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Stochastic process N(t) is a Counting Process (point process) if

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

2] in each random scenario, the trajectory of N(t) is piece-wise constant,

3] N(t) is non-decreasing,

4] in each random scenario, the trajectory of N(t) is right-continuous,

5] for any moment of time *s* there is time t > s such that N(t) = N(s) + 1.

Counting processes are 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.

The intensity of counting process N(t) is defined to be a stochastic process 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. The intensity stands for the expected number of jumps per unit of time in a very small time interval.

Counting process is a generalization of a non-homogeneous Poisson 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. Trajectories of counting process are visualized in this plot.

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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