. Suppose N1(t) and N2(t) are Poisson processes defined on...

. Suppose $N_1(t)$ and $N_2(t)$ are Poisson processes defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$ relative to the same filtration $\mathcal{F}(t), t \geq 0$. Assume that almost surely $N_1(t)$ and $N_2(t)$ have no simultaneous jump. Show that, for each fixed $t$, the random variables $N_1(t)$ and $N_2(t)$ are independent. (Hint: Adapt the proof of Corollary 11.5.3.) (In fact, the whole path of $N_1$ is independent of the whole path of $N_2$, although you are not being asked to prove this stronger statement.)

Key Concepts

Almost Sure Non-Simultaneous Jumps

This concept refers to the condition that two processes do not jump at exactly the same time with probability one. In proving the independence of two processes, ensuring that they have no simultaneous jumps prevents any dependency that could arise from shared events, as each jump (or event) in one process is distinct from the events in the other.

Filtration

A filtration is an increasing sequence of sigma-algebras that models the evolution of information over time in a probability space. It is used to formalize concepts like adapted processes and conditional independence, allowing one to rigorously analyze dependencies and independence between different stochastic processes with respect to the same underlying information structure.

Independence of Processes

The independence of processes means that the entire path or counts of events of one process are statistically independent of those of another. Establishing this independence typically involves showing that the joint probability law of the two processes decomposes into the product of their marginal probability laws, often by leveraging properties like independent increments and the absence of simultaneous jumps.

Poisson Process

A Poisson process is a stochastic process that counts the number of events occurring in fixed intervals of time or space, where these events happen with a constant mean rate and independently of the time since the last event. Its defining properties include independent increments and the fact that the number of events in any interval follows a Poisson distribution with a parameter proportional to the interval’s length.

Independent Increments

This property of a stochastic process means that the number of events occurring in disjoint time intervals are statistically independent. In the context of Poisson processes, it is crucial for demonstrating that events in non-overlapping intervals are independent, which underpins many proofs involving the independence of counts over time.

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