32 research outputs found
Existence of an endogenously complete equilibrium driven by a diffusion
The existence of complete Radner equilibria is established in an economy
which parameters are driven by a diffusion process. Our results complement
those in the literature. In particular, we work under essentially minimal
regularity conditions and treat time-inhomogeneous case.Comment: minor changes to make it identical to the version accepted by Finance
and Stochastic
Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model
We obtain stability estimates and derive analytic expansions for local
solutions of multi-dimensional quadratic BSDEs. We apply these results to a
financial model where the prices of risky assets are quoted by a representative
dealer in such a way that it is optimal to meet an exogenous demand. We show
that the prices are stable under the demand process and derive their analytic
expansions for small risk aversion coefficients of the dealer.Comment: Final version, 28 page
Density of the set of probability measures with the martingale representation property
Let be a multi-dimensional random variable. We show that the set of
probability measures such that the -martingale
has the Martingale Representation Property (MRP) is either empty or dense in
-norm. The proof is based on a related result involving
analytic fields of terminal conditions and probability
measures over an open set . Namely, we show that
the set of points such that does not
have the MRP, either coincides with or has Lebesgue measure zero. Our study
is motivated by the problem of endogenous completeness in financial economics.Comment: 24 pages, forthcoming in Annals of Probabilit
A system of quadratic BSDEs arising in a price impact model
We consider a financial model where the prices of risky assets are quoted by
a representative market maker who takes into account an exogenous demand. We
characterize these prices in terms of a system of BSDEs with quadratic growth.
We show that this system admits a unique solution for every bounded demand if
and only if the market maker's risk-aversion is sufficiently small. The
uniqueness is established in the natural class of solutions, without any
additional norm restrictions. To the best of our knowledge, this is the first
study that proves such (global) uniqueness result for a system of fully coupled
quadratic BSDEs.Comment: Published at http://dx.doi.org/10.1214/15-AAP1103 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Muckenhoupt's condition and the existence of the optimal martingale measure
In the problem of optimal investment with utility function defined on
, we formulate sufficient conditions for the dual optimizer to be a
uniformly integrable martingale. Our key requirement consists of the existence
of a martingale measure whose density process satisfies the probabilistic
Muckenhoupt condition for the power , where is
a lower bound on the relative risk-aversion of the utility function. We
construct a counterexample showing that this condition is sharp.Comment: 24 page
Sensitivity analysis of utility-based prices and risk-tolerance wealth processes
In the general framework of a semimartingale financial model and a utility
function defined on the positive real line, we compute the first-order
expansion of marginal utility-based prices with respect to a ``small'' number
of random endowments. We show that this linear approximation has some important
qualitative properties if and only if there is a risk-tolerance wealth process.
In particular, they hold true in the following polar cases:
\begin{tabular}@p97mm@ for any utility function , if and only if the set of
state price densities has a greatest element from the point of view of
second-order stochastic dominance;for any financial model, if and only if
is a power utility function ( is an exponential utility function if it is
defined on the whole real line). \end{tabular}Comment: Published at http://dx.doi.org/10.1214/105051606000000529 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets
We study the two-times differentiability of the value functions of the primal
and dual optimization problems that appear in the setting of expected utility
maximization in incomplete markets. We also study the differentiability of the
solutions to these problems with respect to their initial values. We show that
the key conditions for the results to hold true are that the relative risk
aversion coefficient of the utility function is uniformly bounded away from
zero and infinity, and that the prices of traded securities are sigma-bounded
under the num\'{e}raire given by the optimal wealth process.Comment: Published at http://dx.doi.org/10.1214/105051606000000259 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
An optimal transport problem with backward martingale constraints motivated by insider trading
We study a single-period optimal transport problem on with a
covariance-type cost function and a backward
martingale constraint. We show that a transport plan is optimal if and
only if there is a maximal monotone set that supports the -marginal of
and such that for every in the
support of . We obtain sharp regularity conditions for the uniqueness
of an optimal plan and for its representation in terms of a map. Our study is
motivated by a variant of the classical Kyle model of insider trading from
Rochet and Vila (1994).Comment: 46 page