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POISSON PROCESS

Stochastic process N(t) is a Poisson Process if

1] N(t) takes non-negative integer values;

2] N(0) = 0;

3] N(t) has independent increments: for any moments of time s < t < u, random variables N(t) - N(s) and N(u) - N(s) are independent;

4] there exists a deterministic function L(t) such that, for any time t and shift h,

P( N(t+h) - N(t) = 1 | N(s), 0 ≤ s  t ) = L(t) * h + o(h),

where o(h) is a function of h which converges to 0 when h converges to 0;

5] for any time t and shift h,

P( N(t+h) - N(t) ≥ 2 | N(s), 0  t ) = o(h).


Function L(t) is called the intensity of N(t). It indicates how often process N(t) jumps. In fact, it stands for the expected number of jumps per unit of time in a very small time interval. If L(t) is a constant, process N(t) is called homogeneous Poisson process. If L(t) varies with time, process N(t) is called non-homogeneous Poisson process.

It can be shown that the increment of a homogeneous Poisson process on an interval of length h has Poisson distribution with parameter L * h. This is the connection with Poisson random variables. From here it follows that, for any time t, the distribution of N(t) is Poisson with parameter L * t.

Poisson process is a particular case of counting process. Poisson process jumps with a deterministic intensity, which is allowed to vary with time. In a more complex fashion, counting process jumps with a stochastic intensity, which is allowed to vary with time.

Poisson process can be used to model the number of events of a particular type in a given time interval. Examples: defaults of companies registered in the US, deaths of people based in Los Angeles, hurricanes in the Caribbean basin, etc. Trajectories of Poisson process are visualized in this plot.


POISSON PROCESS REFERENCES

Lawler, G. F. (1995). Introduction to Stochastic Processes. New York: Chapman and Hall/CRC.

Ross, S. M. (1995). Stochastic Processes (2nd ed). New York: Wiley.

Karlin, S., & Taylor, H. M. (1975). A First Course in Stochastic Processes (2nd ed). New York: Academic Press.

Protter, P. E. (2005). Stochastic Integration and Differential Equations (2nd ed). Springer.

Bremaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer.


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