S TOCHASTIC M ODELS L ECTURE 2 P ART II P OISSON P ROCESSES Nan Chen MSc Program in Financial Engineering The Chinese University of Hong Kong (ShenZhen) Sept. 30, 2015

Outline 1.Time-inhomogeneous Poisson process 2.Compound Poisson 3.Mixed Poisson 4.Random Intensity and Hawkes Process

2.4 T IME -I NHOMOGENEOUS P OISSON P ROCESSES

Time Homogeneity The arrival intensity of a regular Poisson process is time homogenous. In other words, is a Poisson random variable with mean for any time But, in engineering practices, we do need non-homogenous Poisson processes to model non-stationary arrivals.

Arrival Intensity to a Call Center

Non-homogeneous Poisson Processes We may introduce a time-dependent intensity function to indicate the arrival rates at each moment For a non-homogeneous Poisson process, its increment should be a random variable with mean

Example I: Siegbert’s Hot Dog Stand Siegbert runs a hot dog stand that opens at 8am and closes at 5pm. He observes the following pattern about customer arrivals. – 8am to 11am: on average, a steadily increasing rate that starts with an initial rate of 5 customers per hour at 8am and reaches a maximum of 20 customers per hour at 11am – 11am to 1pm: remaining at 20 per hour

Example I: Siegbert’s Hot Dog Stand (Continued) (Continued) – 1pm to 5pm: arrival rates drop steadily from 20 customers per hour to 12 customers per hour. What is the probability that no customers arrive between 8:30am to 9:30am? What is the expected number of arrivals in this period?

Generating Nonhomogeneous Poisson Time sampling an ordinary Poisson process generates a non-homogeneous Poisson process. Let be a Poisson process with rate, and suppose that an event occurring at time, independently of what has occurred prior to, counted with probability

Generating Nonhomogeneous Poisson (Continued) Let denote the counting process of counted events. Then, it is a non- homogeneous Poisson process with intensity function

2.5 C OMPOUND P OISSON P ROCESS

Compound Poisson Processes A stochastic process is said to be a compound Poisson process if it can be represented as where is a Poisson process, and is a family of independent and identically distributed random variables.

Example II: Buses and Fans Suppose that buses arrive at a sporting event in accordance with a Poisson process, and suppose that the numbers of fans in each bus are assumed to be i.i.d.

Example II: Buses and Fans (Continued) Then is a compound Poisson process, where denotes the number of fans who have arrived by and represents the number of fans in the ith bus.

Mean and Variance Mean: Variance:

Compound Poisson Process as a Superposition of Poisson Processes There is a very nice presentation of the compound Poisson process when the set of possible values of the is finite. Let Let us say that the event is a type event whenever the value of this arrival is Then,, the number of type events by time, follows a Poisson process with rate

Compound Poisson Process as a Superposition of Poisson Processes Furthermore,

Example III: Jump Model for Security Price Let be the price process of a financial security. A popular model supposes that the price remains unchanged until a “shock” occurs, at which time the price is multiplied by a random factor. That is, where is equal to 1 when

Example III: Jump Model for Security Price (Continued) Suppose that the are independent exponential random variable with rate is a Poisson process with rate and independent of Find and

2.6 M IXED P OISSON P ROCESSES

Definition Let be a counting process constructed as follows. There is a positive random variable such that, conditional on the counting process is a Poisson process with rate Such a counting process is called a conditional or mixed Poisson process.

Distribution of Increments of Mixed Poisson Process The distribution of the increment

Conditional Distribution of

Conditional Distribution of (Continued) The conditional density function of is thus given by

2.7 R ANDOM I NTENSITY AND H AWKES P ROCESSES

Random Intensity Whereas the intensity function of a non- homogeneous Poisson process is a deterministic function, there are counting process whose intensity function value at time, denoted by, is a random variable whose value depends on the history of the process up to time

Hawkes Processes The Hawkes process is an example of a counting process with random intensity. Its intensity function is given by

Intensity Function of Hawkes A sample trajectory of

Self-Exciting Feature A Hawkes process is a counting process in which – – whenever an event occurs, the random intensity increases; that is, it is self-exciting; – if there is no events between and, then

Homework Assignments Read Ross Chapter 5.4 and 5.5. Answer Questions: – Exercises 38 (Page 344, Ross) – Exercises 46 (Page 345, Ross) – Exercises 59 (Page 348, Ross) – Exercise 71 (Page 350, Ross) – Exercise 78 (Page 352, Ross) – (Optional, Extra Bonus) Exercise 81 (Page 352), 87 (page 353, Ross).