arXiv subjects

Alexander Schied

Publications and source records attributed to Alexander Schied.

At least 19 recordsLinked to original sources

arXiv2026

Sample Path Properties of the Fractional Wiener--Weierstrass Bridge II

Fractional Wiener--Weierstrass bridges are a class of Gaussian processes obtained by replacing trigonometric functions in the construction of classical Weierstrass functions by fractional Brownian bridges. A number of their sample path properties were derived in Schied--Zhang (2024,2026). The analysis in these papers left several open questions, most of which are addressed here. Specifically, we prove that, in the regime in which the Weierstrass mechanism dominates the underlying fractional Brownian bridge, the limiting $b$-adic variation coefficient has an absolutely continuous distribution and is therefore genuinely random. At the critical point between the two roughness regimes, we establish the power-variation formula and the critical $\Phi$-variation limit conjectured in Schied--Zhang (2024). Finally, we derive the Hausdorff dimension for the graphs of the sample paths by proving a conjecture from Schied--Zhang (2026) for the missing high-Hurst case.

math.PR↗

arXiv2026

Dynamic Lagrange Multipliers in a Non-concave Utility Framework

In continuous-time portfolio selection for non-concave utility functions, the martingale duality approach is widely adopted in complete markets, while the dynamic programming approach may sometimes lead to singular solutions of the Hamilton-Jacobi-Bellman (HJB) equation. We propose "dynamic Lagrange multipliers" in a non-concave utility framework, bridging two approaches and demonstrating that the Lagrangian multiplier function (in the martingale duality approach) equals the conjugate dual point related to the value function (in dynamic programming), which is exactly its partial derivative with respect to wealth. Moreover, the dynamic multiplier process exhibits homogeneity via the optimal wealth and pricing kernel processes, offering intuitive economic interpretations as a dynamic shadow price of the envelope theorem. Finally, classical optimal results are recovered and numerically validated by non-concave utility examples.

math.OC↗

arXiv2025

Lambda Value-at-Risk under ambiguity and risk sharing

In this paper, we investigate the Lambda Value-at-Risk ($\Lambda$VaR) under ambiguity, where the ambiguity is represented by a family of probability measures. We establish that for increasing Lambda functions, the robust (i.e., worst-case) $\Lambda$VaR under such an ambiguity set is equivalent to $\Lambda$VaR computed with respect to a capacity, a novel extension in the literature. This framework unifies and extends both traditional $\Lambda$VaR and Choquet quantiles (Value-at-Risk under ambiguity). We analyze the fundamental properties of this extended risk measure and establish a novel equivalent representation for $\Lambda$VaR under capacities with monotone Lambda functions in terms of families of downsets. Moreover, explicit formulas are derived for robust $\Lambda$VaR when ambiguity sets are characterized by $\phi$-divergence and the likelihood ratio constraints, respectively. We further explore the applications in risk sharing among multiple agents. We demonstrate that the family of risk measures induced by families of downsets is closed under inf-convolution. In particular, we prove that the inf-convolution of $\Lambda$VaR with capacities and monotone Lambda functions is another$\Lambda$VaR under a capacity. The explicit forms of optimal allocations are also derived. Moreover, we obtain more explicit results for risk sharing under ambiguity sets characterized by $\phi$-divergence and likelihood ratio constraints. Finally, we explore comonotonic risk-sharing for $\Lambda$VaR under ambiguity.

q-fin.RM↗

arXiv2025

Universal portfolios in continuous time: an approach in pathwise It\^o calculus

We provide a simple and straightforward approach to a continuous-time version of Cover's universal portfolio strategies within the model-free context of F\"ollmer's pathwise It\^o calculus. We establish the existence of the universal portfolio strategy and prove that its portfolio value process is the average of all values of constant rebalanced strategies. This result relies on a systematic comparison between two alternative descriptions of self-financing trading strategies within pathwise It\^o calculus. We moreover provide a comparison result for the performance and the realized volatility and variance of constant rebalanced portfolio strategies.

q-fin.MF↗

arXiv2025

On the rate of convergence of estimating the Hurst parameter of rough stochastic volatility models

In [Han \& Schied, 2023, \textit{arXiv 2307.02582}], an easily computable scale-invariant estimator $\widehat{\mathscr{R}}^s_n$ was constructed to estimate the Hurst parameter of the drifted fractional Brownian motion $X$ from its antiderivative. This paper extends this convergence result by proving that $\widehat{\mathscr{R}}^s_n$ also consistently estimates the Hurst parameter when applied to the antiderivative of $g \circ X$ for a general nonlinear function $g$. We also establish an almost sure rate of convergence in this general setting. Our result applies, in particular, to the estimation of the Hurst parameter of a wide class of rough stochastic volatility models from discrete observations of the integrated variance, including the rough fractional stochastic volatility model.

q-fin.ST↗

arXiv2024

Sample Path Properties of the Fractional Wiener--Weierstrass Bridge

Fractional Wiener--Weierstrass bridges are a class of Gaussian processes that arise from replacing the trigonometric function in the construction of classical Weierstrass functions by a fractional Brownian bridge. We investigate the sample path properties of such processes, including local and uniform moduli of continuity, $\Phi$-variation, Hausdorff dimension, and location of the maximum. Our analysis relies heavily on upper and lower bounds of fractional integrals, where we establish a novel improvement of the classical Hardy--Littlewood inequality for fractional integrals of a special class of step functions.

math.PR↗

arXiv2024

The roughness exponent and its model-free estimation

Motivated by pathwise stochastic calculus, we say that a continuous real-valued function $x$ admits the roughness exponent $R$ if the $p^{\text{th}}$ variation of $x$ converges to zero if $p>1/R$ and to infinity if $p<1/R$. For the sample paths of many stochastic processes, such as fractional Brownian motion, the roughness exponent exists and equals the standard Hurst parameter. In our main result, we provide a mild condition on the Faber--Schauder coefficients of $x$ under which the roughness exponent exists and is given as the limit of the classical Gladyshev estimates $\widehat R_n(x)$. This result can be viewed as a strong consistency result for the Gladyshev estimators in an entirely model-free setting, because it works strictly trajectory-wise and requires no probabilistic assumptions. Nonetheless, our proof is probabilistic and relies on a martingale that is hidden in the Faber--Schauder expansion of $x$. Since the Gladyshev estimators are not scale-invariant, we construct several scale-invariant estimators that are derived from the sequence $(\widehat R_n)_{n\in\mathbb N}$. We also discuss how a dynamic change in the roughness parameter of a time series can be detected. Finally, we extend our results to the case in which the $p^{\text{th}}$ variation of $x$ is defined over a sequence of unequally spaced partitions. Our results are illustrated by means of high-frequency financial time series.

math.ST↗

arXiv2024

Multiarmed Bandits Problem Under the Mean-Variance Setting

The classical multi-armed bandit (MAB) problem involves a learner and a collection of K independent arms, each with its own ex ante unknown independent reward distribution. At each one of a finite number of rounds, the learner selects one arm and receives new information. The learner often faces an exploration-exploitation dilemma: exploiting the current information by playing the arm with the highest estimated reward versus exploring all arms to gather more reward information. The design objective aims to maximize the expected cumulative reward over all rounds. However, such an objective does not account for a risk-reward tradeoff, which is often a fundamental precept in many areas of applications, most notably in finance and economics. In this paper, we build upon Sani et al. (2012) and extend the classical MAB problem to a mean-variance setting. Specifically, we relax the assumptions of independent arms and bounded rewards made in Sani et al. (2012) by considering sub-Gaussian arms. We introduce the Risk Aware Lower Confidence Bound (RALCB) algorithm to solve the problem, and study some of its properties. Finally, we perform a number of numerical simulations to demonstrate that, in both independent and dependent scenarios, our suggested approach performs better than the algorithm suggested by Sani et al. (2012).

math.OC↗

arXiv2024

The perturbation method applied to a robust optimization problem with constraint

The present paper studies a kind of robust optimization problems with constraint. The problem is formulated through Backward Stochastic Differential Equations (BSDEs) with quadratic generators. A necessary condition is established for the optimal solution using a terminal perturbation method and properties of Bounded Mean Oscillation (BMO) martingales. The necessary condition is further proved to be sufficient for the existence of an optimal solution under an additional convexity assumption. Finally, the optimality condition is applied to discuss problems of partial hedging with ambiguity, fundraising under ambiguity and randomized testing problems for a quadratic $g$-expectation.

math.OC↗

arXiv2023

Weierstrass Bridges

We introduce a new class of stochastic processes called fractional Wiener-Weierstrass bridges. They arise by applying the convolution from the construction of the classical, fractal Weierstrass functions to an underlying fractional Brownian bridge. By analyzing the $p$-th variation of the fractional Wiener-Weierstrass bridge along the sequence of $b$-adic partitions, we identify two regimes in which the processes exhibit distinct sample path properties. We also analyze the critical case between those two regimes for Wiener-Weierstrass bridges that are based on standard Brownian bridge. We furthermore prove that fractional Wiener-Weierstrass bridges are never semimartingales, and we show that their covariance functions are typically fractal functions. Some of our results are extended to Weierstrass bridges based on bridges derived from a general continuous Gaussian martingale.

math.PR↗

arXiv2023

Estimating the roughness exponent of stochastic volatility from discrete observations of the integrated variance

We consider the problem of estimating the roughness of the volatility process in a stochastic volatility model that arises as a nonlinear function of fractional Brownian motion with drift. To this end, we introduce a new estimator that measures the so-called roughness exponent of a continuous trajectory, based on discrete observations of its antiderivative. The estimator has a very simple form and can be computed with great efficiency on large data sets. It is not derived from distributional assumptions but from strictly pathwise considerations. We provide conditions on the underlying trajectory under which our estimator converges in a strictly pathwise sense. Then we verify that these conditions are satisfied by almost every sample path of fractional Brownian motion (with drift). As a consequence, we obtain strong consistency theorems in the context of a large class of rough volatility models, such as the rough fractional volatility model and the rough Bergomi model. We also demonstrate that our estimator is robust with respect to proxy errors between the integrated and realized variance, and that it can be applied to estimate the roughness exponent directly from the price trajectory. Numerical simulations show that our estimation procedure performs well after passing to a scale-invariant modification of our estimator.

q-fin.ST↗

arXiv2023

A criterion for absolute continuity relative to the law of fractional Brownian motion

Let $X$ be the sum of a fractional Brownian motion with Hurst parameter $H$ and an absolutely continuous and adapted drift process. We establish a simple criterion that guarantees that the law of $X$ is absolutely continuous with respect to the law of the original fractional Brownian motion. For $H<1/2$, the trajectories of the derivative of the drift need to be bounded by an almost surely finite random variable; for $H>1/2$, they need to satisfy a H\"older condition with some exponent larger than $2H-1$. These are almost-sure conditions, and no expectation requirements are imposed. For the case in which $X$ arises as the solution of a nonlinear stochastic integral equation driven by fractional Brownian motion, we provide a simple criterion on the drift coefficient under which the law of $X$ is automatically equivalent to the one of fractional Brownian motion.

math.PR↗

arXiv2022

Robust Faber--Schauder approximation based on discrete observations of an antiderivative

We study the problem of reconstructing the Faber--Schauder coefficients of a continuous function $f$ from discrete observations of its antiderivative $F$. For instance, this question arises in financial mathematics when estimating the roughness of volatility from the integrated volatility of an asset price trajectory. Our approach starts with mathematically formulating the reconstruction problem through piecewise quadratic spline interpolation. We then provide a closed-form solution and an in-depth error analysis. These results lead to some surprising observations, which also throw new light on the classical topic of quadratic spline interpolation itself: They show that the well-known instabilities of this method can be located exclusively within the final generation of estimated Faber--Schauder coefficients, which suffer from non-locality and strong dependence on the initial value. By contrast, all other Faber--Schauder coefficients depend only locally on the data, are independent of the initial value, and admit uniform error bounds. We thus conclude that a robust and well-behaved estimator for our problem can be obtained by simply dropping the final-generation coefficients from the estimated Faber--Schauder coefficients.

math.NA↗

arXiv2021

A limit theorem for Bernoulli convolutions and the $Φ$-variation of functions in the Takagi class

We consider a probabilistic approach to compute the Wiener--Young $Φ$-variation of fractal functions in the Takagi class. Here, the $Φ$-variation is understood as a generalization of the quadratic variation or, more generally, the $p^{\text{th}}$ variation of a trajectory computed along the sequence of dyadic partitions of the unit interval. The functions $Φ$ we consider form a very wide class of functions that are regularly varying at zero. Moreover, for each such function $Φ$, our results provide in a straightforward manner a large and tractable class of functions that have nontrivial and linear $Φ$-variation. As a corollary, we also construct stochastic processes whose sample paths have nontrivial, deterministic, and linear $Φ$-variation for each function $Φ$ from our class. The proof of our main result relies on a limit theorem for certain sums of Bernoulli random variables that converge to an infinite Bernoulli convolution.

math.PR↗

arXiv2021

Protecting Pegged Currency Markets from Speculative Investors

We consider a stochastic game between a trader and a central bank in a target zone market with a lower currency peg. This currency peg is maintained by the central bank through the generation of permanent price impact, thereby aggregating an ever increasing risky position in foreign reserves. We describe this situation mathematically by means of two coupled singular control problems, where the common singularity arises from a local time along a random curve. Our first result identifies a certain local time as that central bank strategy for which this risk position is minimized. We then consider the worst-case situation the central bank may face by identifying that strategy of the strategic investor that maximizes the expected inventory of the central bank under a cost criterion, thus establishing a Stackelberg equilibrium in our model.

q-fin.MF↗

arXiv2021

Robustness in the Optimization of Risk Measures

We study issues of robustness in the context of Quantitative Risk Management and Optimization. We develop a general methodology for determining whether a given risk measurement related optimization problem is robust, which we call "robustness against optimization". The new notion is studied for various classes of risk measures and expected utility and loss functions. Motivated by practical issues from financial regulation, special attention is given to the two most widely used risk measures in the industry, Value-at-Risk (VaR) and Expected Shortfall (ES). We establish that for a class of general optimization problems, VaR leads to non-robust optimizers whereas convex risk measures generally lead to robust ones. Our results offer extra insight on the ongoing discussion about the comparative advantages of VaR and ES in banking and insurance regulation. Our notion of robustness is conceptually different from the field of robust optimization, to which some interesting links are derived.

q-fin.RM↗

arXiv2020

A probabilistic approach to the $Φ$-variation of classical fractal functions with critical roughness

We consider Weierstraß and Takagi-van der Waerden functions with critical degree of roughness. In this case, the functions have vanishing $p^{\text{th}}$ variation for all $p>1$ but are also nowhere differentiable and hence not of bounded variation either. We resolve this apparent puzzle by showing that these functions have finite, nonzero, and linear Wiener--Young $Φ$-variation along the sequence of $b$-adic partitions, where $Φ(x)=x/\sqrt{-\log x}$. For the Weierstraß functions, our proof is based on the martingale central limit theorem (CLT). For the Takagi--van der Waerden functions, we use the CLT for Markov chains if a certain parameter $b$ is odd, and the standard CLT for $b$ even.

math.PR↗

arXiv2020

A central bank strategy for defending a currency peg

We consider a central bank strategy for maintaining a two-sided currency target zone, in which an exchange rate of two currencies is forced to stay between two thresholds. To keep the exchange rate from breaking the prescribed barriers, the central bank is generating permanent price impact and thereby accumulating inventory in the foreign currency. Historical examples of failed target zones illustrate that this inventory can become problematic, in particular when there is an adverse macroeconomic trend in the market. We model this situation through a continuous-time market impact model of Almgren--Chriss-type with drift, in which the exchange rate is a diffusion process controlled by the price impact of the central bank's intervention strategy. The objective of the central bank is to enforce the target zone through a strategy that minimizes the accumulated inventory. We formulate this objective as a stochastic control problem with random time horizon. It is solved by reduction to a singular boundary value problem that was solved by Lasry and Lions (1989). Finally, we provide numerical simulations of optimally controlled exchange rate processes and the corresponding evolution of the central bank inventory.

q-fin.TR↗