Stochastic Process 2 -- Poisson Process

Definition: Stopping Time

A stopping time with respect to a filtration \(\{\mathcal{F}_t, t \in \Sigma \subset [0, \infty)\}\) is a mapping \(\tau: \Omega \rightarrow \Sigma \cup \{\infty\}\) such that \(\{\tau \le t\} \in \mathcal{F}_t\) for every \(t \in \Sigma.\)

This definition seems really abstract. When we describe it using simple word, we can view a stopping time as the time that a process stops because it satisfies some properties.

Examples of stopping time:

1. First hitting time: \(H_z := \min\{s : X(s) = z\}\) is the first time \(X(s)\) hits level \(z.\)
2. First exit time: \(H_{a,b} := \min\{s: X(s) \notin (a,b)\}.\)
3. Moment inverse of integral functional: \(v(t) := \min \{s: \int_0^s g(X(r)) dr = t\}\) given \(g\) as non-negative measurable function.
4. Inverse range time: \(\theta_c = \min\{t: \underset{0\le s \le t}{\sup} X(s) - \underset{0\le s \le t}{\inf} X(s) \ge v\}\) given value \(v > 0.\)

Definition: Process with Independent Increments

A process \(X(t)\) is called a process with independent increments if \(\forall 0=t_0<t_1<\cdots<t_n\) the variables \(X(t_0), X(t_1)-X(t_0),...,X(t_n)-X(t_{n-1})\) are independent.

Definition: Poisson Process

A Poisson process with intensity \(\lambda > 0\) is a process \(N(t), t \in [0, \infty), N(0) = 0,\) with independent increments having the Poisson distribution \[ P(N(t) - N(s) = k) = \frac{(\lambda(t-s))^k}{k!}e^{-\lambda(t-s)}. \]

Specially, we have that \[ P(N(t) = k) = \frac{(\lambda t)^k}{k!}e^{-\lambda t}. \]

Here are some properties of Poisson process:

1. \(E(N(t)) = \lambda t.\)
2. \(D(N(t)) = \lambda t.\)
3. \(Cov(N(s), N(t)) = \lambda\min(s,t).\)
4. \(R_N(s,t) = \lambda^2 st+ \lambda\min(s,t).\)

Definition: Compound Poisson Process

The compound Poisson process is the process that \[ N_c(t) = \sum\limits_{k=1}^{N(t)} Y_k, t\ge0, \] where \(Y_k, k = 1,2,...\) are i.i.d. random variables independent of the process \(N.\)

Here are some properties of Compound Poisson Process

1. \(E(N_c(t)) = E(Y)E(N(t)).\)
2. \(Cov(N_c(s), N_c(t)) = \lambda (s\land t) [D(X)+(E(X))^2].\)

Stopping time of Poisson Process [[TODO]]