312 research outputs found
Convex Duality in Constrained Portfolio Optimization
We study the stochastic control problem of maximizing expected utility from terminal wealth and/or consumption, when the portfolio is constrained to take values in a given closed, convex subset of R^d. The setting is that of a continuous-time, Itô process model for the underlying asset prices. General existence results are established for optimal portfolio/consumption strategies, by suitably embedding the constrained problem in an appropriate family of unconstrained ones, and finding a member of this family for which the corresponding optimal policy obeys the constraints. Equivalent conditions for optimality are obtained, and explicit solutions
leading to feedback formulae are derived for special utility functions and for deterministic coefficients. Results on incomplete markets, on short-selling constraints and on different interest rates for borrowing and lending are covered as special cases. The mathematical tools are those of continuous-time martingales, convex analysis and duality theory
Optimal arbitrage under model uncertainty
In an equity market model with "Knightian" uncertainty regarding the relative
risk and covariance structure of its assets, we characterize in several ways
the highest return relative to the market that can be achieved using
nonanticipative investment rules over a given time horizon, and under any
admissible configuration of model parameters that might materialize. One
characterization is in terms of the smallest positive supersolution to a fully
nonlinear parabolic partial differential equation of the
Hamilton--Jacobi--Bellman type. Under appropriate conditions, this smallest
supersolution is the value function of an associated stochastic control
problem, namely, the maximal probability with which an auxiliary
multidimensional diffusion process, controlled in a manner which affects both
its drift and covariance structures, stays in the interior of the positive
orthant through the end of the time-horizon. This value function is also
characterized in terms of a stochastic game, and can be used to generate an
investment rule that realizes such best possible outperformance of the market.Comment: Published in at http://dx.doi.org/10.1214/10-AAP755 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Testing composite hypotheses via convex duality
We study the problem of testing composite hypotheses versus composite
alternatives, using a convex duality approach. In contrast to classical results
obtained by Krafft and Witting (Z. Wahrsch. Verw. Gebiete 7 (1967) 289--302),
where sufficient optimality conditions are derived via Lagrange duality, we
obtain necessary and sufficient optimality conditions via Fenchel duality under
compactness assumptions. This approach also differs from the methodology
developed in Cvitani\'{c} and Karatzas (Bernoulli 7 (2001) 79--97).Comment: Published in at http://dx.doi.org/10.3150/10-BEJ249 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Trading Strategies Generated Pathwise by Functions of Market Weights
Almost twenty years ago, E.R. Fernholz introduced portfolio generating
functions which can be used to construct a variety of portfolios, solely in the
terms of the individual companies' market weights. I. Karatzas and J. Ruf
recently developed another methodology for the functional construction of
portfolios, which leads to very simple conditions for strong relative arbitrage
with respect to the market. In this paper, both of these notions of functional
portfolio generation are generalized in a pathwise, probability-free setting;
portfolio generating functions are substituted by path-dependent functionals,
which involve the current market weights, as well as additional
bounded-variation functions of past and present market weights. This
generalization leads to a wider class of functionally-generated portfolios than
was heretofore possible, and yields improved conditions for outperforming the
market portfolio over suitable time-horizons.Comment: 45 pages, 3 figure
The numeraire portfolio in semimartingale financial models
We study the existence of the numeraire portfolio under predictable convex
constraints in a general semimartingale model of a financial market. The
numeraire portfolio generates a wealth process, with respect to which the
relative wealth processes of all other portfolios are supermartingales.
Necessary and sufficient conditions for the existence of the numeraire
portfolio are obtained in terms of the triplet of predictable characteristics
of the asset price process. This characterization is then used to obtain
further necessary and sufficient conditions, in terms of a no-free-lunch-type
notion. In particular, the full strength of the "No Free Lunch with Vanishing
Risk" (NFLVR) is not needed, only the weaker "No Unbounded Profit with Bounded
Risk" (NUPBR) condition that involves the boundedness in probability of the
terminal values of wealth processes. We show that this notion is the minimal
a-priori assumption required in order to proceed with utility optimization. The
fact that it is expressed entirely in terms of predictable characteristics
makes it easy to check, something that the stronger NFLVR condition lacks.Comment: 43 page
Pathwise solvability of stochastic integral equations with generalized drift and non-smooth dispersion functions
We study one-dimensional stochastic integral equations with non-smooth
dispersion coefficients, and with drift components that are not restricted to
be absolutely continuous with respect to Lebesgue measure. In the spirit of
Lamperti, Doss and Sussmann, we relate solutions of such equations to solutions
of certain ordinary integral equations, indexed by a generic element of the
underlying probability space. This relation allows us to solve the stochastic
integral equations in a pathwise sense.Comment: Accepted for publication: Annales de l'Institut Henri Poincar\'
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