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Register a new userA Framework of Multivariate Utility Optimization with General Benchmarks
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Benchmarks in the utility function have various interpretations, including performance guarantees and risk constraints in fund contracts and reference levels in cumulative prospect theory. In most literature, benchmarks are a deterministic constant or a fraction of the underlying wealth; as such, the utility is still a univariate function of the wealth. In this paper, we propose a framework of multivariate utility optimization with general benchmark variables, which include stochastic reference levels as typical examples. The utility is state-dependent and the objective is no longer distribution-invariant. We provide the optimal solution(s) and fully investigate the issues of well-posedness, feasibility, finiteness and attainability. The discussion does not require many classic conditions and assumptions, e.g., the Lagrange multiplier always exists. Moreover, several surprising phenomena and technical difficulties may appear: (i) non-uniqueness of the optimal solutions, (ii) various reasons for non-existence of the Lagrangian multiplier and corresponding results on the optimal solution, (iii) measurability issues of the concavification of a multivariate utility and the selection of the optimal solutions, and (iv) existence of an optimal solution not decreasing with respect to the pricing kernel. These issues are thoroughly addressed, rigorously proved, completely summarized and insightfully visualized. As an application, the framework is adopted to model and solve a constraint utility optimization problem with state-dependent performance and risk benchmarks.
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We introduce and treat a class of Multi Objective Risk-Sensitive Markov Decision Processes (MORSMDPs), where the optimality criteria are generated by a multivariate utility function applied on a finite set of emph{different running costs}. To illustrate our approach, we study the example of a two-armed bandit problem. In the sequel, we show that it is possible to reformulate standard Risk-Sensitive Partially Observable Markov Decision Processes (RSPOMDPs), where risk is modeled by a utility function that is a emph{sum of exponentials}, as MORSMDPs that can be solved with the methods described in the first part. This way, we extend the treatment of RSPOMDPs with exponential utility to RSPOMDPs corresponding to a qualitatively bigger family of utility functions.
We propose a Markov regime switching GARCH model with multivariate normal tempered stable innovation to accommodate fat tails and other stylized facts in returns of financial assets. The model is used to simulate sample paths as input for portfolio optimization with risk measures, namely, conditional value at risk and conditional drawdown. The motivation is to have a portfolio that avoids left tail events by combining models that incorporates fat tail with optimization that focuses on tail risk. In-sample test is conducted to demonstrate goodness of fit. Out-of-sample test shows that our approach yields higher performance measured by Sharpe-like ratios than the market and equally weighted portfolio in recent years which includes some of the most volatile periods in history. We also find that suboptimal portfolios with higher return constraints tend to outperform optimal portfolios.
In this paper, we consider a multistage expected utility maximization problem where the decision makers utility function at each stage depends on historical data and the information on the true utility function is incomplete. To mitigate the risk arising from ambiguity of the true utility, we propose a maximin robust model where the optimal policy is based on the worst sequence of utility functions from an ambiguity set constructed with partially available information about the decision makers preferences. We then show that the multistage maximin problem is time consistent when the utility functions are state-dependent and demonstrate with a counter example that the time consistency may not be retained when the utility functions are state-independent. With the time consistency, we show the maximin problem can be solved by a recursive formula whereby a one-stage maximin problem is solved at each stage beginning from the last stage. Moreover, we propose two approaches to construct the ambiguity set: a pairwise comparison approach and a $zeta$-ball approach where a ball of utility functions centered at a nominal utility function under $zeta$-metric is considered. To overcome the difficulty arising from solving the infinite dimensional optimization problem in computation of the worst-case expected utility value, we propose piecewise linear approximation of the utility functions and derive error bound for the approximation under moderate conditions. Finally we develop a scenario tree-based computational scheme for solving the multistage preference robust optimization model and report some preliminary numerical results.
In many smart infrastructure applications flexibility in achieving sustainability goals can be gained by engaging end-users. However, these users often have heterogeneous preferences that are unknown to the decision-maker tasked with improving operational efficiency. Modeling user interaction as a continuous game between non-cooperative players, we propose a robust parametric utility learning framework that employs constrained feasible generalized least squares estimation with heteroskedastic inference. To improve forecasting performance, we extend the robust utility learning scheme by employing bootstrapping with bagging, bumping, and gradient boosting ensemble methods. Moreover, we estimate the noise covariance which provides approximated correlations between players which we leverage to develop a novel correlated utility learning framework. We apply the proposed methods both to a toy example arising from Bertrand-Nash competition between two firms as well as to data from a social game experiment designed to encourage energy efficient behavior amongst smart building occupants. Using occupant voting data for shared resources such as lighting, we simulate the game defined by the estimated utility functions to demonstrate the performance of the proposed methods.
We consider an optimal stochastic impulse control problem over an infinite time horizon motivated by a model of irreversible investment choices with fixed adjustment costs. By employing techniques of viscosity solutions and relying on semiconvexity arguments, we prove that the value function is a classical solution to the associated quasi-variational inequality. This enables us to characterize the structure of the continuation and action regions and construct an optimal control. Finally, we focus on the linear case, discussing, by a numerical analysis, the sensitivity of the solution with respect to the relevant parameters of the problem.
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