Pstat 160B                                                             Programming Assignment 1

Instructions:

You can use either R or Python. I strongly suggest you to use notebooks (either R Markdown or Jupyter).

The code must be annotated. Show your results and make your conclusion.

Submit these files on Gauchospace:

– pdf file, which should contain annotated code, results, graphics (if any) , conclusion.

– source file (.ipynb, .Rmd, .py or .R) in case  we  need  to  rerun  your  code.  Therefore  clearly state the seed you’re using for your simulations.

The problems below are due Sunday January 27th at 11:59pm.

1. (Sarantsev 160 Notes, Problem 21.16 page 119) Simulate 1000 times the first 50 jumps of a Poisson process (Nt)t>0 with intensity λ = 2. Calculate the empirical expectation E[N1N2], and compare it with the true value.

2.(Dobrow, Problem 6.44 page 264) Investors purchase \$1000 dollar bonds at the random times of a Poisson process with parameter λ. If the interest rate is r, then the present value of an investment purchased at time t is 1000e−rt. Then the expected total present value of the bonds purchased by time t is 1000(1 − e−rt) λ .

Simulate the expected total present value of bonds if the interest rate is 4%, the Poisson parameter is λ = 50, and t = 10. Compare with the exact value.

3.Compound Poisson process. We modify the Poisson process to let it jump not only by 1 upward, but in a more general way. Define a sequence of i.i.d. (independent identically distributed) random variables Z1, Z2, . . . , independent of (Nt)t≥0. Then a compound Poisson process is defined as

It starts from C0 = 0, then waits time X1 and jumps to CX1 = Z1. Next, it waits additional time X2  (for the total time S1 = X1 + X2) and jumps to CS1  = Z1 + Z2, then waits time X3  and jumps to CS3 = Z1 + Z2 + Z3, etc. For a Compound Poisson process the mean and the variance are given by:

E(Ct) = E(Nt)E(Zk)

V ar(Ct) = E(Nt)V ar(Zk) + V ar(Nt)(E(Zk))2

(Sarantsev 160 Notes, Problem 21.17 page 119) Simulate 1000 times the first 50 jumps of a compound Poisson process (Ct)t≥0 with increments Z1, Z2, . . . distributed as N (2.5, 4), and intensity λ = 0.5. Use this to find empirical value V ar(C4), and compare this with the true  value.

1 of 1