Chapter Iv - Applications to Mathematical Finance

where u1 is the proportion of the portfolio held in stocks, and u2 is the consumption rate. Evidently it is natural to take 0 ≤ u1 ≤ 1 and u2 ≥ 0, but in fact we shall allow −∞ < u1 <∞. If u1 > 1 then the cash part of the portfolio is negative, so we are borrowing money to buy stocks. If u1 < 0 then the stock part of the portfolio is negative, so we are so called shorting the stock. The goal is to maximize expected consumption over some time period. If σ << 1 it seems clear since μ > 0 that we should put most of our assets into stocks. Of course there is the possibility that the stock price could fall a lot, so a decision to put most of our assets into stocks is dependent on our attitude to risk. A general principle in finance is risk aversion, which means that one does not agree to play a fair game. In the current situation risk aversion implies that if μ = 0 then we set u1 = 0 so all our assets are in cash. More generally risk aversion is measured by a utility function U(v), where v > 0 is the value of our portfolio. Evidently U(v) should be an increasing function of v. We also have from risk aversion that