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COUNTING PROCESS

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  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.

COUNTING PROCESS SUBCATEGORIES

COUNTING 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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