American Options, Multi–armed Bandits, and Optimal Consumption Plans: A Unifying View

In this survey, we show that various stochastic optimization problems arising in option theory, in dynamical allocation problems, and in the microeconomic theory of intertemporal consumption choice can all be reduced to the same problem of representing a given stochastic process in terms of running maxima of another process. We describe recent results of Bank and El Karoui (2002) on the general stochastic representation problem, derive results in closed form for Levy processes and diffusions, present an algorithm for explicit computations, and discuss some applications.

[1]  J. Snell.  Applications of martingale system theorems , 1952 .

[2]  P. Meyer,et al.  Probabilités et potentiel , 1966 .

[3]  S. Taylor.  DIFFUSION PROCESSES AND THEIR SAMPLE PATHS , 1967 .

[4]  T. Liggett,et al.  Optimal Stopping for Partial Sums , 1972 .

[5]  J. Gittins.  Bandit processes and dynamic allocation indices , 1979 .

[6]  P. Whittle.  Multi‐Armed Bandits and the Gittins Index , 1980 .

[7]  N. Karoui.  Les Aspects Probabilistes Du Controle Stochastique , 1981 .

[8]  I. Karatzas.  Gittins Indices in the Dynamic Allocation Problem for Diffusion Processes , 1984 .

[9]  A. Mandelbaum.  CONTINUOUS MULTI-ARMED BANDITS AND MULTIPARAMETER PROCESSES , 1987 .

[10]  I. Karatzas.  On the pricing of American options , 1988 .

[11]  Hiroaki Morimoto,et al.  On average cost stopping time problems , 1991 .

[12]  I. Karatzas,et al.  Dynamic Allocation Problems in Continuous Time , 1994 .

[13]  F. Delbaen,et al.  A general version of the fundamental theorem of asset pricing , 1994 .

[14]  The Optimal Stopping Problem for a General American Put-Option , 1995 .

[15]  N. Karoui,et al.  Dynamic Programming and Pricing of Contingent Claims in an Incomplete Market , 1995 .

[16]  E. Eberlein,et al.  Hyperbolic distributions in finance , 1995 .

[17]  D. Kramkov.  Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets , 1996 .

[18]  H. McKean,et al.  Diffusion processes and their sample paths , 1996 .

[19]  Hans Föllmer,et al.  Optional decomposition and Lagrange multipliers , 1997, Finance Stochastics.

[20]  Ole E. Barndorff-Nielsen,et al.  Processes of normal inverse Gaussian type , 1997, Finance Stochastics.

[21]  P. Bank.  Singular control of optional random measures , 2000 .

[22]  R. Wolpert.  Lévy Processes , 2000 .

[23]  M. Schweizer,et al.  Stochastic Optimization and Representation Problems Arising in the Microeconomic Theory of Intertemporal Consumption Choice , 2000 .

[24]  Kenneth H. Karlsen,et al.  Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach , 2001, Finance Stochastics.

[25]  Frank Riedel,et al.  Optimal consumption choice with intertemporal substitution , 2001 .

[26]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[27]  S. Levendorskii,et al.  Perpetual American options under Levy processes , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[28]  Ernesto Mordecki,et al.  Optimal stopping and perpetual options for Lévy processes , 2002, Finance Stochastics.

[29]  Hans Föllmer,et al.  American Options, Multi–armed Bandits, and Optimal Consumption Plans , 2003 .

[30]  N. Karoui,et al.  A stochastic representation theorem with applications to optimization and obstacle problems , 2004 .

[31]  S. Asmussen,et al.  Russian and American put options under exponential phase-type Lévy models , 2004 .