Xxxxxx’x Setting Sample Clauses

Xxxxxx’x Setting. Xxxxxx’x problem [16] assumes that an investor starts with initial wealth X0 and at any time t, the wealth process follows the following stochastic differential equation dXt = r + πt(µ − r) Xt − ct dt + XtπtσdWt, (2.10) where r is the risk free interest rate, µ and σ are the expected return and volatility of the risky asset and Wt denotes a standard Brownian motion. Investors are able to choose the consumption of wealth ct and investment proportion in stock πt. Their objective is to choose a proper control (π, c) to maximise the expected value of a utility function U . The objective is to maximise the accumulated consumption and terminal wealth, i.e. maximising E h ∫ T 0 U1(t, ct)dt + U2(XT )i. Two particular cases are the infinite horizon problem t maximising Eh ∫ ∞ U (t, c )dti and the terminal wealth problem maximising E U (XT ) . There are two main approaches to solve Xxxxxx’x problem [18]. First is the stochastic control approach [17]. Merton derives the Xxxxxxx equation for the value function. By solving the Xxxxxxxxx-Xxxxxx-Xxxxxxx equations, the result can be cal- culated in terms of the derivatives of the utility function. Although the analytic form of these derivatives is rarely known, one can turn to numerical methods in solving the PDEs. Thanks to the well-developed PDE techniques and numerical methods, explicit solutions to some special cases of utility functions have already been derived in [1][24]. Another approach is based on the martingale theory [3][7]. The central idea here is to evaluate suboptimal dynamic portfolio strategies by computing the upper and lower bounds on the original expected utility function. In general, a better subopti- mal solution implies a narrower gap from optimality. In this case, the problem can be formulated as a dual problem to solve the original problem by the convex duality [4][9]. Here in the following sections, we would like to consider the case of exponential utility using the Martingale approach. We start by defining the market below.