Counting Process

Study guide: Exam MAS-I

A Counting Process $\{N(t), t \geq 0\}$ is a stochastic process in which $N(t)$ is the total number of events — claims reported, policies lapsing, storms making landfall — that have occurred by time $t$ . It is the frame on which the Poisson Process and all its variants are built.

$N(t) - N(s) = \text{number of events in } (s, t], \quad s < t$

Four defining conditions (Ross): $N(t) \geq 0$ ; $N(t)$ is integer-valued; $N(t)$ is nondecreasing, so $s < t \Rightarrow N(s) \leq N(t)$ ; and the increment $N(t) - N(s)$ counts the events in $(s, t]$ . A quantity that can fall — open claims, surplus — is not a counting process.
Independent increments: counts in disjoint intervals are independent. Stationary increments: the distribution of $N(t+s) - N(s)$ depends only on the length $t$ , not on where the interval sits.
The Poisson Process is the counting process with $N(0) = 0$ , independent increments and $N(t+s) - N(s) \sim \text{Poi}(\lambda t)$ . Let the rate vary over time and stationarity is lost — the Nonhomogeneous Poisson Process; let the rate vary randomly across risks and independence is lost — the Mixed Poisson Process.
Counting and waiting carry the same information: $N(t) \geq n \iff S_n \leq t$ , where $S_n$ is the arrival time of the $n$ -th event (see Interarrival Time).
Exam technique: rewrite a joint probability such as $P(N(1) = a,\ N(3) = b)$ in terms of non-overlapping increments — $N(1)$ and $N(3) - N(1)$ — so that independence lets it factor.

Which Claim Measures Are Counting Processes? A claims department tracks four quantities through a policy year:

$A(t)$ — the cumulative number of claims reported by time $t$

$B(t)$ — the number of claims open at time $t$

$C(t)$ — the cumulative dollars paid on claims by time $t$

$D(t)$ — the cumulative number of claims over $\$50{,}000$ reported by time $t$

Which are counting processes, and when would they have stationary increments?

Answer

$A(t)$ : yes — nonnegative, integer-valued and nondecreasing. Its increments are stationary only if the reporting rate is constant through the year; a hail-exposed book with a summer peak breaks that.

$B(t)$ : no — it falls every time a claim closes.

$C(t)$ : no — it is nondecreasing but records dollars, not events. Aggregate amounts are modelled on top of a counting process, as in the Compound Poisson Process.

$D(t)$ : yes — it is $A(t)$ with each claim kept only if it is large, a thinned count.

Joint and Conditional Claim Counts Claims are reported as a Poisson process at $\lambda = 2$ per week. Find $P(N(1) = 1,\ N(3) = 4)$ and $P(N(1) = 1 \mid N(3) = 4)$ .
Answer Split into the disjoint increments $N(1)$ and $N(3) - N(1)$ , which are independent with means $2$ and $4$ :
 $\begin{align*} P(N(1)=1,\ N(3)=4) &= P(N(1)=1)\,P\big(N(3)-N(1)=3\big) \\ &= \left(2e^{-2}\right)\left(\frac{e^{-4}\,4^3}{3!}\right) \\ &= (0.27067)(0.19537) \\ &= 0.05288 \end{align*}$ 
Dividing by $P(N(3) = 4) = e^{-6}6^4/4! = 0.13385$ , the rate cancels and a binomial is left — given four claims in three weeks, each falls in week 1 independently with probability $1/3$ :
 $\begin{align*} P(N(1)=1 \mid N(3)=4) &= \binom{4}{1}\left(\tfrac{1}{3}\right)\left(\tfrac{2}{3}\right)^3 \\ &= \frac{32}{81} \\ &= 0.3951 \end{align*}$ 
Conditioning on the total turns a Poisson question into a binomial one, and $\lambda$ no longer matters.