Daniel N. Ostrov

Viscosity solutions and convergence of monotone schemes for synthetic aperture radar shape-from-shading equations with discontinuous intensities

The shape-from-shading (SFS) equation relating u(y,r), the unknown (angular) height of a surface, to I(y,r), the known synthetic aperture radar (SAR) intensity data from the surface, is I = \frac{u_r^2}{\sqrt{1+u_r^2+u_y^2}}, where y and r are axial and radial cylindrical coordinates. Unlike the more common eikonal SFS equation which relates surface height in Cartesian coordinates to optical/photographic intensity data, the above radar equation can be transformed into Hamilton--Jacobi Cauchy form: ur+g(I,uy)=0. We explore the case where I is a discontinuous function, which occurs commonly in radar data. By considering sequences of continuous intensity functions that converge to I, we obtain corresponding sequences of viscosity solutions. We prove that these sequences must converge. We also establish conditions that guarantee that these sequences converge to a common limit, which we define as the solution to the radar equation. Finally, we establish and demonstrate that when this common limit exists, monot...

Balancing Small Transaction Costs with Loss of Optimal Allocation in Dynamic Stock Trading Strategies

We discuss optimal trading strategies for general utility functions in portfolios of cash and stocks subject to small proportional transaction costs. We present a new interpretation of scalings found by Soner, Shreve, and others. To leading order in the small transaction cost parameter, the free boundary problem for the expected utilitys value function is shown to be dual, in the sense of Lagrange multipliers for optimal design problems, to a free boundary problem minimizing a cost function. This cost function is the sum of a boundary integral corresponding to the rate of trading and an interior integral corresponding to opportunity loss that results from suboptimal portfolio allocation. Using the dual problems formulation, we show that the quasi-steady state probability density of the optimal portfolio is uniform for a single stock but generally blows up even in the simple case of two uncorrelated stocks.

Evolutionary dynamics over continuous action spaces for population games that arise from symmetric two-player games

Any absolutely continuous, piecewise smooth, symmetric two-player game can be extended to define a population game in which each player interacts with a large representative subset of the entire population. Assuming that players respond to the payoff gradient over a continuous action space, we obtain nonlinear integro-partial differential equations that are numerically tractable and sometimes analytically tractable. Economic applications include oligopoly, growth theory, and financial bubbles and crashes.

Boundary conditions and fast algorithms for surface reconstructions from synthetic aperture radar data

Most attempts to determine surface height from noiseless synthetic aperture radar (SAR) data involve approximating the surface by solving a related standard shape from shading (SFS) problem. Through analysis of the underlying partial differential equations for both the original SAR problem and the approximating standard SFS problem, the authors demonstrate significant differences between them. For example, if it is known that the surface is smooth, the standard SPS problem can generally be uniquely solved from knowledge of the height and concavity at one surface point, whereas for SAR, multiple valid solutions will generally exist unless height information is specified along entire curves on the surface (i.e., boundary conditions). Unlike the standard SFS approximation, the underlying SAR equation can be reexpressed as a time-dependent Hamilton-Jacobi equation. This transformation allows the authors to compute the correct surface topography from noiseless SAR data with boundary conditions extremely quickly. Finally, they consider the effect of radar noise on the computed surface reconstruction and discuss the ability of the presented PDE method to quickly compute an initial surface that will significantly cut the computational time needed by cost minimization algorithms to approximate surfaces from noisy radar data.

An Option to Reduce Transaction Costs

For small transaction costs, we determine the leading order optimal dynamic trading strategy of a portfolio of stock, cash, and options. Except for the transaction costs, our market assumptions are those of Black, Scholes, and Merton. Without transaction costs, the option is redundant in the portfolio. With transaction costs, however, we show that adding the option to the portfolio can significantly reduce overall trading costs compared to optimal strategies that use only stock and cash. The analysis is based on an asymptotic expansion with three scales: macroscopic, mesoscopic, and microscopic. The macroscopic analysis is Mertons optimal investment problem. Within a plane defined by the amount of stock and options held, the macroscopic analysis yields a Merton line of optimal portfolios. We show that there is a particular magic point on the Merton line that minimizes expensive stochastic movement away from the Merton line. The mesoscopic scale governs less expensive deviations of the portfolio away from the magic point but along the Merton line. The microscopic scale governs the more expensive deviations of the portfolio away from the magic point, transverse to the Merton line. The resulting strategy is related to commonly used Delta and Gamma hedging strategies, but our scale analysis implies that some rebalancings are much more effective than others. We do not give rigorous mathematical proofs, only arguments of formal applied mathematics.

Optimal Strategies for Traditional versus Roth IRA/401(k) Consumption During Retirement

We establish an algorithm that produces an optimal strategy for retirees to withdraw funds between their tax-deferred accounts (TDAs), like traditional IRA/401(k) accounts, and their Roth IRA/401(k) accounts, in the context of a financial model based on American tax law. This optimal strategy follows a geometrically simple, intuitive approach that can be used to maximize the size of a retirees bequest to an heir or, alternatively, to maximize a retirees portfolio longevity. We give examples where retirees following the approach currently implemented by major investment firms, like Fidelity and Vanguard, will reduce their bequests by approximately 10% or lose 18 months of portfolio longevity compared to our optimal approach. Further, our strategy and algorithm can be extended to many cases where the retiree has additional, known yearly sources of money, such as income from part-time work, taxable investment accounts, and Social Security.

Dynamic Systemic Risk Networks: A Note

We propose a theory-driven framework for monitoring system-wide risk. Our approach extends the one-firm Merton (1974) credit risk model to a generalized stochastic network-based framework across all financial institutions, comprising a novel approach to measuring systemic risk over time. We develop four desired properties for any systemic risk measure. We also develop measures for the risks created by each individual institution and a measure for risk created by each pairwise connection between institutions. Four specific implementation models are then explored, and brief empirical examples illustrate the ease of implementation of these four models and show general consistency between their results.

Optimal Asset Allocation for Passive Investing with Capital Loss Harvesting

Abstract This article examines how to quantify and optimally utilize the beneficial effect that capital loss harvesting generates in a taxable portfolio. We explicitly determine the optimal initial asset allocation for an investor who follows the continuous time dynamic trading strategy of Constantinides (1983). This strategy sells and re-buys all stocks with losses, but is otherwise passive. Our model allows the use of the stock positions full purchase history for computing the cost basis. The method can also be used to rebalance at later times. For portfolios with one stock position and cash, the probability density function for the portfolios state corresponds to the solution of a 3−space + 1−time dimensional partial differential equation (PDE) with an oblique reflecting boundary condition. Extensions of this PDE, including to the case of multiple correlated stocks, are also presented. We detail a numerical algorithm for the PDE in the single stock case. The algorithm shows the significant effect capital loss harvesting can have on the optimal stock allocation, and it also allows us to compute the expected additional return that capital loss harvesting generates. Our PDE-based algorithm, compared with Monte Carlo methods, is shown to generate much more precise results in a fraction of the time. Finally, we employ Monte Carlo methods to approximate the impact of many of our models assumptions.

Solutions to Scalar Conservation Laws Where the Flux is Discontinuous in Space and Time

We consider the following scalar conservation law where k may be discontinuous along a finite number of possibly intersecting smooth curves in the (x, t) plane: The flux, f, is assumed to be convex in v, locally Lipschitz continuous in k and v, and grow at a superlinear rate; i.e., We will use γ(t) to denote a generic curve of discontinuity in k.These types of discontinuous k.functions occur in many different physical phenomena including traffic flow [1], continuous sedimentation [2], and shape from shading [3]. They also occur in decoupled systems of conservation laws of the form since the second equation has form (1) when the solution of the first equation has a finite number of shocks.

Unique Solutions to Discontinuous Hamilton-Jacobi Equations in Shape-From-Shading

We consider two shape from shading equations that have similar physical motivation but correspond to significantly different Hamilton-Jacobi equations. One is the eikonal equation and has a convex Hamiltonian; the other has a time dependent form and the Hamiltonian is not convex. We analyze both cases when the Hamiltonian is discontinuous with respect to its spatial data by considering solutions to sequences of equations with continuous Hamiltonian that converge from below and above to the discontinuous Hamiltonian. When the corresponding solutions converge to a common limit, we define this limit to be the unique solution to the discontinuous problem. We discuss when the unique solution can be established, its behavior, and the convergence of monotone schemes to this notion of solution.