A Poisson process is a continuous-time stochastic process which counts the arrival of randomly occurring events. Commonly cited examples which can be modeled by a Poisson process include radioactive decay of atoms and telephone calls arriving at an exchange, in which the number of events occurring in each consecutive time interval are assumed to be independent. Being piecewise constant, Poisson processes have very simple pathwise properties. However, they are very important to the study of stochastic calculus and, together with Brownian motion, forms one of the building blocks for the much more general class of Lévy processes. I will describe some of their properties in this post.

A random variable *N* has the Poisson distribution with parameter , denoted by , if it takes values in the set of nonnegative integers and

(1) |

for each . The mean and variance of *N* are both equal to , and the moment generating function can be calculated,

which is valid for all . From this, it can be seen that the sum of independent Poisson random variables with parameters and is again Poisson with parameter . The Poisson distribution occurs as a limit of binomial distributions. The binomial distribution with success probability *p* and *m* trials, denoted by , is the sum of *m* independent -valued random variables each with probability *p* of being 1. Explicitly, if then

In the limit as and such that , it can be verified that this tends to the Poisson distribution (1) with parameter .

Poisson *processes* are then defined as processes with independent increments and Poisson distributed marginals, as follows.

Definition 1A Poisson processXof rate is a cadlag process with and independently of for all .

An immediate consequence of this definition is that, if *X* and *Y* are independent Poisson processes of rates and respectively, then their sum is also Poisson with rate .

Note that this definition does give consistent distributions for the increments across different time intervals. That is, if is an increasing sequence of times and the increments are independent, then the sum will have the distribution. The processes given by Definition 1 are also called *homogeneous* Poisson processes, to distinguish them from the inhomogeneous case where the rate is time-dependent. I will also consider the inhomogeneous case below.

As the Poisson distribution is supported by the nonnegative integers, it follows from the definition that Poisson processes are almost surely increasing and integer valued. In fact, an alternative definition can be given as processes which count a sequence of random times occurring at exponentially distributed intervals. A random variable *T* has the exponential distribution with parameter (or rate) , denoted by , if it takes values in the nonnegative real numbers and

for all . The exponential distribution satisfies the memoryless property,

for all .

I will say that stochastic process *X* is a *counting process* if there exists an increasing sequence of random variables taking values in such that whenever is finite and

(2) |

for all . That is, counts the number of times in which are no greater than *t*. This is equivalent to *X* being right-continuous, piecewise constant, starting at zero, and having jump size at each discontinuity. Then, Poisson processes count the arrival of times at exponentially distributed intervals.

Lemma 2If are independent random variables with the distribution, and , thenXdefined by (2) is a Poisson process of rate .

*Proof:* Let be the natural filtration of *X*. We start by showing that, for each time *s*, the random variable

is exponential of rate independently of . In fact, restricting to the event

the memoryless property of the exponential distribution gives

As this holds on , which has probability 1, is indeed exponential of rate .

This shows that, for , the -valued random variable is independent of and equals 0 with probability . For any setting it follows that

(3) |

has the binomial distribution, , with success probability . Letting *n* increase to infinity, tends to . So, taking the limit, (3) tends to the distribution independently of . Furthermore, (3) equals for large *n*, giving . ⬜

As it is such a fundamental process, it is not surprising that there are in fact many ways in which Poisson processes can be characterized. In fact, they are the *only* counting processes with stationary independent increments.

Theorem 3Any counting process with stationary independent increments is a homogeneous Poisson process.

Compare this with the characterization of Brownian motion as the only continuous process with stationary independent increments (up to a scaling factor and drift term). I give a proof of Theorem 3 below in the more general context of non-stationary increments and inhomogeneous Poisson processes (Theorem 9 and Corollary 10).

Now, suppose that we have a filtered probability space . Then, a Poisson process, *X*, with respect to this space is defined to be an adapted process satisfying Definition 1 and such that is independent of for all . This is also referred to as an -Poisson process. Note, in particular, if *X* satisfies Definition 1 then it is automatically a Poisson process with respect to its natural filtration . There are several useful characterizations of -Poisson processes, which I list below.

Theorem 4LetXbe a cadlag adapted process with . Then, for any , the following are equivalent.

Xis an -Poisson process of rate .- At each time
t,

(4) for .

Xis a counting process and, for each timet, the random variablehas the distribution independently of .

Xis a counting process such that is a martingale.Xis a counting process such that is a local martingale.- For all bounded measurable functions , the process
is a martingale.

- For each ,
is a martingale.

The proof of these statements is given below in the more general, but no more difficult, context of inhomogeneous Poisson processes. For now, I briefly go through each of these statements. The second condition gives the intuitive property of a Poisson process being such that the probability of an event occurring in each interval of size being equal to to leading order. The terms in (4) just denote something which is equal to multiplied by a term which tends to zero (in probability) as .

The third statement of Theorem 4 says that, at any time, the time to wait until the next event is always distributed as . This was the idea used above in the proof of Lemma 2.

The martingale given by statement 4 above is known as a *compensated Poisson process*. Note also that, being an FV process, *M* has quadratic variation . Therefore, is also a martingale. In fact,

is a martingale. Taking provides a counterexample showing that, in Lévy’s characterization of Brownian motion as a continuous local martingale *W* such that is a local martingale, the continuity of *W* is indeed a necessary condition. In fact, in the limit , it can be shown that *W* does converge in distribution to a Brownian motion.

A consequence of the compensated process being a martingale is that the jump times of *X* are totally inaccessible. That is, for each predictable stopping time .

Lemma 5The jump times of a Poisson process are totally inaccessible.

*Proof:* Let be a predictable stopping time and be stopping times announcing . Replacing by if necessary, we can suppose that is bounded. Then, by optional sampling,

So, as is a nonnegative random variable with zero expectation, . ⬜

In fact, as I will show later in these notes, the jump times of an adapted counting process *X* are totally inaccessible if and only if it has a continuous compensator *V*. That is, *V* is a continuous process making *X*–*V* a local martingale.

Condition 6 of Theorem 4 says that the infinitesimal generator of the Poisson process of rate is . In fact, the following is an immediate consequence of this statement,

where (in probability) as .

The final condition of Theorem 4 is equivalent to the statement that is independent of with characteristic function . It is, however, often more useful to express it in this martingale form. For example, below I use it to show that a finite collection of Poisson processes is independent if and only if none of their jump times coincide (almost surely).

#### Inhomogeneous Poisson Processes

The definition of Poisson processes given above is easily extended to rates which are time-dependent,

As long as is locally integrable, we instead require that has the Poisson distribution with parameter . Equivalently, letting be the *cumulative rate* (or *cumulative intensity*), we can write this as . One advantage of writing things in terms of instead of the `instantaneous rate’ is that we can generalize to the situation where is not differentiable. In that case, need not be well defined (except in the sense of distributions).

Then, an inhomogeneous (or, non-homogeneous) Poisson process is defined as follows.

Definition 6Let be a continuous increasing function. Then,Xis a Poisson process of cumulative rate if and independently of for all .

It follows from this definition that if *X* is a Poisson process of cumulative rate and is a continuous increasing process with , then the time-changed process will be Poisson with cumulative rate . This gives an easy way of constructing inhomogeneous Poisson processes. First, use Lemma 2 to construct a homogeneous Poisson process *X* of rate 1 and set to transform it into a Poisson process with cumulative rate .

I now state and prove the equivalence of each of the statements of Theorem 4 in the more general context of inhomogeneous Poisson processes. In equation (5) below, denotes terms which are a product of with a term which goes to zero in probability as .

Theorem 7LetXbe a cadlag adapted process with . Then, for any continuous increasing function , the following are equivalent.

Xis an -Poisson process of cumulative rate .- At each time
t,

(5) for , setting .

Xis a counting process and, for each timet, the random variableis independent of and satisfies .

Xis a counting process such that is a martingale.Xis a counting process such that is a local martingale.- For all bounded measurable functions , the process
is a martingale.

- For each ,
is a martingale.

*Proof:* I start by proving equivalence of 1,4,5,6,7, and will then prove equivalence of the second and third statements.

*(1) implies (4):* The definition of a Poisson process with cumulative rate immediately implies that has mean independently of , for . So, setting ,

and *M* is a martingale as required.

*(4) implies (5):* This is trivial, as all cadlag martingales are also local martingales.

*(5) implies (6):* By assumption, is a local martingale. As *X* is a counting process, will be piecewise constant and,

So, the process can be written as a stochastic integral with respect to the local martingale *M*

By preservation of the local martingale property, is a local martingale. As *f* is assumed bounded, is also uniformly bounded over each finite time interval and is therefore a proper martingale.

*(6) implies (7):* Property (6) generalizes to complex-valued functions *f*, simply by applying it to the real and imaginary parts separately. Then, using shows that

is a martingale. Setting and gives . Using integration by parts,

By preservation of the local martingale property, this shows that is a local martingale. Furthermore, as it is uniformly bounded over each finite time interval, *UV* is a proper martingale.

*(7) implies (1):* Letting *M* be the martingale gives

for all . Comparing this with the characteristic function of the Poisson distribution shows that has the distribution independently of .

This completes the proof of equivalence of 1,4,5,6 and 7. We now move on to the equivalence of these properties with 2 and 3.

*(1) implies (2):* If *X* is Poisson with cumulative rate then

as required.

*(2) implies (7):* Fixing any , we need to show that

is a martingale. Applying (5) to the expectation of ,

Rearranging this a bit gives the following,

Fixing a time , an -measurable and bounded random variable *Z*, taking expectations gives

Here, bounded convergence ensures that expectations of the terms are themselves . So, the derivative is zero over , and is constant over this range, giving . Therefore, *M* is a martingale.

*(1) implies (3):* If *X* is a Poisson process with cumulative rate then,

as required.

*(3) implies (2):* This follows along the same lines as the proof of Lemma 2 given above. Condition (3) says that for . In particular, almost surely, whenever . Consequently, *X* is constant over all intervals for which is constant.

Fix times and, for each *n*, choose times such that . Then,

(6) |

has the distribution independently of , with success probability . Letting *n* increase to infinity, *np* tends to . So, (6) tends to the Poisson distribution with parameter .

Finally, as *X* is constant on periods of constancy of , for large enough *n* it will contain at most one discontinuity in each of the intervals , in which case (6) is equal to . So, has the Poisson distribution with parameter independently of . ⬜

An immediate consequence of Definition 6 is that the sum of a finite set of independent Poisson processes is itself Poisson, and their cumulative rates add. There is also a very simple condition for such processes to be independent — their jump times must be almost surely distinct. The necessity of this condition follows from the fact that any pair of independent and continuously distributed random variables are almost surely distinct. We now prove that, for Poisson processes, having disjoint sets of jump times is also a *sufficient* condition.

Lemma 8Let be Poisson processes with cumulative rates respectively, and defined on the same filtered probability space.

Then, they are independent if and only if their jump times are almost surely disjoint. In that case, is a Poisson process with cumulative rate .

*Proof:* As mentioned above, if they are independent, then the jump times will be disjoint. We just need to show the converse. I make use of property 7 of Theorem 7 stating that, for real numbers , the processes

are martingales, for *k*=1,2,…,*n*. Next, I make use of the result that, if *M*, *N* are local martingales then is a local martingale. In particular, if they are FV processes whose jump times are almost surely disjoint, the quadratic covariation is zero, so *MN* is a local martingale. Inductively, this shows that the product of any finite number of FV local martingales with pairwise disjoint sets of jump times is itself a local martingale. In our case,

is a local martingale and, as it is uniformly bounded over each finite time interval, this is a proper martingale. Taking conditional expectations , for , gives the following

(7) |

This computes the joint characteristic function of in terms of the product of the individual characteristic functions, showing that they are independent. A simple induction extends this to a sequence of times ,

This is just the result of taking the expectation conditional on and applying (7) across the interval , successively for . This shows that the characteristic function of the joint process at finite sets of times is the product of the characteristic functions of the individual processes. So, they are independent. ⬜

Finally, for this post, we characterize Poisson processes as the counting processes with independent increments property. Compare with the previous post characterizing continuous processes with independent increments.

Theorem 9Any counting process with the independent increments property, and which is continuous in probability, is an (inhomogenous) Poisson process.

*Proof:* Suppose that *X* is a counting process with the independent increments property, and set

for . This is a nonnegative decreasing function, and the aim is to show that is the cumulative rate for *X*. By continuity in probability, in probability for any sequence , so , and is continuous. Next, suppose that for some *t*. We may choose *t* minimal such that this is the case. So, by the independent increments property,

for any . By the choice of *t*, . Also, by continuity in probability, is nonzero for *s* close to *t*. Therefore, , contradicting the assumption.

We have shown that is continuous, decreasing and nonzero. Let , which is continuous and increasing. Finally, by the independent increments property,

That *X* is a Poisson process with cumulative rate is now given by condition 3 of Theorem 7. ⬜

In particular, if *X* has *stationary* independent increments, then it is a homogeneous Poisson process. This was stated above in Theorem 3.

Corollary 10Any counting process with stationary independent increments is a homogeneous Poisson process.

*Proof:* In order to apply Theorem 9 we must show that the counting process *X* with stationary independent increments is continuous in probability. For a sequence of times , stationarity of the increments implies that is distributed as whenever and as when . In either case, which, by right-continuity of *X*, tends to zero in probability as . So, *X* is continuous in probability.

By Theorem 9, *X* is a Poisson process with some cumulative rate . By stationarity of the increments,

So, , giving for some constant . Then, *X* is a Poisson process of rate . ⬜

Dear Mr.George Lowther ,

This is one of the best explanation i had come across so far.Self explanatory.Like to receive this article.

Look forward to see in my mail box.

Keep doing best work like this.

All the best.

Hi,

I was wondering why you stopped your generalization to inhomogeneous Poisson processes i.e. those for which is a deterministic increasing function, and did not include the case of increasing processes ?

If I put it in another way, what equivalences still hold in theorem 7, if we ask for to be only an increasing process adapted (with respect to some large enough filtration), and maybe how general are those processes in the area of counting processes ?

Best regards

Hi. The case where is a continuous deterministic increasing process is relatively simple to deal with, and is the only case where you are assured that the counting process has the Poisson distribution.

The more general case where is just assumed to be continuous, adapted and increasing involves more advanced ideas. Actually, in this case will be the

compensatorofX, which is something I mentioned in my recent post on special semimartingales. I am planning on doing a post on compensators (the next post, I think) and I could include a generalization of Theorem 7. The case where is continuous corresponds toXbeing quasi-left-continuous. More generally, you can take to be a right-continuous and increasing predictable process with the constraint ΔΛ ≤ 1 (assuming thatXcan only have jumps of size 1).Ok thanks for this answer

Best regards

Hello,

Thank you for the great explanation. I wondered if you could let me know a textbook or paper where I coulf find a proof for what you say just before Lemma 5, namely: “A consequence of the compensated process {M_t=X_t-\lambda t} being a martingale is that the jump times of X are totally inaccessible”.

Thanks for your help in advance.

All the best.

This is quite quick to prove. Let T be the time of the n’th jump of X. Also let S be a predictable stopping time. As M is a martingale, E[ΔM

_{S}]=0 (predictable times are fair). Then, as the jumps of X and M are all of size 1,So T satisfies the definition of a totally inaccessible stopping time.

Alternatively, use the result that a cadlag increasing integrable process is quasi-left-continuous if and only if its compensator is continuous (quasi-left-continuous = jump times are totally inaccessible). Any textbook which covers similar material to these notes should also include these results.

(apologies for the very slow response here, I’ve been busy lately and not had much time to update the blog).

Hi, I think there is a typo on the first line of the second paragraph. latex N\sim {rm Po(\lambda)}

Fixed. Thanks!