33 research outputs found
Asymptotic arbitrage and num\'eraire portfolios in large financial markets
This paper deals with the notion of a large financial market and the concepts
of asymptotic arbitrage and strong asymptotic arbitrage (both of the first
kind), introduced by Yu.M. Kabanov and D.O. Kramkov. We show that the arbitrage
properties of a large market are completely determined by the asymptotic
behavior of the sequence of the num\'eraire portfolios, related to the small
markets. The obtained criteria can be expressed in terms of contiguity, entire
separation and Hellinger integrals, provided these notions are extended to
sub-probability measures. As examples we consider market models on finite
probability spaces, semimartingale and diffusion models. Also a discrete-time
infinite horizon market model with one log-normal stock is examined.Comment: 18 page
Stochastic Perron's method for optimal control problems with state constraints
We apply the stochastic Perron method of Bayraktar and S\^irbu to a general
infinite horizon optimal control problem, where the state is a controlled
diffusion process, and the state constraint is described by a closed set. We
prove that the value function is bounded from below (resp., from above) by
a viscosity supersolution (resp., subsolution) of the related state constrained
problem for the Hamilton-Jacobi-Bellman equation. In the case of a smooth
domain, under some additional assumptions, these estimates allow to identify
with a unique continuous constrained viscosity solution of this equation.Comment: 14 page
Central limit theorem under uncertain linear transformations
We prove a variant of the central limit theorem (CLT) for a sequence of
i.i.d. random variables , perturbed by a stochastic sequence of linear
transformations , representing the model uncertainty. The limit,
corresponding to a "worst" sequence , is expressed in terms of the
viscosity solution of the -heat equation. In the context of the CLT under
sublinear expectations this nonlinear parabolic equation appeared previously in
the papers of S.Peng. Our proof is based on the technique of half-relaxed
limits from the theory of approximation schemes for fully nonlinear partial
differential equations.Comment: 11 page
Resource allocation in communication networks with large number of users: the stochastic gradient descent method
We consider a communication network with fixed number of links, shared by
large number of users. The resource allocation is performed on the basis of an
aggregate utility maximization in accordance with the popular approach,
proposed by Kelly and coauthors (1998). The problem is to construct a pricing
mechanism for transmission rates to stimulate an optimal allocation of the
available resources.
In contrast to the usual approach, the proposed algorithm does not use the
information on the aggregate traffic over each link. Its inputs are the total
number of users, the link capacities and optimal myopic reactions of
randomly selected users to the current prices. The dynamic pricing scheme is
based on the dual projected stochastic gradient descent method. For a special
class of utility functions we obtain upper bounds for the amount of
constraint violation and the deviation of the objective function from the
optimal value. These estimates are uniform in and are of order
in the number of reaction measurements. We present some
computer experiments for quadratic utility functions .Comment: 19 page
Martingale selection problem and asset pricing in finite discrete time
Given a set-valued stochastic process , we say that the
martingale selection problem is solvable if there exists an adapted sequence of
selectors , admitting an equivalent martingale measure. The aim
of this note is to underline the connection between this problem and the
problems of asset pricing in general discrete-time market models with portfolio
constraints and transaction costs. For the case of relatively open convex sets
we present effective necessary and sufficient conditions for the
solvability of a suitably generalized martingale selection problem. We show
that this result allows to obtain computationally feasible formulas for the
price bounds of contingent claims. For the case of currency markets we also
give a comment on the first fundamental theorem of asset pricing.Comment: 6 page
Kreps-Yan theorem for Banach ideal spaces
Let be a closed convex cone in a Banach ideal space on a measurable
space with a -finite measure. We prove that conditions and imply the existence of a strictly positive
continuous functional on , whose restriction to is non-positive.Comment: 6 page
Lower bounds of martingale measure densities in the Dalang-Morton-Willinger theorem
For a -dimensional stochastic process we obtain criteria
for the existence of an equivalent martingale measure, whose density , up to
a normalizing constant, is bounded from below by a given random variable .
We consider the case of one-period model (N=1) under the assumptions ; , , where , and the case of
-period model for . The mentioned criteria are expressed in terms
of the conditional distributions of the increments of , as well as in terms
of the boundedness from above of an utility function related to some optimal
investment problem under the loss constraints. Several examples are presented.Comment: 19 page
Verification by stochastic Perron's method in stochastic exit time control problems
We apply the Stochastic Perron method, created by Bayraktar and S\^irbu, to a
stochastic exit time control problem. Our main assumption is the validity of
the Strong Comparison Result for the related Hamilton-Jacobi-Bellman (HJB)
equation. Without relying on Bellman's optimality principle we prove that
inside the domain the value function is continuous and coincides with a
viscosity solution of the Dirichlet boundary value problem for the HJB
equation.Comment: 14 page
Asymptotic sequential Rademacher complexity of a finite function class
For a finite function class we describe the large sample limit of the
sequential Rademacher complexity in terms of the viscosity solution of a
-heat equation. In the language of Peng's sublinear expectation theory, the
same quantity equals to the expected value of the largest order statistics of a
multidimensional -normal random variable. We illustrate this result by
deriving upper and lower bounds for the asymptotic sequential Rademacher
complexity.Comment: 10 page
A proof of the Dalang-Morton-Willinger theorem
We give a new proof of the Dalang-Morton-Willinger theorem, relating the
no-arbitrage condition in stochastic securities market models to the existence
of an equivalent martingale measure with bounded density for a -dimensional
stochastic sequence of stock prices. Roughly speaking, the
proof is reduced to the assertion that under the no-arbitrage condition for N=1
and there exists a strictly positive linear fucntional on ,
which is bounded from above on a special subset of the subspace
of investor's gains.Comment: 9 page