nov 24 '18 at 3:37
MakinaMakina
1,158316
add a comment |
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1 Answer
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1 Answer
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active
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It is a martingale if the integrand is adapted and the expectation of the square of this expression (using ito's isometry) is finite. That is the general approach to such question.
answered Nov 24 '18 at 3:37
MakinaMakina
1,158316
add a comment |
It is a martingale if the integrand is adapted and the expectation of the square of this expression (using ito's isometry) is finite. That is the general approach to such question.
answered Nov 24 '18 at 3:37
MakinaMakina
1,158316
add a comment |
It is a martingale if the integrand is adapted and the expectation of the square of this expression (using ito's isometry) is finite. That is the general approach to such question.
answered Nov 24 '18 at 3:37
MakinaMakina
1,158316
It is a martingale if the integrand is adapted and the expectation of the square of this expression (using ito's isometry) is finite. That is the general approach to such question.
answered Nov 24 '18 at 3:37
MakinaMakina
1,158316
answered Nov 24 '18 at 3:37
MakinaMakina
1,158316
answered Nov 24 '18 at 3:37
MakinaMakina
1,158316
answered Nov 24 '18 at 3:37
MakinaMakina
1,158316
1,158316
add a comment |
add a comment |
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have a look here math.stackexchange.com/questions/38908/…
– TheBridge
Nov 21 '18 at 22:05