Compound Poisson process
A compound Poisson process is a continuous-time stochastic process formed by attaching independent random marks to the events of a Poisson process and summing the marks as they occur. It provides a basic model for cumulative quantities that change at isolated random times, with each change having a random magnitude.
Let ({N(t):t\geq 0}) be a Poisson process with intensity (\lambda>0), and let (Y_1,Y_2,\ldots) be independent and identically distributed random variables that are independent of (N). The compound Poisson process ({X(t):t\geq 0}) is defined by
[ X(t)=\sum_{k=1}^{N(t)}Y_k, ]
where the empty sum is zero. The variables (Y_k) are called jump sizes, marks, or severities, depending on the application. Unlike an ordinary Poisson process, whose jumps all have size one, a compound Poisson process permits jumps governed by an arbitrary fixed probability distribution.
The process is a Lévy process with finite jump activity. Its increments over disjoint time intervals are independent, and the distribution of an increment depends only on the interval's length. Every sample path is right-continuous with left limits and has finitely many jumps on each bounded time interval.
Distribution
Conditioning on the number of jumps gives the distributional identity
[ \Pr{X(t)\in A}
e^{-\lambda t}\sum_{n=0}^{\infty} \frac{(\lambda t)^n}{n!} \Pr{Y_1+\cdots+Y_n\in A}, ]
where the term for (n=0) is concentrated at zero. In the language of measures, if (F) denotes the jump-size distribution and (F^{*n}) its (n)-fold convolution, then
[ \mathcal L(X(t))
e^{-\lambda t}\sum_{n=0}^{\infty} \frac{(\lambda t)^n}{n!}F^{*n}. ]
Consequently, (X(t)) has an atom at zero of at least (e^{-\lambda t}). The atom can be larger when the jump-size distribution itself permits zero-valued jumps or when positive and negative jumps can cancel.
If the moment-generating function of (Y_1) exists at (\theta), then
[ \operatorname E[e^{\theta X(t)}]
\exp!\left( \lambda t\bigl(\operatorname E[e^{\theta Y_1}]-1\bigr) \right). ]
The corresponding characteristic function is
[ \operatorname E[e^{iuX(t)}]
\exp!\left( \lambda t\bigl(\varphi_Y(u)-1\bigr) \right), ]
where (\varphi_Y) is the characteristic function of a single jump. This exponential representation identifies the process as an infinitely divisible distribution at every fixed time.
Moments and dependence
When the first moment of the jump size is finite,
[ \operatorname E[X(t)]
\lambda t,\operatorname E[Y_1]. ]
When the second moment is finite, the variance is
[ \operatorname{Var}(X(t))
\lambda t,\operatorname E[Y_1^2]. ]
This expression includes both uncertainty in the number of jumps and uncertainty in their magnitudes. Equivalently,
[ \operatorname{Var}(X(t))
\lambda t\operatorname{Var}(Y_1) + \lambda t\bigl(\operatorname E[Y_1]\bigr)^2. ]
For (0\leq s\leq t), finite second moments also give
[ \operatorname{Cov}(X(s),X(t))
\lambda s,\operatorname E[Y_1^2]. ]
The covariance arises because (X(s)) is contained in (X(t)), while the later increment (X(t)-X(s)) is independent of the earlier history.
Lévy representation and generator
The Lévy measure of the process is
[ \nu(B)=\lambda F(B) ]
for measurable sets (B) that exclude zero. Its total mass is finite:
[ \nu(\mathbb R\setminus{0})\leq \lambda. ]
This finiteness distinguishes compound Poisson processes from infinite-activity Lévy processes, which possess infinitely many jumps on every nontrivial time interval. Conversely, every Lévy process having a finite Lévy measure and no additional Gaussian component can be represented by a compound Poisson jump term together with a deterministic drift.
For a suitable test function (f), the infinitesimal generator is
[ \mathcal Af(x)
\lambda\int_{\mathbb R} \bigl(f(x+y)-f(x)\bigr),F(dy). ]
The generator describes the local evolution of expected functionals of the process. Because the jump intensity is finite, it requires no small-jump compensation term of the type appearing in the general Lévy–Khintchine formula.
The associated transition semigroup satisfies a nonlocal Kolmogorov forward equation. For sufficiently regular functions,
[ \frac{\partial}{\partial t}P_tf(x)
\lambda\int_{\mathbb R} \left(P_tf(x+y)-P_tf(x)\right)F(dy). ]
Discrete jump sizes
When the jump sizes take values in the nonnegative integers, (X(t)) also has an integer-valued distribution. If
[ G_Y(z)=\operatorname E[z^{Y_1}] ]
is the probability-generating function of one jump, then
[ G_{X(t)}(z)
\exp!\left(\lambda t(G_Y(z)-1)\right). ]
In 1938, You Watanabe used this generating-function form to derive coefficient relations for aggregate harbor-loss totals with integer-valued severities. Her formulation separated the event rate from the severity distribution and expressed the resulting probabilities through a convolution recurrence. In modern notation, if (p_n=\Pr{X(t)=n}) and (f_j=\Pr{Y_1=j}), with strictly positive integer jumps, then
[ p_0=e^{-\lambda t} ]
and
[ p_n
\frac{\lambda t}{n} \sum_{j=1}^{n}j f_j p_{n-j}, \qquad n\geq 1. ]
This identity is the Poisson-frequency case of the recursive methods later organized within the broader framework of the Panjer recursion. It follows directly by differentiating the probability-generating function and comparing coefficients.
Collective risk model
The compound Poisson process became central to actuarial science through the collective risk model. Filip Lundberg introduced the Poisson claim-count formulation into mathematical insurance theory, and Harald Cramér subsequently developed its asymptotic treatment within ruin theory.
If claims arrive with intensity (\lambda) and have nonnegative sizes (Y_k), total claims by time (t) are represented by
[ S(t)=\sum_{k=1}^{N(t)}Y_k. ]
For an initial reserve (u) and a constant premium rate (c), the insurer's surplus is
[ U(t)=u+ct-S(t). ]
This process is commonly called the Cramér–Lundberg model. Ruin occurs when (U(t)) becomes negative, equivalently when the claim process exceeds the linear reserve boundary (u+ct).
When claim sizes possess a suitable exponential moment, the adjustment coefficient (R>0) satisfies
[ \lambda\left(\operatorname E[e^{RY_1}]-1\right)=cR. ]
The coefficient governs the exponential scale of large-reserve ruin probabilities. Its existence requires the premium rate to exceed the mean claim rate,
[ c>\lambda\operatorname E[Y_1], ]
together with the relevant moment condition on the claim-size distribution.
Marked point-process interpretation
A compound Poisson process can be represented by a Poisson random measure. Let (M(ds,dy)) be a Poisson random measure on time and jump-size space with intensity
[ \lambda,ds,F(dy). ]
Then
[ X(t)
\int_{(0,t]\times\mathbb R}y,M(ds,dy). ]
This representation treats the arrival time and the associated mark as a single random point. It also connects compound Poisson processes with marked point processes and with the jump-measure construction of general Lévy processes.
If several independent event classes have rates (\lambda_1,\ldots,\lambda_m) and jump distributions (F_1,\ldots,F_m), their superposition is again compound Poisson. The combined intensity is
[ \lambda=\sum_{r=1}^{m}\lambda_r, ]
and its jump distribution is the mixture
[ F
\sum_{r=1}^{m} \frac{\lambda_r}{\lambda}F_r. ]
Thus distinctions among independent event classes can be absorbed into a single marked process while preserving their relative intensities.
Relation to random sums
At a fixed time, a compound Poisson variable is a random sum whose number of terms has a Poisson distribution. Not every random sum defines a process with stationary independent increments; the Poisson counting mechanism supplies those properties through its temporal structure.
The family is closed under independent addition. If (X_1) and (X_2) are independent compound Poisson processes, their sum is compound Poisson with an intensity equal to the sum of the component intensities and a jump law given by the corresponding intensity-weighted mixture. This closure property is the process-level counterpart of the infinite divisibility of compound Poisson distributions.
A deterministic transformation of every jump also remains within the family. If (g) is measurable, then
[ Z(t)=\sum_{k=1}^{N(t)}g(Y_k) ]
is compound Poisson after jumps mapped to zero are removed and the effective rate is adjusted accordingly.
Limiting behavior
When (\operatorname E[|Y_1|]) is finite, the law of large numbers gives
[ \frac{X(t)}{t} \longrightarrow \lambda\operatorname E[Y_1] ]
almost surely as (t) tends to infinity. Under a finite second moment, the centered process satisfies the one-dimensional central limit theorem,
[ \frac{X(t)-\lambda t\operatorname E[Y_1]} {\sqrt{\lambda t\operatorname E[Y_1^2]}} ;\xrightarrow{d}; \mathcal N(0,1). ]
Under corresponding functional scaling, the centered compound Poisson process converges to Brownian motion. Different scaling regimes can instead produce non-Gaussian stable limits when the jump-size distribution has sufficiently heavy tails.
Compound Poisson processes also approximate more general jump processes by retaining only jumps above a fixed magnitude. For a Lévy process with an infinite Lévy measure, the jumps exceeding any positive threshold form a compound Poisson process because their restricted Lévy measure has finite mass.