The Pearson system of utility functions

This paper describes a parametric family of utility functions for decision analysis. The parameterization is obtained by embedding the HARA class in a four-parameter representation for the risk aversion function. The resulting utility functions have only four shapes: concave, convex, S-shaped, and reverse S-shaped. This makes the family suited for both expected utility and prospect theory. We also describe an alternative technique to estimate the four parameters from elicited utilities, which is simpler and easier to implement than standard fitting by minimization of the mean quadratic error.coefficient of risk aversion, elicitation of preferences under risk, expected utility, HARA utility functions, Pearson system of distributions, prospect theory, probability weighting function, target- based decisions.

The Pearson system of utility functions

This paper describes a parametric family of utility functions for decision analysis. The parameterization embeds the HARA class in a four-parameter representation for the risk aversion function. The resulting utility functions can have only four shapes: concave, convex, S-shaped, and reverse S-shaped. This makes the family suited for both expected utility and prospect theory. The paper also describes an alternative technique to estimate the four parameters from elicited utilities, which is simpler than standard fitting by minimization of the mean quadratic error

The Generalised Extreme Value Distribution as Utility Function

The idea that probability distribution functions could provide appropriate mathematical forms for utility functions representing risk aversion is of respectable antiquity. But the relatively few examples that have appeared in the economics literature have displayed quite restrictive risk aversion properties. This paper examines the potential of the generalised extreme value (GEV) distribution as utility function, showing it possesses considerable flexibility as regards risk aversion properties, even in its single parameter form. The paper concludes that the GEV utility function is worth considering for applications in cases where parametric parsimony matters.

The Generalised Extreme Value Distribution as Utility Function

The idea that probability distribution functions could provide appropriate mathematical forms for utility functions representing risk aversion is of respectable antiquity. But the relatively few examples that have appeared in the economics literature have displayed quite restrictive risk aversion properties. This paper examines the potential of the generalised extreme value (GEV) distribution as utility function, showing it possesses considerable flexibility as regards risk aversion properties, even in its single parameter form. The paper concludes that the GEV utility function is worth considering for applications in cases where parametric parsimony matters.

FPGA Acceleration of Mean Variance Framework for Optimal Asset Allocation

Asset classes respond differently to shifts in financial markets, thus an investor can minimize the risk of loss and maximize return of his portfolio by diversification of assets. Increasing the number of diversified assets in a financial portfolio significantly improves the optimal allocation of different assets giving better investment opportunities. However, a large number of assets require a significant amount of computation that only high performance computing can currently provide. Because of the highly parallel nature of Markowitzpsila mean variance framework (the most popular approximation approach for optimal asset allocation) an FPGA implementation of the framework can also provide the performance necessary to compute the optimal asset allocation with a large number of assets. In this work, we propose an FPGA implementation of Markowitzpsila mean variance framework and show it has a potential performance ratio of 221 times over a software implementation

Representing Risk Preferences in Expected Utility Based Decision Models

The application and estimation of expected utility based decision models would benefit from having additional simple and flexible functional forms to represent risk preferences. The literature so far has provided these functional forms for the utility function itself. This work shows that functional forms for the marginal utility function are as useful, are easier to provide, and can represent a larger set of risk preferences. Several functional forms for marginal utility are suggested, and how they can be used is discussed. These marginal utility functions represent risk preferences that cannot be represented by any functional form for the utility function.Risk and Uncertainty,

Momentary Lapses of Reason: The Psychophysics of Law and Behavior

Article published in the Michigan State Law Review

Probabilistic regional ocean predictions : stochastic fields and optimal planning

Thesis: Ph. D. in Mechanical Engineering and Computation, Massachusetts Institute of Technology, Department of Mechanical Engineering, 2018.Cataloged from PDF version of thesis. "Submitted to the Department of Mechanical Engineering and Center for Computational Engineering."Includes bibliographical references (pages 253-268).The coastal ocean is a prime example of multiscale nonlinear fluid dynamics. Ocean fields in such regions are complex, with multiple spatial and temporal scales and nonstationary heterogeneous statistics. Due to the limited measurements, there are multiple sources of uncertainties, including the initial conditions, boundary conditions, forcing, parameters, and even the model parameterizations and equations themselves. To reduce uncertainties and allow long-duration measurements, the energy consumption of ocean observing platforms need to be optimized. Predicting the distributions of reachable regions, time-optimal paths, and risk-optimal paths in uncertain, strong and dynamic flows is also essential for their optimal and safe operations. Motivated by the above needs, the objectives of this thesis are to develop and apply the theory, schemes, and computational systems for: (i) Dynamically Orthogonal ocean primitive-equations with a nonlinear free-surface, in order to quantify uncertainties and predict probabilities for four-dimensional (time and 3-d in space) coastal ocean states, respecting their nonlinear governing equations and non-Gaussian statistics; (ii) Stochastic Dynamically Orthogonal level-set optimization to rigorously incorporate realistic ocean flow forecasts and plan energy-optimal paths of autonomous agents in coastal regions; (iii) Probabilistic predictions of reachability, time-optimal paths and risk-optimal paths in uncertain, strong and dynamic flows. For the first objective, we further develop and implement our Dynamically Orthogonal (DO) numerical schemes for idealized and realistic ocean primitive equations with a nonlinear free-surface. The theoretical extensions necessary for the free-surface are completed. DO schemes are researched and DO terms, functions, and operations are implemented, focusing on: state variable choices; DO norms; DO condition for flows with a dynamic free-surface; diagnostic DO equations for pressure, barotropic velocities and density terms; non-polynomial nonlinearities; semi-implicit time-stepping schemes; and re-orthonormalization consistent with leap-frog time marching. We apply the new DO schemes, as well as their theoretical extensions and efficient serial implementation to forecast idealized-to-realistic stochastic coastal ocean dynamics. For the realistic simulations, probabilistic predictions for the Middle Atlantic Bight region, Northwest Atlantic, and northern Indian ocean are showcased. For the second objective, we integrate data-driven ocean modeling with our stochastic DO level-set optimization to compute and study energy-optimal paths, speeds, and headings for ocean vehicles in the Middle Atlantic Bight region. We compute the energy-optimal paths from among exact time-optimal paths. For ocean currents, we utilize a data-assimilative multiscale re-analysis, combining observations with implicit two-way nested multi-resolution primitive-equation simulations of the tidal-to-mesoscale dynamics in the region. We solve the reduced-order stochastic DO level-set partial differential equations (PDEs) to compute the joint probability of minimum arrival-time, vehicle-speed time-series, and total energy utilized. For each arrival time, we then select the vehicle-speed time-series that minimize the total energy utilization from the marginal probability of vehicle-speed and total energy. The corresponding energy-optimal path and headings be obtained through a particle backtracking equation. For the missions considered, we analyze the effects of the regional tidal currents, strong wind events, coastal jets, shelfbreak front, and other local circulations on the energy-optimal paths. For the third objective, we develop and apply stochastic level-set PDEs that govern the stochastic time-optimal reachability fronts and paths for vehicles in uncertain, strong, and dynamic flow fields. To solve these equations efficiently, we again employ their dynamically orthogonal reduced-order projections. We develop the theory and schemes for risk-optimal planning by combining decision theory with our stochastic time-optimal planning equations. The risk-optimal planning proceeds in three steps: (i) obtain predictions of the probability distribution of environmental flows, (ii) obtain predictions of the distribution of exact time-optimal paths for the forecast flow distribution, and (iii) compute and minimize the risk of following these uncertain time-optimal paths. We utilize the new equations to complete stochastic reachability, time-optimal and risk-optimal path planning in varied stochastic quasi-geostrophic flows. The effects of the flow uncertainty on the reachability fronts and time-optimal paths is explained. The risks of following each exact time-optimal path is evaluated and risk-optimal paths are computed for different risk tolerance measures. Key properties of the risk-optimal planning are finally discussed. Theoretically, the present methodologies are PDE-based and compute stochastic ocean fields, and optimal path predictions without heuristics. Computationally, they are several orders of magnitude faster than direct Monte Carlo. Such technologies have several commercial and societal applications. Specifically, the probabilistic ocean predictions can be input to a technical decision aide for a sustainable fisheries co-management program in India, which has the potential to provide environment friendly livelihoods to millions of marginal fishermen. The risk-optimal path planning equations can be employed in real-time for efficient ship routing to reduce greenhouse gas emissions and save operational costs.by Deepak Narayanan Subramani.Ph. D. in Mechanical Engineering and Computatio