Martingale and Duality Methods for Utility Maximization in an Incomplete Market

Abstract

Abstract. The problem of maximizing the expected utility from terminal wealth is well understood in the context of a complete financial market. This paper studies the same problem in an incomplete market containing a bond and a finite number of stocks whose prices are driven by a multidimensional Brownian motion process W. The coefficients of the bond and stock processes are adapted to the filtration (history) of W, and incompleteness arises when the number of stocks is strictly smaller than the dimension of W. It is shown that there is a way to complete the market by introducing additional "fictitious" stocks so that the optimal portfolio for the thus completed market coincides with the optimal portfolio for the original incomplete market. The notion of a "least favorable" completion is introduced and is shown to be closely related to the existence question for an optimal portfolio in the incomplete market. This notion is expounded upon using martingale techniques; several equivalent characterizations are provided for it, examples are studied in detail, and a fairly general existence result for an optimal portfolio is established based on convex duality theory.Key words, incomplete markets, portfolio processes, stochastic control, convex duality, utility maximization AMS(MOS) subject classifications, primary 93E20; secondary 60G44, 90A16, 49B601. Introduction. This paper studies the problem of an agent who receives a deterministic initial capital, which he must then invest in an incomplete market so as to maximize the expected utility of his wealth at a prespecified final time. The market consists of a bond and m stocks, the latter being driven by a d-dimensional Brownian motion. In such a model, incompleteness arises when m is strictly smaller than d. The market coefficients, i.e., the interest rate, the rates of stock appreciation, and the stock volatility coefficients, are random processes adapted to the full d-dimensional Brownian

Risk- and ambiguity-averse portfolio optimization with quasiconcave utility functionals

“…for some function U : R → R. This result forms the axiomatic basis for the problem of maximization of (subjective) expected utility, 2 which since the seminal contributions of [60] has played a prominent role within mathematical finance; see, among others, [16,45,46,67]. The question of model uncertainty was incorporated in the axiomatic approaches starting with the work of Gilboa and Schmeidler [35], later followed by, among others, Maccheroni et al [57] and Cerreia-Vioglio et al [11]; see also [10,13].…”

Generalised Sharpe Ratios and Asset Pricing in Incomplete Markets *

“…This argument is inspired by '…ctitious completions' of Karatzas et al (1991). In a …nite state model the in…mum is always attained by at least one pricing kernel.…”

The financial value of a weak information on a financial market

“…is complete in the sense that the martingale (S t ) 0≤t≤T enjoys the following predictable representation property (in abbreviate PRP) : For each F-adapted local martingale (M t ) 0≤t≤T there exists a predictable process Θ locally in L 2 such that In what follows, we use the following notion of utility functions (see [23]): Definition 6. A utility function is a strictly increasing, strictly concave and twice continuously differentiable function…”

“…In the case of an incomplete market, by using duality methods introduced in [22], [23], and further developed in [25] for utility maximization, we obtain the fact that in some sense the dual problem associated with the computation of the financial value of the information is also independent of the utility function. Furthermore, we give an interesting lower bound for the financial value of a weak information on the value of the price at a given date.…”