arXiv subjects

Daniel Bartl

Publications and source records attributed to Daniel Bartl.

At least 37 records · Page 2Linked to original sources

arXiv2022

Random embeddings with an almost Gaussian distortion

Let $X$ be a symmetric, isotropic random vector in $\mathbb{R}^m$ and let $X_1...,X_n$ be independent copies of $X$. We show that under mild assumptions on $\|X\|_2$ (a suitable thin-shell bound) and on the tail-decay of the marginals $\langle X,u\rangle$, the random matrix $A$, whose columns are $X_i/\sqrt{m}$ exhibits a Gaussian-like behaviour in the following sense: for an arbitrary subset of $T\subset \mathbb{R}^n$, the distortion $\sup_{t \in T} | \|At\|_2^2 - \|t\|_2^2 |$ is almost the same as if $A$ were a Gaussian matrix. A simple outcome of our result is that if $X$ is a symmetric, isotropic, log-concave random vector and $n \leq m \leq c_1(α)n^α$ for some $α>1$, then with high probability, the extremal singular values of $A$ satisfy the optimal estimate: $1-c_2(α) \sqrt{n/m} \leq λ_{\rm min} \leq λ_{\rm max} \leq 1+c_2(α) \sqrt{n/m}$.

math.FA↗

arXiv2022

On Monte-Carlo methods in convex stochastic optimization

We develop a novel procedure for estimating the optimizer of general convex stochastic optimization problems of the form $\min_{x\in\mathcal{X}} \mathbb{E}[F(x,ξ)]$, when the given data is a finite independent sample selected according to $ξ$. The procedure is based on a median-of-means tournament, and is the first procedure that exhibits the optimal statistical performance in heavy tailed situations: we recover the asymptotic rates dictated by the central limit theorem in a non-asymptotic manner once the sample size exceeds some explicitly computable threshold. Additionally, our results apply in the high-dimensional setup, as the threshold sample size exhibits the optimal dependence on the dimension (up to a logarithmic factor). The general setting allows us to recover recent results on multivariate mean estimation and linear regression in heavy-tailed situations and to prove the first sharp, non-asymptotic results for the portfolio optimization problem.

math.ST↗

arXiv2021

Sensitivity analysis of Wasserstein distributionally robust optimization problems

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints. We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we provide explicit first-order approximation for square-root LASSO regression coefficients and deduce coefficient shrinkage compared to the ordinary least squares regression. We consider robustness of call option pricing and deduce a new Black-Scholes sensitivity, a non-parametric version of the so-called Vega. We also compute sensitivities of optimized certainty equivalents in finance and propose measures to quantify robustness of neural networks to adversarial examples.

math.OC↗

arXiv2021

Duality Theory for Robust Utility Maximisation

In this paper we present a duality theory for the robust utility maximisation problem in continuous time for utility functions defined on the positive real axis. Our results are inspired by -- and can be seen as the robust analogues of -- the seminal work of Kramkov & Schachermayer [18]. Namely, we show that if the set of attainable trading outcomes and the set of pricing measures satisfy a bipolar relation, then the utility maximisation problem is in duality with a conjugate problem. We further discuss the existence of optimal trading strategies. In particular, our general results include the case of logarithmic and power utility, and they apply to drift and volatility uncertainty.

q-fin.MF↗

arXiv2021

Estimating processes in adapted Wasserstein distance

A number of researchers have independently introduced topologies on the set of laws of stochastic processes that extend the usual weak topology. Depending on the respective scientific background this was motivated by applications and connections to various areas (e.g. Plug-Pichler - stochastic programming, Hellwig - game theory, Aldous - stability of optimal stopping, Hoover-Keisler - model theory). Remarkably, all these seemingly independent approaches define the same adapted weak topology in finite discrete time. Our first main result is to construct an adapted variant of the empirical measure that consistently estimates the laws of stochastic processes in full generality. A natural compatible metric for the weak adapted topology is the given by an adapted refinement of the Wasserstein distance, as established in the seminal works of Pflug-Pichler. Specifically, the adapted Wasserstein distance allows to control the error in stochastic optimization problems, pricing and hedging problems, optimal stopping problems, etc. in a Lipschitz fashion. The second main result of this article yields quantitative bounds for the convergence of the adapted empirical measure with respect to adapted Wasserstein distance. Surprisingly, we obtain virtually the same optimal rates and concentration results that are known for the classical empirical measure wrt. Wasserstein distance.

math.PR↗

arXiv2021

Limits of random walks with distributionally robust transition probabilities

We consider a nonlinear random walk which, in each time step, is free to choose its own transition probability within a neighborhood (w.r.t. Wasserstein distance) of the transition probability of a fixed Lévy process. In analogy to the classical framework we show that, when passing from discrete to continuous time via a scaling limit, this nonlinear random walk gives rise to a nonlinear semigroup. We explicitly compute the generator of this semigroup and corresponding PDE as a perturbation of the generator of the initial Lévy process.

math.PR↗

arXiv2020

All Adapted Topologies are Equal

A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. Aldous defines an extended weak topology based on the weak convergence of prediction processes. In the economic literature, Hellwig introduced the information topology to study the stability of equilibrium problems. Bion-Nadal and Talay introduce a version of the Wasserstein distance between the laws of diffusion processes. Pflug and Pichler consider the nested distance (and the weak nested topology) to obtain continuity of stochastic multistage programming problems. These distances can be seen as a symmetrization of Lassalle's causal transport problem, but there are also further natural ways to derive a topology from causal transport. Our main result is that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover we show that this 'weak adapted topology' is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions.

math.PR↗

arXiv2020

Adapted Wasserstein Distances and Stability in Mathematical Finance

Assume that an agent models a financial asset through a measure Q with the goal to price / hedge some derivative or optimize some expected utility. Even if the model Q is chosen in the most skilful and sophisticated way, she is left with the possibility that Q does not provide an "exact" description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of Q? If we measure proximity with the usual Wasserstein distance (say), the answer is NO. Models which are similar w.r.t. Wasserstein distance may provide dramatically different information on which to base a hedging strategy. Remarkably, this can be overcome by considering a suitable "adapted" version of the Wasserstein distance which takes the temporal structure of pricing models into account. This adapted Wasserstein distance is most closely related to the nested distance as pioneered by Pflug and Pichler \cite{Pf09,PfPi12,PfPi14}. It allows us to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time. Notably, these abstract results are sharp already for Brownian motion and European call options.

q-fin.MF↗

arXiv2019

Pathwise superhedging on prediction sets

In this paper we provide a pricing-hedging duality for the model-independent superhedging price with respect to a prediction set $Ξ\subseteq C[0,T]$, where the superhedging property needs to hold pathwise, but only for paths lying in $Ξ$. For any Borel measurable claim $ξ$ which is bounded from below, the superhedging price coincides with the supremum over all pricing functionals $\mathbb{E}_{\mathbb{Q}}[ξ]$ with respect to martingale measures $\mathbb{Q}$ concentrated on the prediction set $Ξ$. This allows to include beliefs in future paths of the price process expressed by the set $Ξ$, while eliminating all those which are seen as impossible. Moreover, we provide several examples to justify our setup.

q-fin.MF↗

arXiv2019

Functional inequalities for forward and backward diffusions

In this article we derive Talagrand's $T_2$ inequality on the path space w.r.t. the maximum norm for various stochastic processes, including solutions of one-dimensional stochastic differential equations with measurable drifts, backward stochastic differential equations, and the value process of optimal stopping problems. The proofs do not make use of the Girsanov method, but of pathwise arguments. These are used to show that all our processes of interest are Lipschitz transformations of processes which are known to satisfy desired functional inequalities.

math.PR↗

arXiv2019

Stochastic integration and differential equations for typical paths

The goal of this paper is to define stochastic integrals and to solve stochastic differential equations for typical paths taking values in a possibly infinite dimensional separable Hilbert space without imposing any probabilistic structure. In the spirit of [33, 37] and motivated by the pricing duality result obtained in [4] we introduce an outer measure as a variant of the pathwise minimal superhedging price where agents are allowed to trade not only in $ω$ but also in $\intω\,dω:=ω^2 -\langle ω\rangle$ and where they are allowed to include beliefs in future paths of the price process expressed by a prediction set. We then call a property to hold true on typical paths if the set of paths where the property fails is null with respect to our outer measure. It turns out that adding the second term $ω^2 -\langle ω\rangle$ in the definition of the outer measure enables to directly construct stochastic integrals which are continuous, even for typical paths taking values in an infinite dimensional separable Hilbert space. Moreover, when restricting to continuous paths whose quadratic variation is absolutely continuous with uniformly bounded derivative, a second construction of model-free stochastic integrals for typical paths is presented, which then allows to solve in a model-free way stochastic differential equations for typical paths.

math.PR↗

arXiv2019

Duality for pathwise superhedging in continuous time

We provide a model-free pricing-hedging duality in continuous time. For a frictionless market consisting of $d$ risky assets with continuous price trajectories, we show that the purely analytic problem of finding the minimal superhedging price of a path dependent European option has the same value as the purely probabilistic problem of finding the supremum of the expectations of the option over all martingale measures. The superhedging problem is formulated with simple trading strategies, the claim is the limit inferior of continuous functions, which allows for upper and lower semi-continuous claims, and superhedging is required in the pathwise sense on a $σ$-compact sample space of price trajectories. If the sample space is stable under stopping, the probabilistic problem reduces to finding the supremum over all martingale measures with compact support. As an application of the general results we deduce dualities for Vovk's outer measure and semi-static superhedging with finitely many securities.

q-fin.MF↗

arXiv2019

Conditional nonlinear expectations

Let $Ω$ be a Polish space with Borel $σ$-field $\mathcal{F}$ and countably generated sub $σ$-field $\mathcal{G}\subset\mathcal{F}$. Denote by $\mathcal{L}(\mathcal{F})$ the set of all bounded $\mathcal{F}$-upper semianalytic functions from $Ω$ to the reals and by $\mathcal{L}(\mathcal{G})$ the subset of $\mathcal{G}$-upper semianalytic functions. Let $\mathcal{E}(\cdot|\mathcal{G})\colon\mathcal{L}(\mathcal{F})\to\mathcal{L}(\mathcal{G})$ be a sublinear increasing functional which leaves $\mathcal{L}(\mathcal{G})$ invariant. It is shown that there exists a $\mathcal{G}$-analytic set-valued mapping $\mathcal{P}_{\mathcal{G}}$ from $Ω$ to the set of probabilities which are concentrated on atoms of $\mathcal{G}$ with compact convex values such that $\mathcal{E}(X|\mathcal{G})(ω)=$ $\sup_{P\in\mathcal{P}_{\mathcal{G}}(ω)} E_P[X]$ if and only if $\mathcal{E}(\cdot |\mathcal{G})$ is pointwise continuous from below and continuous from above on the continuous functions. Further, given another sublinear increasing functional $\mathcal{E}(\cdot)\colon\mathcal{L}(\mathcal{F})\to\mathbb{R}$ which leaves the constants invariant, the tower property $\mathcal{E}(\cdot)=\mathcal{E}(\mathcal{E}(\cdot|\mathcal{G}))$ is characterized via a pasting property of the representing sets of probabilities, and the importance of analytic functions is explained. Finally, it is characterized when a nonlinear version of Fubini's theorem holds true and when the product of a set of probabilities and a set of kernels is compact.

math.PR↗

arXiv2019

Computational aspects of robust optimized certainty equivalents and option pricing

Accounting for model uncertainty in risk management and option pricing leads to infinite dimensional optimization problems which are both analytically and numerically intractable. In this article we study when this hurdle can be overcome for the so-called optimized certainty equivalent risk measure (OCE) -- including the average value-at-risk as a special case. First we focus on the case where the uncertainty is modeled by a nonlinear expectation penalizing distributions that are "far" in terms of optimal-transport distance (Wasserstein distance for instance) from a given baseline distribution. It turns out that the computation of the robust OCE reduces to a finite dimensional problem, which in some cases can even be solved explicitly. This principle also applies to the shortfall risk measure as well as for the pricing of European options. Further, we derive convex dual representations of the robust OCE for measurable claims without any assumptions on the set of distributions. Finally, we give conditions on the latter set under which the robust average value-at-risk is a tail risk measure.

q-fin.RM↗

arXiv2019

Exponential utility maximization under model uncertainty for unbounded endowments

We consider the robust exponential utility maximization problem in discrete time: An investor maximizes the worst case expected exponential utility with respect to a family of nondominated probabilistic models of her endowment by dynamically investing in a financial market, and statically in available options. We show that, for any measurable random endowment (regardless of whether the problem is finite or not) an optimal strategy exists, a dual representation in terms of (calibrated) martingale measures holds true, and that the problem satisfies the dynamic programming principle (in case of no options). Further it is shown that the value of the utility maximization problem converges to the robust superhedging price as the risk aversion parameter gets large, and examples of nondominated probabilistic models are discussed.

q-fin.PM↗

arXiv2018

Robust expected utility maximization with medial limits

In this paper we study a robust expected utility maximization problem with random endowment in discrete time. We give conditions under which an optimal strategy exists and derive a dual representation for the optimal utility. Our approach is based on a general representation result for monotone convex functionals, a functional version of Choquet's capacitability theorem and medial limits. The novelty is that it works under nondominated model uncertainty without any assumptions of time-consistency. As applications, we discuss robust utility maximization problems with moment constraints, Wasserstein constraints and Wasserstein penalties.

q-fin.PM↗

arXiv2018

A pointwise bipolar theorem

We provide a pointwise bipolar theorem for liminf-closed convex sets of positive Borel measurable functions on a sigma-compact metric space without the assumption that the polar is a tight set of measures. As applications we derive a version of the transport duality under non-tight marginals, and a superhedging duality for semistatic hedging in discrete time.

math.FA↗

arXiv2018

Marginal and dependence uncertainty: bounds, optimal transport, and sharpness

Motivated by applications in model-free finance and quantitative risk management, we consider Fréchet classes of multivariate distribution functions where additional information on the joint distribution is assumed, while uncertainty in the marginals is also possible. We derive optimal transport duality results for these Fréchet classes that extend previous results in the related literature. These proofs are based on representation results for increasing convex functionals and the explicit computation of the conjugates. We show that the dual transport problem admits an explicit solution for the function $f=1_B$, where $B$ is a rectangular subset of $\mathbb R^d$, and provide an intuitive geometric interpretation of this result. The improved Fréchet--Hoeffding bounds provide ad-hoc upper bounds for these Fréchet classes. We show that the improved Fréchet--Hoeffding bounds are pointwise sharp for these classes in the presence of uncertainty in the marginals, while a counterexample yields that they are not pointwise sharp in the absence of uncertainty in the marginals, even in dimension 2. The latter result sheds new light on the improved Fréchet--Hoeffding bounds, since Tankov [30] has showed that, under certain conditions, these bounds are sharp in dimension 2.

math.PR↗