Optimal Investment and Consumption with Borrowing

Abstract

The basic problem of optimal investment and consumption with borrowing is studied in this thesis. Throughout the chapters, the optimisation problems are formulated as optimal stochastic control problems with nonlinear system dynamics. This nonlinearity results from considering markets with borrowing where the borrowing interest rate is higher than the bond interest rate. In addition to this nonlinearity, the optimal investment solutions turn out to be of linear state-feedback form the gain of which can have three different values, depending on the values of the market coefficients. While this is not the case in markets with the same interest rates. This problem is proposed under different market settings and solved by different approaches. Namely, completion of squares, backward stochastic differential equations (BSDEs) (in Chapter 4 and Chapter 5) and a combination of the dynamic programming/variational inequalities (in Chapter 7). Chapter 1 introduces the problem of optimal investment and consumption. It also presents the most important findings of the literature and highlights the importance of considering such markets with different interest rates for lending and borrowing. Other applications of this more realistic assumption are mentioned in the literature, such as in option pricing and the optimal investment problems with mean-variance criteria. Additionally, Chapter 2 formulates the basic problem of optimal investment and consumption with borrowing and presents some mathematical preliminaries that we employed throughout this thesis. As a contribution to the literature on optimal investment and consump- tion with a stochastic interest rate, we propose a market with borrowing and Hull-White modulated interest rates. This is in Chapter 3 when the criteria of optimality are proposed in the finite time horizon. Chapter 4 deals with market with borrowing and possibly unbounded random coefficients. By a comparison theorem, the existence of a non-negative solution pair of BSDE with a certain nonlinear growth is proved. It turns out that the quadratic-affine model for the bond rate is a special case of this market. Under some sufficient conditions, the admissibility of the optimal controls is achieved. A more general BSDE is derived in Chapter 5. This is when the stock and the BSDE are driven by the same source of uncertainty which is a multi-dimensional Brownian motion. In Chapter 6, the problem of optimal investment and consumption in a market with Markovian switching coefficients and borrowing is considered. The crite- ria of optimality are proposed in the finite and infinite horizon cases. Optimal investment with discretionary stopping in a market with borrowing is studied in Chapter 7. This is when the discounted logarithmic utility from terminal wealth is maximised in an infinite horizon. Explicit closed form solutions are found in this thesis. In Chapters 3 and 4, the explicit solutions to the case of the power utility function are achieved under some solvability assumptions of certain nonlinear differential equations. Although, some special cases under which these differential equations admit a solution are proved. Furthermore, in Chapter 5 and (Chapter 6) the explicit solutions to the case of the power utility function are obtained subject to solvability assumptions to a certain nonlinear BSDE (systems of coupled Bernoulli equations). A summary of the main contributions of the thesis is outlined in Chapter 8. It also points out some interesting open questions for future research in markets with borrowing.