Utility Maximization

Utility Maximisation with Model-independent Constraints ArXiv ID: 2512.24371 “View on arXiv” Authors: Alexander M. G. Cox, Daniel Hernandez-Hernandez Abstract We consider an agent who has access to a financial market, including derivative contracts, who looks to maximise her utility. Whilst the agent looks to maximise utility over one probability measure, or class of probability measures, she must also ensure that the mark-to-market value of her portfolio remains above a given threshold. When the mark-to-market value is based on a more pessimistic valuation method, such as model-independent bounds, we recover a novel optimisation problem for the agent where the agents investment problem must satisfy a pathwise constraint. For complete markets, the expression of the optimal terminal wealth is given, using the max-plus decomposition for supermartingales. Moreover, for the Black-Scholes-Merton model the explicit form of the process involved in such decomposition is obtained, and we are able to investigate numerically optimal portfolios in the presence of options which are mispriced according to the agent’s beliefs. ...

On the existence of personal equilibria ArXiv ID: 2512.08348 “View on arXiv” Authors: Laurence Carassus, Miklós Rásonyi Abstract We consider an investor who, while maximizing his/her expected utility, also compares the outcome to a reference entity. We recall the notion of personal equilibrium and show that, in a multistep, generically incomplete financial market model such an equilibrium indeed exists, under appropriate technical assumptions. Keywords: Personal Equilibrium, Utility Maximization, Incomplete Market, Reference Dependence, Game Theory, General/Asset Pricing ...

On the utility problem in a market where price impact is transient ArXiv ID: 2511.12093 “View on arXiv” Authors: Lóránt Nagy, Miklós Rásonyi Abstract We consider a discrete-time model of a financial market where a risky asset is bought and sold with transactions having a transient price impact. It is shown that the corresponding utility maximization problem admits a solution. We manage to remove some unnatural restrictions on the market depth and resilience processes that were present in earlier work. A non-standard feature of the problem is that the set of attainable portfolio values may fail the convexity property. ...

Machine-learning a family of solutions to an optimal pension investment problem ArXiv ID: 2511.07045 “View on arXiv” Authors: John Armstrong, Cristin Buescu, James Dalby, Rohan Hobbs Abstract We use a neural network to identify the optimal solution to a family of optimal investment problems, where the parameters determining an investor’s risk and consumption preferences are given as inputs to the neural network in addition to economic variables. This is used to develop a practical tool that can be used to explore how pension outcomes vary with preference parameters. We use a Black-Scholes economic model so that we may validate the accuracy of network using a classical and provably convergent numerical method developed using the duality approach. ...

The Interplay between Utility and Risk in Portfolio Selection ArXiv ID: 2509.10351 “View on arXiv” Authors: Leonardo Baggiani, Martin Herdegen, Nazem Khan Abstract We revisit the problem of portfolio selection, where an investor maximizes utility subject to a risk constraint. Our framework is very general and accommodates a wide range of utility and risk functionals, including non-concave utilities such as S-shaped utilities from prospect theory and non-convex risk measures such as Value at Risk. Our main contribution is a novel and complete characterization of well-posedness for utility-risk portfolio selection in one period that takes the interplay between the utility and the risk objectives fully into account. We show that under mild regularity conditions the minimal necessary and sufficient condition for well-posedness is given by a very simple either-or criterion: either the utility functional or the risk functional need to satisfy the axiom of sensitivity to large losses. This allows to easily describe well-posedness or ill-posedness for many utility-risk pairs, which we illustrate by a large number of examples. In the special case of expected utility maximization without a risk constraint (but including non-concave utilities), we show that well-posedness is fully characterised by the asymptotic loss-gain ratio, a simple and interpretable quantity that describes the investor’s asymptotic relative weighting of large losses versus large gains. ...

Duality and Policy Evaluation in Distributionally Robust Bayesian Diffusion Control ArXiv ID: 2506.19294 “View on arXiv” Authors: Jose Blanchet, Jiayi Cheng, Hao Liu, Yang Liu Abstract We consider a Bayesian diffusion control problem of expected terminal utility maximization. The controller imposes a prior distribution on the unknown drift of an underlying diffusion. The Bayesian optimal control, tracking the posterior distribution of the unknown drift, can be characterized explicitly. However, in practice, the prior will generally be incorrectly specified, and the degree of model misspecification can have a significant impact on policy performance. To mitigate this and reduce overpessimism, we introduce a distributionally robust Bayesian control (DRBC) formulation in which the controller plays a game against an adversary who selects a prior in divergence neighborhood of a baseline prior. The adversarial approach has been studied in economics and efficient algorithms have been proposed in static optimization settings. We develop a strong duality result for our DRBC formulation. Combining these results together with tools from stochastic analysis, we are able to derive a loss that can be efficiently trained (as we demonstrate in our numerical experiments) using a suitable neural network architecture. As a result, we obtain an effective algorithm for computing the DRBC optimal strategy. The methodology for computing the DRBC optimal strategy is greatly simplified, as we show, in the important case in which the adversary chooses a prior from a Kullback-Leibler distributional uncertainty set. ...

Framework for asset-liability management with fixed-term securities ArXiv ID: 2502.19213 “View on arXiv” Authors: Unknown Abstract We consider an optimal investment-consumption problem for a utility-maximizing investor who has access to assets with different liquidity and whose consumption rate as well as terminal wealth are subject to lower-bound constraints. Assuming utility functions that satisfy standard conditions, we develop a methodology for deriving the optimal strategies in semi-closed form. Our methodology is based on the generalized martingale approach and the decomposition of the problem into subproblems. We illustrate our approach by deriving explicit formulas for agents with power-utility functions and discuss potential extensions of the proposed framework. In numerical studies, we substantiate how the parameters of our framework impact the optimal proportion of initial capital allocated to the illiquid asset, the monetary value that the investor subjectively assigns to the fixed-term asset, and the potential of the illiquid asset to increase terminal the terminal value of liabilities without loss in the investor’s expected utility. ...

Optimal portfolio under ratio-type periodic evaluation in stochastic factor models under convex trading constraints ArXiv ID: 2411.13579 “View on arXiv” Authors: Unknown Abstract This paper studies a type of periodic utility maximization problems for portfolio management in incomplete stochastic factor models with convex trading constraints. The portfolio performance is periodically evaluated on the relative ratio of two adjacent wealth levels over an infinite horizon, featuring the dynamic adjustments in portfolio decision according to past achievements. Under power utility, we transform the original infinite horizon optimal control problem into an auxiliary terminal wealth optimization problem under a modified utility function. To cope with the convex trading constraints, we further introduce an auxiliary unconstrained optimization problem in a modified market model and develop the martingale duality approach to establish the existence of the dual minimizer such that the optimal unconstrained wealth process can be obtained using the dual representation. With the help of the duality results in the auxiliary problems, the relationship between the constrained and unconstrained models as well as some fixed point arguments, we finally derive and verify the optimal constrained portfolio process in a periodic manner for the original problem over an infinite horizon. ...

A new approach to the theory of optimal income tax ArXiv ID: 2408.14476 “View on arXiv” Authors: Unknown Abstract The Nobel-price winning Mirrlees’ theory of optimal taxation inspired a long sequence of research on its refinement and enhancement. However, an issue of concern has been always the fact that, as was shown in many publications, the optimal schedule in Mirrlees’ paradigm of maximising the total utility (constructed from individually optimised individual ones) usually did not lead to progressive taxation (contradicting the ethically supported practice in developed economies), and often even assigned minimal tax rates to the higher paid strata of society. The first objective of this paper is to support this conclusion by proving a theorem on optimal tax schedule in (practically most exploited) piecewise-linear environment under a simplest natural utility function. The second objective is to suggest a new paradigm for optimal taxation, where instead of just total average utility maximization one introduces a standard deviation of utility as a second parameter (in some analogy with Marcowitz portfolio optimization). We show that this approach leads to transparent and easy interpreted optimality criteria for income tax. ...

Examples and Counterexamples of Cost-efficiency in Incomplete Markets ArXiv ID: 2407.08756 “View on arXiv” Authors: Unknown Abstract We present a number of examples and counterexamples to illustrate the results on cost-efficiency in an incomplete market obtained in [“BS24”]. These examples and counterexamples do not only illustrate the results obtained in [“BS24”], but show the limitations of the results and the sharpness of the key assumptions. In particular, we make use of a simple 3-state model in which we are able to recover and illustrate all key results of the paper. This example also shows how our characterization of perfectly cost-efficient claims allows to solve an expected utility maximization problem in a simple incomplete market (trinomial model) and recover results from [“DS06, Chapter 3”], there obtained using duality. ...