Introduction to Stochastic Calculus with Applications, Second Edition
Containing many solved examples and exercises, this book gives a simple but rigorous treatment of stochastic calculus and its applications, including a range of advanced topics.
TABLE OF CONTENTS Introduction to Stochastic Calculus with Applications, Second Edition
Chapter 9: Pure Jump Processes
In this Chapter we consider pure jump processes, that is, processes that change only by jumps. Counting processes and Markov Jump processes are defined and their semimartingale representation is given. This representation allows us to see the process as a solution to a stochastic equation driven by discontinuous martingales.
9.1 Definitions
A counting process is determined by a sequence of non-negative random variables T n, satisfying T n < T n+1 if T n < ? and T n = T n+1 if T n = ?. T n can be considered as the time of the n-th occurrence of an event, and they are often referred to as arrival times. N( t) counts the number of events that occurred up to time t, that is,
N( t) is piece-wise constant and has jumps of size one at the points T n. Such processes are also known as simple point processes to distinguish them from more general marked point processes, which are described by a sequence ( T n, Z n) for some random variables Z n. Z n, for example, may describe the size of jump at T n.
The pure jump process X is defined as follows.
Note that X in (9.2) is right-continuous, piece-wise constant with the time of the n-th jump at T n, and Z n
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