Poisson process

Paths of two Poisson processes with constant intensity: 2.4 ( blue ) and 0.6 ( red ). The blue process has a four times higher intensity and shows even with 30 jumps in the drawn time interval [0; 14.9] far more than the red (only 8). This is almost exactly four times as many jumps, which was to be expected.

Paths of two compensated composite Poisson processes. As above, the intensity (frequency of jumps) of the blue process, at 2.4, is exactly four times that of the red process. In the drawn interval [0; 35] the blue process jumps 66 times (expected would be 35 · 2.4 = 84), the red 16 times, that is about four times as often. In both processes, the jumps are normally distributed with a mean of 0.25. These upward jumps are balanced out (compensated) by the negative drift in such a way that both processes are martingales. Since the blue process jumps up more often, its negative drift is stronger.

A Poisson process is a stochastic process named after Siméon Denis Poisson . It is a process of renewal , the gains of which are Poisson distributed .

The rare events described with a Poisson process, however, typically have a high risk (as a product of costs and probability). For this reason, it is often used in the insurance industry to model malfunctions in complex industrial plants, flood disasters, plane crashes, etc.

parameter

The distribution of the increases has a parameter λ, this is referred to as the intensity of the process, since exactly λ jumps are expected per unit of time (the expected value of the Poisson distribution is also λ). The height of each jump is one, the times between the jumps are exponentially distributed . The Poisson process is thus a discrete process in continuous (i.e. continuous) time .

definition

A Poisson point process is a random measure , more precisely a point process , with an s-finite intensity measure on an arbitrary measure space that fulfills the following two conditions: ${\displaystyle \eta }$ ${\displaystyle \lambda }$ ${\displaystyle (\mathbf {X} ,{\mathcal {X}})}$

1. For every measurable amount the random variable is Poisson distributed with parameters . That is, it applies to everyone . ${\displaystyle B}$ ${\displaystyle \eta (B)}$ ${\displaystyle \lambda (B)}$ ${\displaystyle \mathbb {P} (\eta (B)=k)=P_{\lambda (B)}(k)}$ ${\displaystyle k\in \mathbb {N} _{0}}$
2. For any number of pairwise disjoint sets the random variables are independent. ${\displaystyle B_{1},\dots ,B_{n}\in {\mathcal {X}}}$ ${\displaystyle \eta (B_{1}),\dotsc ,\eta (B_{n})}$

Shorthand is also used for a Poisson point process . If it is a homogeneous (i.e. also stationary ) Poisson point process, one also writes , whereby the -fold Lebesgue measure is meant. The following applies to the intensity measure . ${\displaystyle \eta \sim {\text{PPP}}(\lambda )}$ ${\displaystyle \eta \sim {\text{PPP}}(\lambda \mathrm {d} x)}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda (B)=\operatorname {E} (\eta (B))}$

Poisson point processes can be viewed in any space. Often one is interested in space or in the positive real axis . In particular, when one speaks of a Poisson point process on the real axis, the second property is also called independent increments . ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle \mathbb {R} _{+}}$

Unfortunately, the terminology is not standardized. Some authors speak of the Poisson process and thus mean the Poisson point process, others mean by the Poisson process Poisson counting process , ie . The latter counts the number of points of the Poisson point process up to the point in time . ${\displaystyle N(t)_{t\geq 0}:=\eta ([0,t])_{t\geq 0}}$ ${\displaystyle t}$

Definition on ${\displaystyle \mathbb {R} _{+}}$

A stochastic process with càdlàg paths over a probability space is called a (homogeneous) Poisson process with intensity and , if the following three conditions are met: ${\displaystyle [\Omega ;{\mathfrak {A}};\mathbb {P} ]}$ ${\displaystyle P_{\lambda ,t}\,}$ ${\displaystyle \lambda \,}$ ${\displaystyle t\in [0;\infty )}$

• ${\displaystyle P_{\lambda ,0}=0{\text{ }}\left(\mathbb {P} -f.s.\right)\,}$(See Almost Safe Properties ).
• ${\displaystyle P_{\lambda ,t}-P_{\lambda ,s}\sim {\mathcal {P}}_{\lambda \cdot (t-s)}\,}$ ${\displaystyle \forall \,s<t}$. The Poisson distribution denotes with parameter . ${\displaystyle {\mathcal {P}}_{\lambda \cdot (t-s)}}$ ${\displaystyle \lambda \cdot (t-s)}$
• Be given for a sequence . Then the family is stochastically independent of random variables . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle 0<t_{1}<\dotsb <t_{n}\,}$ ${\displaystyle \langle P_{\lambda ,t_{i}}-P_{\lambda ,t_{i-1}}\mid 2\leq i\leq n\rangle }$

For the definition of the inhomogeneous Poisson process see Poisson process # Inhomogeneous Poisson process .

properties

• By definition, a Poisson process is a stochastic process with independent increments.
• A homogeneous Poisson process is a Markov process in continuous time with a discrete state space. The Q matrix is . ${\displaystyle p_{ij}=\lambda \mathbf {1} _{\{j=i+1\}}-\lambda \mathbf {1} _{\{j=i\}}}$
• The period between two increases is exponentially distributed with the parameter . The waiting time for the next jump is therefore memoryless , i. H. the remaining waiting time for the next jump is independent of the previous waiting time. It follows from this that the waiting time paradox also applies here. ${\displaystyle \min \left\{t\in [0;\infty )\mid P_{\lambda ,t}=n+1\right\}-\min \left\{s\in [0;\infty )\mid P_{\lambda ,s}=n\right\}\;\;n\geq 0}$ ${\displaystyle \lambda }$
• Accordingly, the waiting time until the -th jump is gamma distributed with parameters and . You can see that clearly when you as writes. ${\displaystyle n}$ ${\displaystyle T_{n}}$ ${\displaystyle n}$ ${\displaystyle \lambda }$ ${\displaystyle T_{n}}$ ${\displaystyle T_{n}=T_{1}+(T_{2}-T_{1})+\cdots +(T_{n}-T_{n-1})}$
• Is a Poisson process and , then for another Poisson process, d. H. the gains of homogeneous Poisson processes are stationary. A homogeneous Poisson process is therefore a special Lévy process . ${\displaystyle P_{\lambda ,t}\,}$ ${\displaystyle s\geq 0}$ ${\displaystyle {\hat {P}}_{\lambda ,t}=P_{\lambda ,s+t}-P_{\lambda ,s}}$ ${\displaystyle t\geq 0}$
• The following applies to the expected value and the variance . ${\displaystyle \operatorname {E} (P_{\lambda ,t})=\operatorname {Var} (P_{\lambda ,t})=\lambda \cdot t}$
• For the quadratic variation, the following applies: the constant martingale component disappears and all jumps have height 1. ${\displaystyle [P_{\lambda }]_{t}=P_{\lambda ,t}}$ ${\displaystyle P_{\lambda }^{c}}$
• As the path of the process increases monotonically, a submartingale is more natural in terms of its filtration . ${\displaystyle P_{\lambda ,t}\,}$
• If one has a stochastic process that fulfills the three defining properties, there is a version of the process with càdlàg paths, i.e. a Poisson process.
• ${\displaystyle M_{\lambda ,t}:=P_{\lambda ,t}-\operatorname {E} (P_{\lambda ,t})=P_{\lambda ,t}-\lambda \cdot t}$is called the compensated Poisson process and is a martingale with regard to its natural filtration.
• Under relatively general assumptions, the superposition of general renewal processes converges asymptotically against a Poisson process ( Palm-Chinchin theorem ).
• The mapping theorem applies , that is, a Poisson point process with intensity in a measurable mapping , again forms a Poisson point process with the intensity . ${\displaystyle \gamma }$ ${\displaystyle f}$ ${\displaystyle f(\gamma )}$

Alternative definition

In the definition above, the Poisson distribution is assumed, but it can also be derived from the fundamental properties of a stochastic process (Poisson assumptions). If these properties can be assigned to an event in good approximation, the event frequency will be Poisson distributed. In 1837 Poisson published his thoughts on this distribution together with his probability theory in the work "Recherches sur la probabilité des jugements en matières criminelles et en matière civile" ("Investigations into the probability of judgments in criminal and civil cases").

One considers a space or time continuum in which countable events take place with a constant mean number per unit interval (a Bernoulli experiment is very often carried out, so to speak, at every point on the continuum). Now one looks at a sufficiently small continuum interval that, depending on the experiment, can represent an area, a time interval, a delimited distance, area or volume. What happens there determines the global distribution on the continuum. ${\displaystyle w}$ ${\displaystyle g}$ ${\displaystyle \Delta w}$

The three Poisson assumptions are:

1. There is at most one event (rarity) within the interval . ${\displaystyle [w,w+\Delta w]}$
2. The probability of finding an event in the interval is proportional to the length of the interval . Since is constant, it is also independent of . ${\displaystyle \Delta w}$ ${\displaystyle g}$ ${\displaystyle w}$
3. The occurrence of an event in the interval is not influenced by events that took place in the prehistory (lack of history). ${\displaystyle \Delta w}$

With assumptions 1 and 2, the probability of finding an event in the interval is given as ${\displaystyle \Delta w}$

${\displaystyle p_{1}(\Delta w)=g\cdot \Delta w,}$

as well as the probability of an empty interval

${\displaystyle p_{0}(\Delta w)=1-p_{1}(\Delta w)=1-g\cdot \Delta w.}$

According to assumption 3, the probability of an empty interval is independent of the occurrence of any events in the area before it. How to calculate the probability of an event to point to ${\displaystyle \Delta w}$ ${\displaystyle w}$ ${\displaystyle w+\Delta w}$

${\displaystyle p_{0}(w+\Delta w)=p_{0}(w)\cdot p_{0}(\Delta w)=p_{0}(w)-g\cdot p_{0}(w)\cdot \Delta w.}$

That approximates the ordinary differential equation with the solution ${\displaystyle {\tfrac {\mathrm {d} p_{0}(w)}{\mathrm {d} w}}=-g\cdot p_{0}(w)}$

${\displaystyle p_{0}(w)=\mathrm {e} ^{-g\cdot w}}$

under the initial condition . Likewise, one finds the probability of events up to the point ${\displaystyle p_{0}(0)=1}$ ${\displaystyle m}$ ${\displaystyle w+\Delta w}$

${\displaystyle p_{m}(w+\Delta w)=p_{m}(w)\cdot p_{0}(\Delta w)+p_{m-1}(w)\cdot p_{1}(\Delta w)=p_{m}(w)-g\cdot p_{m}(w)\cdot \Delta w+g\cdot p_{m-1}(w)\cdot \Delta w.}$

According to assumption 1, each appended interval may contain either no or one event. The corresponding differential equation has the solution ${\displaystyle \Delta w}$ ${\displaystyle {\tfrac {\mathrm {d} p_{m}(w)}{\mathrm {d} w}}=-g\cdot p_{m}(w)+g\cdot p_{m-1}(w)}$

${\displaystyle p_{m}(w)={\frac {(g\cdot w)^{m}}{m!}}\mathrm {e} ^{-g\cdot w}.}$

If one now identifies the parameters with and with in this expression, which describes the probability of occurrence of events in the continuum region , it agrees with the formula of the Poisson distribution. In many tasks, the number is the product of a rate (number of events per unit interval) and a multiple of the unit interval. ${\displaystyle m}$ ${\displaystyle w}$ ${\displaystyle (g\cdot w)}$ ${\displaystyle \lambda }$ ${\displaystyle m}$ ${\displaystyle k}$ ${\displaystyle \lambda }$

Compound Poisson processes

If a Poisson process with intensity and independently and identically distributed random variables is independent of , then the stochastic process becomes ${\displaystyle N_{t}}$ ${\displaystyle \mu }$ ${\displaystyle Y_{1},Y_{2},\ldots }$ ${\displaystyle N_{t}}$

${\displaystyle X_{t}:=\sum _{n=1}^{N_{t}}Y_{n}}$

referred to as the compound Poisson process . is then composed of a Poisson distribution . Like the original Poisson process, X is also a jump process of independent increases and exponentially (µ) -distributed distances between the jumps, with jump heights that are distributed according to Y. If fs applies , a Poisson process is obtained again. ${\displaystyle X_{t}}$ ${\displaystyle Y_{1}=1}$

The formula of Wald (after the mathematician Abraham Wald ) applies to the expected value ,

${\displaystyle \mathbb {E} (X_{t})=\mathbb {E} (N_{t})\mathbb {E} (Y_{1})=\mu t\mathbb {E} (Y_{1})}$.

The Blackwell-Girshick equation applies to the variance :

${\displaystyle \operatorname {Var} (X_{t})=\mu t\operatorname {E} (Y_{1})^{2}+\mu t\operatorname {Var} (Y_{1})}$.

Compound Poisson processes are Lévy processes.

Inhomogeneous Poisson process

Graph of an inhomogeneous Poisson process. The events are marked as black crosses. The rate that changes over time is shown in red. ${\displaystyle \lambda (t)}$

In some cases it can make sense to view it not as a constant but as a function of time. must thereby meet the two conditions ${\displaystyle \lambda }$ ${\displaystyle \lambda (t)}$

• ${\displaystyle \lambda (t)>0}$for everyone and ${\displaystyle t\in \mathbb {R} _{+}}$
• ${\displaystyle \int _{\tau _{1}}^{\tau _{2}}\lambda (t)\,\mathrm {d} t<\infty }$ For ${\displaystyle \tau _{1},\tau _{2}\in \mathbb {R} _{+}}$

fulfill.

In contrast to a homogeneous Poisson process, the following applies to an inhomogeneous Poisson process: ${\displaystyle (P_{\lambda (t),t})_{t\geq 0}}$

• ${\displaystyle P_{t}-P_{s}\sim {\mathcal {P}}_{\int _{s}^{t}\lambda (u)\,\mathrm {d} u}}$, again denoting the Poisson distribution with the parameter . ${\displaystyle {\mathcal {P}}}$ ${\displaystyle \int _{s}^{t}\lambda (u)\,\mathrm {d} u}$
• The following applies to the expected value . ${\displaystyle \operatorname {E} (P_{t})=\int _{0}^{t}\lambda (u)\,\mathrm {d} u}$
• The same applies to the variance . ${\displaystyle \operatorname {Var} (P_{t})=\int _{0}^{t}\lambda (u)\,\mathrm {d} u}$
• If and are two jump points of the inhomogeneous Poisson process, then is exponentially distributed with the parameter 1. ${\displaystyle \tau _{1}}$ ${\displaystyle \tau _{2}}$ ${\displaystyle \int _{\tau _{1}}^{\tau _{2}}\lambda (t)\,\mathrm {d} t}$

Cox trial

An inhomogeneous Poisson process with a stochastic intensity function is called a double stochastic Poisson process or, according to the English mathematician David Cox, also the Cox process . If one considers a certain realization of , a Cox process behaves like an inhomogeneous Poisson process. For the expected value of true ${\displaystyle \lambda (t)}$ ${\displaystyle \lambda (t)}$ ${\displaystyle P_{\lambda (t),t}}$

${\displaystyle \operatorname {E} (P_{\lambda (t),t})=\operatorname {E} \left(\int _{0}^{t}\lambda (u)\,\mathrm {d} u\right)}$.

Application examples

• General:
  • Counting evenly distributed events per area, space or time measure (e.g. number of raindrops on a street; number of stars in a volume V is a three-dimensional Poisson process)
  • Determination of the frequency of rare events such as insurance claims, decay processes, repair orders or the number of goals in a football match (see Metin Tolan's football book )
• Control systems:
  • the random number of phone calls per unit of time
  • the random number of customers at a counter per unit of time
  • the times at which requests (persons, jobs, telephone calls, heap, ...) are received by an operator (bank, server, switchboard, memory management, ...)
• Errors, failures, quality control:
  • the random number of non-germinating seeds from a package
  • the places where a thread has pimples
  • Number of dead pixels on an LCD
  • Number of potholes on a country road
  • Number of misprints in a book
  • Number of accidents per unit of time at an intersection
  • On [1] (PDF; 35 kB) an attempt is made to model the sequence of suicides at the Massachusetts Institute of Technology as a Poisson process.
• Physics:
  • the times at which a radioactive substance emits a particle ${\displaystyle \alpha }$
  • random number of particles emitted by a radioactive substance in a given period of time ${\displaystyle \alpha }$
• Actuarial :
  • the timing of major insurance losses. In the financial and non-life insurance mathematics occurrence is usually described by damage to be covered by a composite Poisson process, in which the individual occurring independently damage after Y are distributed. If you add a deterministic, negative drift (insurance premiums) to this damage process, you get the insurance company's asset process, also known as the risk process. This is followed by questions such as: How likely is it that the asset process will exceed a certain threshold value x , i.e. the insurance reserves, and thus suffer bankruptcy (so-called ruin problem)? How strong does the negative drift or the contribution rate have to be in order to keep the probability of bankruptcy (so-called ruin probability) below a given threshold?
• Financial Mathematics :
  • Models for the prices of stocks , with jumps also allowed. Lévy processes are often used for this, but since infinite activity is often difficult to measure, compound Poisson processes are also used.
  • Credit risk models help to evaluate and model CDS , spreads and other credit derivatives.

literature

• Sheldon M. Ross: Stochastic Processes. Wiley, New York NY et al. 1983, ISBN 0-471-09942-2 (2nd edition. Ibid. 1996, ISBN 0-471-12062-6 ).

Individual evidence

1. ↑ ^{a b} Günther Last, Mathew Penrose: Lectures on Poisson Process July 5, 2017.