11,152 research outputs found
Incorporating statistical model error into the calculation of acceptability prices of contingent claims
The determination of acceptability prices of contingent claims requires the
choice of a stochastic model for the underlying asset price dynamics. Given
this model, optimal bid and ask prices can be found by stochastic optimization.
However, the model for the underlying asset price process is typically based on
data and found by a statistical estimation procedure. We define a confidence
set of possible estimated models by a nonparametric neighborhood of a baseline
model. This neighborhood serves as ambiguity set for a multi-stage stochastic
optimization problem under model uncertainty. We obtain distributionally robust
solutions of the acceptability pricing problem and derive the dual problem
formulation. Moreover, we prove a general large deviations result for the
nested distance, which allows to relate the bid and ask prices under model
ambiguity to the quality of the observed data.Comment: 27 pages, 2 figure
Online Estimation and Optimization of Utility-Based Shortfall Risk
Utility-Based Shortfall Risk (UBSR) is a risk metric that is increasingly
popular in financial applications, owing to certain desirable properties that
it enjoys. We consider the problem of estimating UBSR in a recursive setting,
where samples from the underlying loss distribution are available
one-at-a-time. We cast the UBSR estimation problem as a root finding problem,
and propose stochastic approximation-based estimations schemes. We derive
non-asymptotic bounds on the estimation error in the number of samples. We also
consider the problem of UBSR optimization within a parameterized class of
random variables. We propose a stochastic gradient descent based algorithm for
UBSR optimization, and derive non-asymptotic bounds on its convergence
On the Minimization of Convex Functionals of Probability Distributions Under Band Constraints
The problem of minimizing convex functionals of probability distributions is
solved under the assumption that the density of every distribution is bounded
from above and below. A system of sufficient and necessary first-order
optimality conditions as well as a bound on the optimality gap of feasible
candidate solutions are derived. Based on these results, two numerical
algorithms are proposed that iteratively solve the system of optimality
conditions on a grid of discrete points. Both algorithms use a block coordinate
descent strategy and terminate once the optimality gap falls below the desired
tolerance. While the first algorithm is conceptually simpler and more
efficient, it is not guaranteed to converge for objective functions that are
not strictly convex. This shortcoming is overcome in the second algorithm,
which uses an additional outer proximal iteration, and, which is proven to
converge under mild assumptions. Two examples are given to demonstrate the
theoretical usefulness of the optimality conditions as well as the high
efficiency and accuracy of the proposed numerical algorithms.Comment: 13 pages, 5 figures, 2 tables, published in the IEEE Transactions on
Signal Processing. In previous versions, the example in Section VI.B
contained some mistakes and inaccuracies, which have been fixed in this
versio
Mixed-Integer Convex Nonlinear Optimization with Gradient-Boosted Trees Embedded
Decision trees usefully represent sparse, high dimensional and noisy data.
Having learned a function from this data, we may want to thereafter integrate
the function into a larger decision-making problem, e.g., for picking the best
chemical process catalyst. We study a large-scale, industrially-relevant
mixed-integer nonlinear nonconvex optimization problem involving both
gradient-boosted trees and penalty functions mitigating risk. This
mixed-integer optimization problem with convex penalty terms broadly applies to
optimizing pre-trained regression tree models. Decision makers may wish to
optimize discrete models to repurpose legacy predictive models, or they may
wish to optimize a discrete model that particularly well-represents a data set.
We develop several heuristic methods to find feasible solutions, and an exact,
branch-and-bound algorithm leveraging structural properties of the
gradient-boosted trees and penalty functions. We computationally test our
methods on concrete mixture design instance and a chemical catalysis industrial
instance
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