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Nate River
Nate River
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Let $B$ be a standard Brownian motion, and $A$ a process of finite variation on compacts almost surely, not necessarily adapted to the Brownian filtration.

Question: Denoting by $\mathcal L$ the Lebesgue measure, is it true that $$\mathcal L(\{t \, | \, B_t = A_t \}) = 0$$

almost surely?

Remark: In the case where $A$ is adapted, one can use the theory of local times to show that the statement is true. However the non adapted case seems subtle.

Let $B$ be a standard Brownian motion, and $A$ a process of finite variation on compacts almost surely, not necessarily adapted to the Brownian filtration.

Question: Denoting by $\mathcal L$ the Lebesgue measure, is it true that $$\mathcal L(\{t \, | \, B_t = A_t \}) = 0$$

almost surely?

Remark: In the case where $A$ is adapted, one can use the theory of local times to show that the statement is true. However the non adapted case seems subtle.

Let $B$ be a standard Brownian motion, and $A$ a process of finite variation on compacts almost surely, not necessarily adapted to the Brownian filtration.

Question: Denoting by $\mathcal L$ the Lebesgue measure, is it true that $$\mathcal L(\{t \, | \, B_t = A_t \}) = 0$$

almost surely?

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Nate River
Nate River
  • 6.2k
  • 2
  • 23
  • 99

Let $B$ be a standard Brownian motion, and $A$ a procedsprocess of finite variation on compacts almost surely, not necessarily adapted to the Brownian filtration.

Question: Denoting by $\mathcal L$ the Lebesgue measure, is it true that $$\mathcal L(\{t \, | \, B_t = A_t \}) = 0$$

almost surely?

Remark: In the case where $A$ is adapted, one can use the theory of local times to show that the statement is true. However the non adapted case seems subtle.

Let $B$ be a standard Brownian motion, and $A$ a proceds of finite variation on compacts almost surely, not necessarily adapted to the Brownian filtration.

Question: Denoting by $\mathcal L$ the Lebesgue measure, is it true that $$\mathcal L(\{t \, | \, B_t = A_t \}) = 0$$

almost surely?

Remark: In the case where $A$ is adapted, one can use the theory of local times to show that the statement is true. However the non adapted case seems subtle.

Let $B$ be a standard Brownian motion, and $A$ a process of finite variation on compacts almost surely, not necessarily adapted to the Brownian filtration.

Question: Denoting by $\mathcal L$ the Lebesgue measure, is it true that $$\mathcal L(\{t \, | \, B_t = A_t \}) = 0$$

almost surely?

Remark: In the case where $A$ is adapted, one can use the theory of local times to show that the statement is true. However the non adapted case seems subtle.

Source Link
Nate River
Nate River
  • 6.2k
  • 2
  • 23
  • 99

Intersection of Brownian motion and finite variation process

Let $B$ be a standard Brownian motion, and $A$ a proceds of finite variation on compacts almost surely, not necessarily adapted to the Brownian filtration.

Question: Denoting by $\mathcal L$ the Lebesgue measure, is it true that $$\mathcal L(\{t \, | \, B_t = A_t \}) = 0$$

almost surely?

Remark: In the case where $A$ is adapted, one can use the theory of local times to show that the statement is true. However the non adapted case seems subtle.