## Description

The story to be told in this book is in the style of a musical theme-and-variations; the main theme is stated, and then a sequence of variations is played, bearing more or less resemblance to the main theme, yet always derived from it. For us, the theme is the Merton problem, to be presented in this chapter, and the variations will follow in the next chapter.

What is the Merton problem? I use the title loosely to describe a collection of stochastic optimal control problems ﬁrst analyzed by Merton [28]. The common theme is of an agent investing in one or more risky assets so as to optimize some objective. We can characterise the dynamics of the agent’s wealth through the equation for some given initial wealth w0. In this equation, the asset price process S is a d-dimensional semimartingale, the portfolio process n is a d-dimensional previsible process, and the dividend process δ is a d-dimensional adapted process.2 The adapted scalar processes e and c are respectively an endowment stream, and a consumption stream.The process r is an adapted scalar process, interpreted as the riskless rate of interest. The processes δ, S, r and e will generally be assumed given, as will the initial wealth w0, and the agent must choose the portfolio process n and the consumption process c.

To explain a little how the wealth equation (1.1) arises, think what would happen if you invested nothing in the risky assets, that is, n ≡ 0; your wealth, invested in a bank account, would grow at the riskless rate r, with addition of your endowment e and withdrawal of your consumption c. If you chose to hold a ﬁxed number nt = n0 of units of the risky assets, then your wealth wt at time t would be made up of the market values ni0Sti of your holding of asset i, i = 1,…,d, together with the cash you hold in the bank, equal to wt − n0 · St. The cash in the bank is growing at rate

r—which explains the ﬁrst term on the right in (1.2)—and the ownership of ni0 units of asset i delivers you a stream ni0δti of dividends.

Next, if you were to follow a piecewise constant investment strategy, where you just change your portfolio in a non-anticipating way at a ﬁnite set of stopping times, then the evolution between change times is just as we have explained it; at change times, the new portfolio you choose has to be funded from your existing resources, so there is no jump in your wealth. Thus we see that the evolution (1.1) is correct for any (left-continuous, adapted) piecewise constant portfolio process n, and by extension for any previsible portfolio process.

If we allow completely arbitrary previsible n, we immediately run into absurdities. For this reason, we usually restrict attention to portfolio processes n and consumption processes c such that the pair (n, c) is admissible.

Deﬁnition 1.1 The pair (nt, ct)t≥0 is said to be admissible for initial wealth w0 if the wealth process wt given by (1.1) remains non-negative at all times. We use the notation