Assignment- optimal stopping rule for a gambler or city

Question

Assignment- Optimal Stopping Rule for a Gambler (or City Trader)

Objective - To practice MATLAB and basic Simulation Techniques.

Problem - A gambler makes a series of plays with outcomes X₁, X₂,.... where the X_i, i ≥ 1 are independent, identically distributed random variables. The cost of each play is c. The gambler can stop at any time n ≥ 1 with the fortune:

Y_n = max_1≤i≤n X_i - cn

The gambler will stop if X_n > β. The gambler wants to maximize the mean of his fortune. What are the best values of β in the following two cases?

1. Case A: c = 0.2 and the distribution of X_n is the exponential distribution, i.e., its density function is

2. Case B: c = 0.1 and X_n is a discrete random variable obeying Poisson distribution:

P(x = k) = 1/ek!    for k = 0, 1, 2,.....,

Task:

1. Find the best of values of β by using Monte Carlo simulation.

2. Write a report with the following sections:

- Problem: You should explain this problem in a way such that a 16-year old high school student can understand it.

- Method: You should describe and explain your method (not your matlab code) in details.

- Code: well-documented code.

- Experimental Results and Discussion: You report your experimental results here and give some discussions. You should use figures to show your results.

- Conclusion: Summarize what you have done and your reflection.