Compound Poisson Processes

In this section we will remove the restriction (1.1a) in the definition of a Poisson process, and thus allow jumps of any size.

Throughout we will take Z = {Z_t; t ≥ 0} to be a stochastic process such that for any ω ∈ Ω, the mapping t → Z_t(ω) is right continuous and real-valued, has Z₀(ω) = 0, and increases or decreases by jumps only.

(6.1) D^EFINITION. Z is said to be a compound Poisson process provided that

(a) for almost all ω ∈ Ω, the function t → Z_t(ω) has only finitely many jumps in any finite interval;

(b) for any t and s ≥ 0, Z_{t + s} − Z_t is independent of the past history {Z_u; u ≤ t} until t;

(c) for any t, s ≥ 0, the distribution of Z_{t + s} − Z_t, depends only on s (independent of t).

Let Z be a compound Poisson process. By axiom (6.1a) it has only finitely many jumps in any finite interval; therefore, the jump times can be ordered. For each ω ∈ Ω, let T₁(ω), T₂(ω), . . . be the times of the first jump, the second jump, . . ., and let X₁(ω), X₂(ω), . . . be the magnitudes of the corresponding jumps.

It follows from (6.1b) that the number of jumps in an interval (t, t + s] is independent of the past history in [0, t], and by (6.1c), the distribution of the number of jumps in (t, t + s] is independent of t.

Hence, T₁, T₂, . . . must be the arrival times in a Poisson process N = {N_t; t ≥ 0}; that is, if N_t is the number of jumps of Z in (0, t], then N is a Poisson process. The process Z differs from N by the fact that the magnitudes of the jumps of Z are random variables instead of being all equal to one.

Moreover, it follows again from (6.1b) and (6.1c) that the jump sizes X₁, X₂, . . . are independent and identically distributed random variables which are, furthermore, independent of T₁, T₂, . . . .

Conversely, if T₁, T₂, . . . are the arrival times in a Poisson process and if X₁, X₂, . . . are independent and identically distributed random variables which are also independent of the T_n, then the process Z obtained by summing all the X_i for which T_i ≤ t to make up Z_t is a compound Poisson process. Thus, we have the following qualitative characterization. See Figure 4.6.1 for a possible realization.

(6.2) PROPOSITION. Z is a compound Poisson process if and only if its jump times form a Poisson process and the magnitudes of its successive jumps are independent and identically distributed random variables independent of the jump times.

(6.3) E^XAMPLE. Arrivals of customers into a store form a Poisson process N. The amount of money spent by the nth customer is a random variable X_n which is independent of all the arrival times (including his own) and all the amounts spent by others. The total amount spent by the first n customers is Y_n = X₁ + · · · + X_n if n ≥ 1, and we set Y₀ = 0. Since the number of customers arriving in (0, t] is N_t, the sales to customers arriving in (0, t] total Z_t, = Y_N

t. Our hypotheses on N and the X_n are

such that the process Z = {Z_t; t ≥ 0} is a compound Poisson process.

Figure 4.6.1 A compound Poisson process increases or decreases, by random amounts, at the times of arrivals of a Poisson process. For the realization ω pictured, arrival times are T1(ω) = 1.5, T2(ω) = 3.7, T3(ω) = 4.2, T4(ω) = 4.5, . . ., and the jump sizes associated are X1(ω) = 2.2, X2(ω) = −0.8, X3(ω) = −1.2, X4(ω) = 0.5, . . . .

(6.4) EXAMPLE. The times between the successive failures of a computer are independent and identically distributed exponential variables. With each failure there is associated a cost of repair.

The costs associated with different failures are independent and identically distributed, and furthermore, the cost of a repair is independent of the times of failures. If Z, is the cumulative cost in (0, t], then Z = {Z_t; t ≥ 0} is a compound Poisson process.

Suppose Z is a compound Poisson process whose jumps occur at rate λ and by discrete amounts, that is, the random variables X₁, X₂, . . . take values in a countable set . For each ω ∈ Ω, let , , . . . be the number of jumps of Z in (0, t] whose magnitudes were exactly equal to a, equal to b, . . . (see Example (5.5) for the approach we are following). Then Theorem (5.3) applies to show that the processes N^a, N^b, . . . are independent Poisson processes with rates λ(a) = λP{X_i = a}, λ(b) = λP{X_i = b}, . . . This yields the following representation of Z: for any t ≥ 0 and ω ∈ Ω,

where N^a, N^b, . . . are independent Poisson processes with rates λ(a), λ(b), . . . .

The representation (6.5) can be used to obtain any quantity concerning Z from the results already obtained for Poisson processes. For example, suppose a, b, . . . ≥ 0, and consider the Laplace transform of Z_t. By the independence of N^a, N^b, . . . in (6.5), Proposition (2.1.26) applies, and

For a Poisson process M with rate μ, for any β ≥ 0,

Using this result repeatedly with β = αa, αb, . . . and μ = λ(a), λ(b), . . ., we get

Conversely, if a process Z has the Laplace transform (6.6) for Z_t for all t ≥ 0, then Z is a weighted sum of independent Poisson processes with rates λ(i), i ∈ E.

A somewhat more general approach makes use of the structure of Z outlined in Proposition (6.2) by conditioning on the number of jumps, as was done previously in Example (2.2.27).

If the number of jumps N_t of Z in (0, t] is n, then Z_t is the sum of n independent and identically distributed random variables. Hence, if E[X_i] = μ and the rate of jumps is λ, then

which gives, by proposition (2.2.16),

This approach can also be used to compute the Laplace transform of Z_t in the case where all the jumps are upward. We then get

(6.9) PROPOSITION. Suppose the jump times of Z form a Poisson process with rate λ and that the magnitudes of jumps are independent and identically distributed non-negative random variables, independent of the jump times, with distribution φ. Then for any α ≥ 0 and t ≥ 0,

where

Proof. By the independence of the X_i from N, since Z_t = X₁ + · · · + X_n when N_t = n, n ≥ 1,

Since the X_i are independent of each other, by Proposition (2.1.26),

We have thus shown that

and by Proposition (2.2.16),

since for any β ≥ 0,

Note that we can write

by letting λ(u) = λφ(u). Then (6.10) becomes

which is the continuous analog of (6.6). Analogous to the case where the jump sizes are discrete random rariables, we have the following result. Consider those jumps of Z whose magnitudes fall in the interval (a, b]; then the instants at which those jumps occur form a Poisson process with rate

independent of the jumps with other sizes.

It follows from axiom (6.1c) that the distribution of any increment Z_{t + s} − Z_t is the same as that of Z_s. By axiom (6.1b), the increments Z_t

1 − Z_t

0, . . ., Z_t

n − Z_t

n−1 are independent of each other for any n ≥ 1 and 0 ≤ t₀ < t₁ < · · · < t_n. Thus, once the distribution of Z_t is known for all t, the joint distribution of any number m of variables Z_s

1, . . ., Z_s

m can be computed easily. Hence, the probability law of the process Z is completely specified once we are given the rate of jumps λ and the distribution φ of the jump sizes.