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Question 1. Stochastic integration: A small company is investing resources in a risky project that it hopes will be profitable. The project could, for example, represent the manufacturing and selling of a gadget in a local market. The performance of the market is measured by the Brownian motion Zt = ut + oWt, where W+ is a standardized Brownian motion, as discussed in class. Here, the randomness of market performance could for example be driven by uncertain market demand. The small change Z+++ - Zt = u At + o (Witst - Wt), or on differential form: d2=putt+odW, represents how well the market performs over a short time period at time t. We call the variable Z the "market performance measure." We assume that u > 0 and o > 0, representing that the measure is (weakly) expected to increase and that is has a random component. At each point in time, the company chooses how much to invest in the market, I. Here, I can in general be a function of t, as well as of Wt. We allow I to be negative for technical reasons.¹ Let Vt denote the value created by the project up until time t. The market performance measure Z then has the following interpretation: If the company chooses to invest a constant, k, between t and t + At, i.e., if Is = k for t S + At, then Vi+At-V=(Z+At-Zt), - = i.e., the change in value depends on how well the market performed and how much the company invested. Written as a stochastic integral, this then leads to the formula VT = ItdZt T = It(uct + odWt) = u 1. T Itdt + o to It dWt. Here, the stochastic (second) part of the integral is defined in the sense of Ito, as discussed in class. (a) Given that the company chooses a fixed investment strategy, I = k: between time 0 and T, derive an expression for what the value of the project at time T is. The value should be expressed as a function of WT, k, H and o. The company wants to choose an investment strategy, I, that leads to as high a value of the project as possible, but it is also risk averse, in that it dislikes high variations of the project's value. Specifically, we use the utility model from assignment 6, and assume that the "value" the company associates (at time 0) with an investment strategy leading to the random value VT at time T is where Y > 0 is a risk aversion parameter that governs how the company trades off uncertainty in terms of variance against expected value.² (b) Assume that the company has to determine a constant investment strategy at t = 0, I III k. What value of k: should the company choose? It should follow from your result in (b) that in the special case when H = 0, the company should choose k: = 0. This is not surprising since when 11 = 0, the performance measure has an expected value of 0 and is risky, SO a risk averse investor sees no benefit in investing, but only costs in the form of risk. Question 2. This is a continuation of question 1. In what follows, assume that u = 0. An analyst, who has a soft spot for the project is considering whether there is some other investment strategy that the company can choose that makes investing worthwhile. Specifically, the analyst realizes that the company does not have to choose a fixed investment strategy. Instead, it can let It depend on Wt, and/or t. The analyst has an idea that the company should choose a time varying strategy such that it invests a lot when the performance measure is high, and very little (even negative amounts) when it is low. By using this investment strategy-the analyst argues-the company will invest a lot in favorable times, and little in unfavorable times, and therefore outperform the linear strategy. Along these lines, the analyst proposes that the company should choose the state dependent investment strategy It=aWt, for some constant O > 0. (c) Derive expressions for the expectation and variance of VT, as a function of T, a, and o (recall that H = 0), when the state dependent strategy is chosen. (d) Given the company's objective function, what value of O should it choose? (e) What is wrong with the analyst's argument?

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