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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 utility's value function is… (More)

We formalize Veblen's idea of conspicuous consumption as two alternative forms of interdependent preferences, dubbed envy and pride. Agents adjust consumption patterns gradually, in the direction of increasing utility. From an arbitrary initial state, the distribution of consumption among agents with identical preferences converges to a unique equilibrium… (More)

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 = u 2 r 1 + u 2 r + u 2 y , where y and r are axial and radial cylindrical coordinates. Unlike the more common eikonal SFS equation which relates surface height in… (More)

The height, u(x, y), of a continuous, Lambertian surface of known albedo (i.e., grayness) is related to n(x, y), information recoverable from a black and white flash photograph of the surface, by the partial differential equation u 2 x + u 2 y − n = 0. We review the notion of a unique viscosity solution for this equation when n is continuous and a recent… (More)

We study the short time behavior of the early exercise boundary for American style put options in the Black–Scholes theory. We develop an asymptotic expansion which shows that the simple lower bound of Barles et al. is a more accurate approximation to the actual boundary than the more complex upper bound. Our expansion is obtained through iteration using a… (More)

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 we demonstrate… (More)

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… (More)

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… (More)

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