11,018 research outputs found

### A Note on the Quantile Formulation

Many investment models in discrete or continuous-time settings boil down to
maximizing an objective of the quantile function of the decision variable. This
quantile optimization problem is known as the quantile formulation of the
original investment problem. Under certain monotonicity assumptions, several
schemes to solve such quantile optimization problems have been proposed in the
literature. In this paper, we propose a change-of-variable and relaxation
method to solve the quantile optimization problems without using the calculus
of variations or making any monotonicity assumptions. The method is
demonstrated through a portfolio choice problem under rank-dependent utility
theory (RDUT). We show that this problem is equivalent to a classical Merton's
portfolio choice problem under expected utility theory with the same utility
function but a different pricing kernel explicitly determined by the given
pricing kernel and probability weighting function. With this result, the
feasibility, well-posedness, attainability and uniqueness issues for the
portfolio choice problem under RDUT are solved. It is also shown that solving
functional optimization problems may reduce to solving probabilistic
optimization problems. The method is applicable to general models with
law-invariant preference measures including portfolio choice models under
cumulative prospect theory (CPT) or RDUT, Yaari's dual model, Lopes' SP/A
model, and optimal stopping models under CPT or RDUT.Comment: to appear in Mathematical Financ

### Investment under Duality Risk Measure

One index satisfies the duality axiom if one agent, who is uniformly more
risk-averse than another, accepts a gamble, the latter accepts any less risky
gamble under the index. Aumann and Serrano (2008) show that only one index
defined for so-called gambles satisfies the duality and positive homogeneity
axioms. We call it a duality index. This paper extends the definition of
duality index to all outcomes including all gambles, and considers a portfolio
selection problem in a complete market, in which the agent's target is to
minimize the index of the utility of the relative investment outcome. By
linking this problem to a series of Merton's optimum consumption-like problems,
the optimal solution is explicitly derived. It is shown that if the prior
benchmark level is too high (which can be verified), then the investment risk
will be beyond any agent's risk tolerance. If the benchmark level is
reasonable, then the optimal solution will be the same as that of one of the
Merton's series problems, but with a particular value of absolute risk
aversion, which is given by an explicit algebraic equation as a part of the
optimal solution. According to our result, it is riskier to achieve the same
surplus profit in a stable market than in a less-stable market, which is
consistent with the common financial intuition.Comment: 17 pages, 1 figur

### A Note on the Monge-Kantorovich Problem in the Plane

The Monge-Kantorovich mass-transportation problem has been shown to be
fundamental for various basic problems in analysis and geometry in recent
years. Shen and Zheng (2010) proposed a probability method to transform the
celebrated Monge-Kantorovich problem in a bounded region of the Euclidean plane
into a Dirichlet boundary problem associated to a nonlinear elliptic equation.
Their results are original and sound, however, their arguments leading to the
main results are skipped and difficult to follow. In the present paper, we
adopt a different approach and give a short and easy-followed detailed proof
for their main results

### Optimal stopping under probability distortion

We formulate an optimal stopping problem for a geometric Brownian motion
where the probability scale is distorted by a general nonlinear function. The
problem is inherently time inconsistent due to the Choquet integration
involved. We develop a new approach, based on a reformulation of the problem
where one optimally chooses the probability distribution or quantile function
of the stopped state. An optimal stopping time can then be recovered from the
obtained distribution/quantile function, either in a straightforward way for
several important cases or in general via the Skorokhod embedding. This
approach enables us to solve the problem in a fairly general manner with
different shapes of the payoff and probability distortion functions. We also
discuss economical interpretations of the results. In particular, we justify
several liquidation strategies widely adopted in stock trading, including those
of "buy and hold", "cut loss or take profit", "cut loss and let profit run" and
"sell on a percentage of historical high".Comment: Published in at http://dx.doi.org/10.1214/11-AAP838 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Semidefinite approximation for mixed binary quadratically constrained quadratic programs

Motivated by applications in wireless communications, this paper develops
semidefinite programming (SDP) relaxation techniques for some mixed binary
quadratically constrained quadratic programs (MBQCQP) and analyzes their
approximation performance. We consider both a minimization and a maximization
model of this problem. For the minimization model, the objective is to find a
minimum norm vector in $N$-dimensional real or complex Euclidean space, such
that $M$ concave quadratic constraints and a cardinality constraint are
satisfied with both binary and continuous variables. {\color{blue}By employing
a special randomized rounding procedure, we show that the ratio between the
norm of the optimal solution of the minimization model and its SDP relaxation
is upper bounded by \cO(Q^2(M-Q+1)+M^2) in the real case and by
\cO(M(M-Q+1)) in the complex case.} For the maximization model, the goal is
to find a maximum norm vector subject to a set of quadratic constraints and a
cardinality constraint with both binary and continuous variables. We show that
in this case the approximation ratio is bounded from below by
\cO(\epsilon/\ln(M)) for both the real and the complex cases. Moreover, this
ratio is tight up to a constant factor

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