EE 512  Fall 2022 Class ProjectEE 512  Fall 2022 Class Project Solve two the following simulation problems in any computational language of your choice. Submit a report (strongly recommend either a Jupyter notebook or an R Markdown document) with your results, including all relevant captioned figures. 1. [Stochastic Control of Investment Portfolios]: An investor has two kinds of assets available to invest in: a riskfree asset Xt0 (safer) and a riskier asset Xt1 both modeled with Ito processes. At time t, the investor is free to modify how much of their total wealth Zt is allocated to each asset. We use the timevarying policy (or decision) function ut ∈ [0, 1] to encode what fraction of Zt is in Xt0 vs. in Xt1. The investor cannot exceed or withhold any part of their wealth from investment (i.e. no borrowing or short selling). This leads to the following expression for the investor’s wealth process: Assume the investor acts on a monthly clock t for up to t = 500 months. (a) Identify conditions on ut that would make Zt an Ito process. (b) Pick a few (e.g. 10) different risk profiles (a, b, α) for Zt. In real applications, typically b < a. Assume the investor defaults to investing equally in both assets (i.e. ut = 0.5 ∀t). Plot ensembles of sample paths for Zt under your selected risk profiles (a, b, α). (c) To evaluate the quality of a decisionmaking process, it is customary to estimate the average utility derived from following that decision process. Utility functions u(.) are typically concave increasing functions. For this problem, we use the logarithmic utility function u(z) = ln(z) (referred to as the Kelly Criterion). Use your ensembles of sample paths to estimate the average utility in the terminal month t = 500 derived from following default investment policy above ut = 0.5 ∀t. i.e. Derive Monte Carlo estimates for (d) Derive Monte Carlo estimates for the average termninal utility derived from following the following alternative investment policy (also a constant fraction over time): Compare the performance of both investment policies (e) Do one of the following: i. Compare the two investment policies under your chosen generalization of the control problem. E.g. modeling Xt1 with a jump processs, using a different (valid) utility function, adding more investment options, etc. OR ii. Give a detailed outline of how you would apply statistical methods to learn optimal investment policies from data and experimentation (e.g. reinforcement learning). Be sure to characterize the strengths and weakness of your proposal. 2. [Options Pricing via Monte Carlo Estimation]: Generate T = 500day sample paths of the geometric Brownian motion (GBM) for different values of (μ, σ) using the EulerMaruyama simulation scheme. Recall that the GBM is the stochastic process that satisfies the following SDE: dSt =μ(t,St)dt+σ(t,St)dBt =μ·Stdt+σ(t,St)·StdBt .
3. [Optimal City Paths]: The famous Traveling Salesman Problem (TSP) is an NPhard routing problem. The time to find optimal solutions to TSPs grows exponentially with the size of the problem (number of cities). A statement of the TSP goes thus: A salesman needs to visit each of N cities exactly once and in any order. Each city is connected to other cities via an air transportation network. Find a minimum length path on the network that goes through all N cities exactly once (an optimal Hamiltonian cycle). A TSP solution is just an ordered list of the N cities with minimum path length. We will be exploring MCMC solutions to small and larger scale versions of the problem. (a) Pick N = 10 2D points in the [0, 1000] × [0, 1000] rectangle. These 2D samples will represent the locations of N = 10 cities.
(b) Run the Simulated Annealing TSP solver you just developed for N = {40, 400, 1000} cities. Explore the speed and convergence properties at these different problem sizes. You might want to play with the cooling schedules.
