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Ofer
Ofer
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Suppose I have two independent Levy processes $X_t$ and $Y_t$, both not continuous.

Is anyone familiar and can refer me to a result(or a counterexample) which states that

${\displaystyle \sum_{0\leq s\leq t}}|\bigtriangleup X_{s}\bigtriangleup Y_{s}|=0 $${\displaystyle \sum_{0\leq s\leq t}}|\bigtriangleup X_{s}(\omega)\bigtriangleup Y_{s}(\omega)|=0 $ for all $t\in \mathbb{R}$ a.s?

A different yet equivalent formulation of this is

$\bigtriangleup X_{t}=0$ or $\bigtriangleup Y_{t}=0$ a.s. for all $t\in \mathbb{R}$

In words, every two independent Levy processes have no simultaneous jumps a.s. I know it holds for independent Poisson processes and I'm wondering if it generalizes.

Suppose I have two independent Levy processes $X_t$ and $Y_t$, both not continuous.

Is anyone familiar and can refer me to a result(or a counterexample) which states that

${\displaystyle \sum_{0\leq s\leq t}}|\bigtriangleup X_{s}\bigtriangleup Y_{s}|=0 $ for all $t\in \mathbb{R}$ a.s?

A different yet equivalent formulation of this is

$\bigtriangleup X_{t}=0$ or $\bigtriangleup Y_{t}=0$ a.s. for all $t\in \mathbb{R}$

In words, every two independent Levy processes have no simultaneous jumps a.s. I know it holds for independent Poisson processes and I'm wondering if it generalizes.

Suppose I have two independent Levy processes $X_t$ and $Y_t$, both not continuous.

Is anyone familiar and can refer me to a result(or a counterexample) which states that

${\displaystyle \sum_{0\leq s\leq t}}|\bigtriangleup X_{s}(\omega)\bigtriangleup Y_{s}(\omega)|=0 $ for all $t\in \mathbb{R}$ a.s?

A different yet equivalent formulation of this is

$\bigtriangleup X_{t}=0$ or $\bigtriangleup Y_{t}=0$ a.s. for all $t\in \mathbb{R}$

In words, every two independent Levy processes have no simultaneous jumps a.s. I know it holds for independent Poisson processes and I'm wondering if it generalizes.

Source Link
Ofer
Ofer
  • 83
  • 3

simultaneous jumps of independent Levy processes

Suppose I have two independent Levy processes $X_t$ and $Y_t$, both not continuous.

Is anyone familiar and can refer me to a result(or a counterexample) which states that

${\displaystyle \sum_{0\leq s\leq t}}|\bigtriangleup X_{s}\bigtriangleup Y_{s}|=0 $ for all $t\in \mathbb{R}$ a.s?

A different yet equivalent formulation of this is

$\bigtriangleup X_{t}=0$ or $\bigtriangleup Y_{t}=0$ a.s. for all $t\in \mathbb{R}$

In words, every two independent Levy processes have no simultaneous jumps a.s. I know it holds for independent Poisson processes and I'm wondering if it generalizes.