Poisson Processes

Motivation: why this matters in quant finance

Poisson processes model random arrival times: trades hitting a limit-order book, insurance claims, defaults in a reduced-form credit model, jump arrivals in a jump-diffusion model, or operational-loss events through time. The Poisson distribution counts how many events occur in a fixed interval; the Poisson process stitches those counts into a full continuous-time path.

The key modelling claim is stronger than "events are random." It says arrivals have independent increments, stationary increments, and a constant event intensity

\lambda

. That combination makes the process simple enough to simulate, calibrate, and embed inside Lévy processes, while still capturing the discontinuous event risk that Brownian motion cannot represent.

The informal idea

A Poisson process $N_t$ counts how many events have occurred by time $t$ . It starts at zero, stays flat between events, and jumps upward by one at each arrival time. Its paths are right-continuous with left limits:

N_t = N_{t+}, \qquad N_{t-} = \lim_{s \uparrow t}N_s.

At a jump time $\tau$ , $N_\tau = N_{\tau-}+1$ . At all other times, $N_\tau=N_{\tau-}$ .

The intensity $\lambda$ means:

\mathbb{P}(\text{one event in }[t,t+\Delta t]) = \lambda \Delta t + o(\Delta t),

and the probability of two or more events in that tiny interval is $o(\Delta t)$ . So $\lambda$ is the local arrival rate, not a deterministic schedule.

Formal definition

A counting process

(N_t)_{t \ge 0}

is a Poisson process with intensity $\lambda>0$ if:

$N_0=0$ .
$N_t$ has nondecreasing integer-valued paths with jumps of size one.
It has independent increments: counts over disjoint intervals are independent.
It has stationary increments: $N_{t+s}-N_s$ has the same distribution as $N_t$ .
$N_t \sim \text{Poisson}(\lambda t)$ .

Equivalently,

\mathbb{P}(N_{t+s}-N_s=k) = e^{-\lambda t}\frac{(\lambda t)^k}{k!}, \qquad k=0,1,2,\ldots

Lawler derives the same process from waiting times. If

T_1,T_2,\ldots

are i.i.d. exponential random variables with rate

\lambda

, define jump times

\tau_n = T_1+\cdots+T_n,

and set $N_t=n$ for $\tau_n \le t < \tau_{n+1}$ . This construction gives the Poisson process and is also the cleanest simulation method.

Key properties

Exponential waiting times

The first arrival time $\tau_1$ satisfies

\mathbb{P}(\tau_1>t)=e^{-\lambda t}.

Thus $\tau_1\sim\text{Exponential}(\lambda)$ , and the same holds for all inter-arrival times. This is where the memoryless property enters continuous-time event modelling.

Mean and variance

Since $N_t\sim\text{Poisson}(\lambda t)$ ,

\mathbb{E}[N_t]=\lambda t, \qquad \text{Var}(N_t)=\lambda t.

The equality of mean and variance is a diagnostic assumption. If real event counts are overdispersed, a Cox process, Hawkes process, or stochastic-intensity model may be more appropriate.

Martingale compensation

The compensated process

M_t = N_t-\lambda t

is a martingale with respect to the natural filtration of $N$ . The deterministic compensator $\lambda t$ removes the predictable event rate, leaving only surprise arrivals.

Generator

For a function $f$ on the state space, the generator is

Lf(x)=\lambda[f(x+1)-f(x)].

This says: in a very small interval, the only first-order event is a one-step jump from $x$ to $x+1$ .

Worked example: default counting over one year

Suppose defaults in a homogeneous portfolio arrive according to a Poisson process with annual intensity $\lambda=0.08$ . Then the probability of no defaults over one year is

\mathbb{P}(N_1=0)=e^{-0.08}\approx 0.923.

The probability of at least one default is therefore $1-e^{-0.08}\approx 0.077$ . The probability of two or more defaults is

1-\mathbb{P}(N_1=0)-\mathbb{P}(N_1=1) =1-e^{-0.08}(1+0.08)\approx 0.003.

For small

\lambda T

, one event dominates the tail. That is why finite-activity jump models often separate jump arrival frequency from jump severity.

Common confusions and pitfalls

"Poisson process means events happen every $1/\lambda$ units of time." No.

1/\lambda

is the mean waiting time. Actual waiting times are random and can cluster.

"The Poisson distribution and Poisson process are the same object." The distribution describes one count. The process describes counts consistently across all times.

"Independent increments mean events cannot cluster." They can cluster by chance; independent increments only says disjoint future counts do not depend on past counts.

"A constant intensity is always realistic." It is a modelling idealisation. Market arrivals usually have intraday seasonality, self-excitation, and regime dependence.

"The compensated process is always nonnegative."

N_t-\lambda t

can be negative. Compensation removes drift; it does not preserve counting-process monotonicity.

Where this goes next

Jump-Diffusion Processes: Uses Poisson arrivals to decide when asset-price jumps occur.
Lévy Processes: Places the Poisson process inside the broader independent-increment family.
Martingales: The compensated Poisson process is the counting-process analogue of drift-corrected Brownian motion.
Infinitesimal Generators and Kolmogorov Equations: The generator $Lf(x)=\lambda[f(x+1)-f(x)]$ leads to forward equations for event counts.

References

Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 6 §6.2 (Poisson process), §6.3 (Compound Poisson process).