Utility maximization in incomplete markets

Abstract

We consider the problem of utility maximization for small traders on incomplete financial markets. As opposed to most of the papers dealing with this subject, the investors' trading strategies we allow underly constraints described by closed, but not necessarily convex, sets. The final wealths obtained by trading under these constraints are identified as stochastic processes which usually are supermartingales, and even martingales for particular strategies. These strategies are seen to be optimal, and the corresponding value functions determined simply by the initial values of the supermartingales. We separately treat the cases of exponential, power and logarithmic utility.Introduction. In this paper we consider a small trader on an incomplete financial market who can trade in a finite time interval [0, T ] by investing in risky stocks and a riskless bond. He aims at maximizing the utility he draws from his final wealth measured by some utility function. The trading strategies he may choose to attain his wealth underly some restriction formalized by a constraint. For example, he may be forced not to have a negative number of shares or that his investment in risky stocks is not allowed to exceed a certain threshold. We will be interested not only in describing the trader's optimal utility, but also the strategies which he may follow to reach this goal. As opposed to most of the papers dealing so far with the maximization of expected utility under constraints, we essentially relax the hypotheses to be fulfilled by them. They are formulated as usual by the requirement that the strategies take their values in some set, which is supposed to simply

Robust Portfolio Choice and Indifference Valuation

“…In the case that there are no jumps, i.e., n p = 0, and there is no ambiguity, i.e., d + = d − = λ = 0, our general solution above reduces to the solution obtained by Hu, Imkeller and Müller [38]; see also El Karoui and Rouge [24]. These results have been generalized for continuous price processes to continuous and non-continuous filtrations, see for instance, Mania and Schweizer [53] and Becherer [4], in a purely risk-based setting.…”

“…Particularly popular is exponential indifference valuation due to its analytical tractability on the one hand -the exponential form induces a convenient translation invariance property -and its theoretically appealing properties, especially in a dynamic context, on the other (El Karoui and Rouge [24], Delbaen et al [18], Kabanov and Stricker [42], Mania and Schweizer [53]). See also Hu, Imkeller and Müller [38], Becherer [4], Morlais [57,58], and Cheridito and Hu [12] for recent work on the problems of portfolio choice and indifference valuation.…”

“…Applications of BSDEs to utility maximization problems in incomplete markets in a Brownian setting include (with exponential, logarithmic or power utility) Hu, Imkeller and Müller [38], Cheridito and Hu [12], and (with a general utility function) Horst et al [37]; for a setting with continuous filtration or non-continuous filtration (but with exponential utility), see Mania and Schweizer [53], Morlais [57] and Becherer [4]. Morlais [58] generalizes some of these results adopting an exponential utility function and allowing for infinite activity jumps in the asset price processes, in a purely risk-based setting without ambiguity.…”

Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions

“…Quadratic BSDEs have important applications in mathematical finance, for example, in utility optimization in incomplete markets [29,21,12]. Assume additionally that the terminal function Φ is Hölder continuous and bounded.…”

Reflected backward stochastic differential equations and a class of non-linear dynamic pricing rule

“…z). To proceed in the quadratic case and as in [HIM05] and assuming that (Y 1 , Z 1 ) and (Y 2 , Z 2 ) are two solutions of the BSDE(f, B), we apply Itô's formula to Y 1,2 := Y 1 − Y 2 between t and τ ∧ T with an arbitrary stopping time τ (similarly, Z 1,2 stands for…”