On cumulative jump random variables

Abstract

Stochastic models for phenomena that can exhibit sudden changes involve the use of processes whose sample functions may have discontinuities. This paper provides some tools for working with such processes. We develop a sample path formula for the cumulative jump height over a given time interval. From this formula an expression for the expected value of the cumulative jump random variable is developed under reasonable conditions. The results are applied to finding the expected number of failures in the separate maintenance model over a stated time interval and to the expected number of occurrences of a regenerative event over a stated time interval.

Appendix

In this appendix, we show how Marlow’s original computation generalizes to yield the number of jumps, regardless of size, of a cadlag+ step function.

Suppose 0=u ₀<u ₁<⋯<u _n <t and $f(x) = \sum_{i = 1}^{n} \alpha_{i}I_{ [ u_{i - 1},u_{i} )}(x)$ for x∈[0,t]. Then

$$\int \!\!\!\int _{0 \le x \le y \le t} \lambda^{2}e^{ - \lambda ( y - x )} \bigl[ f(y) - f(x) \bigr]^{ \vee } \,dy\, dx = \sum_{i = 1}^{n} \int_{u_{i - 1}}^{u_{i}} \int_{x}^{t} \lambda^{2}e^{ - \lambda ( y - x )} \bigl[ f(y) - \alpha_{i} \bigr]^{ \vee } \,dy\, dx $$

because the integral over [u _n ,t] is zero by virtue of f being a step function. Then this becomes

$$\sum_{i = 1}^{n} \int_{u_{i - 1}}^{u_{i}} \Biggl[ \int_{x}^{u_{i}} + \sum _{j = i + 1}^{n} \int_{u_{j - 1}}^{u_{j}} + \int_{u_{n}}^{t} \Biggr] \lambda^{2}e^{ - \lambda ( y - x )} \bigl[ f(y) - \alpha_{i} \bigr]^{ \vee }\, dy\, dx . $$

The integral from x to u _i is zero because f(y)=α _i for all y∈[u _i−1,u _i ). What remains is equal to

The first term is

As λ→+∞, the only term in the inner sum that survives is the j=i+1 term, and this is equal to

$$\sum_{i = 1}^{n} ( \alpha_{i + 1} - \alpha_{i} )^{ \vee }. $$

The second term is

and this term goes to zero as λ→+∞. Finally, we obtain

$$\lim _{\lambda \to + \infty }\int \!\!\!\int _{0 \le x \le y \le t} \lambda^{2}e^{ - \lambda ( y - x )} \bigl[ f(y) - f(x) \bigr]^{ \vee } \,dy\, dx = \sum_{i = 1}^{n} ( \alpha_{i + 1} - \alpha_{i} )^{ \vee } , $$

as desired.