arXiv subjects

Stefan Gerhold

Publications and source records attributed to Stefan Gerhold.

At least 19 recordsLinked to original sources

arXiv2026

Stochastic Volterra equations: failure of the time-homogeneous Markov property

Path-dependence is a defining feature of many real-world systems, with applications ranging from population dynamics to rough volatility models and electricity spot prices. In stochastic Volterra equations (SVEs), such dependence is encoded in the Volterra kernel, which dictates how past trajectories influence present dynamics on infinitesimal time scales. This structure suggests a breakdown of the Markov property. In this article, we develop computational techniques and methods based on small-time asymptotics for SVEs with Hölder coefficients to rigorously establish that they cannot possess the time-homogeneous Markov property. In particular, for affine drifts, we characterise the time-homogeneous Markov property and show that under natural non-degeneracy conditions, it only holds for exponential Volterra kernels.

math.PR↗

arXiv2026

Boundary behaviour of the Volterra square-root process

In this work, we study the boundary behaviour of the Volterra square-root process on $\R_+$. For regular Volterra kernels, we establish a time-dependent Feller condition that guarantees that the process does not hit zero on $[0,T]$, and prove finiteness of negative $p$-moments. For rough kernels that are regularly varying at zero, we show that the process necessarily hits zero with positive probability, and that its law has an atom at the boundary. Finally, we establish analogous results for the limit distribution. Our proofs are based on comparison principles for Volterra integral equations and generalized Riemann--Liouville fractional equations. The latter provide us with upper and lower bounds for the solution of the associated Volterra Riccati equation, and hence bounds on the Laplace transform. As an application, we study the structure of equivalent martingale measures in the Volterra Heston model. For the rough case, we show that equivalent martingale measures exist only under very restrictive assumptions on the drift under the real-world measure.

math.PR↗

arXiv2026

On the integrability of the supremum of stochastic volatility models and other martingales

We propose a method to bound the expectation of the supremum of the price process in stochastic volatility models. It can be applied, for example, to the rough Bergomi model, avoiding the need to discuss finiteness of higher moments. Our motivation stems from the theory of American option pricing, as an integrable supremum implies the existence of an optimal stopping time for any linearly bounded payoff. Moreover, we survey the literature on martingales with non-integrable supremum, and give a new construction that yields uniformly integrable martingales with this property.

math.PR↗

arXiv2025

Failure of the Markov property for stochastic Volterra equations

Memory-driven stochastic dynamics arise naturally in many applications, and stochastic Volterra equations (SVEs) offer a flexible framework for modeling such systems. Their convolution structure with Volterra kernels endows the dynamics with a formal path-dependency, which suggests the failure of the Markov property. While this has previously been rigorously established only for Gaussian Volterra processes, by constructing nondegenerate admissible perturbations through Markovian lifts, we prove that also general SVEs with Hölder-continuous coefficients do not possess the Markov property for a broad class of Volterra kernels. Moreover, we show that the associated Markovian lift is, in general, necessarily infinite-dimensional. These observations reflect the intrinsic infinite-dimensionality of memory effects in SVEs and underscore the need for analytical and probabilistic tools beyond the classical Markovian framework.

math.PR↗

arXiv2025

On the supremum of a quotient of power sums

We define a function of two real vectors by a certain homogeneous quotient involving power sums, and show that its supremum grows asymptotically linearly w.r.t. the dimension. From this, we deduce a condition under which a parametric set of real matrices satisfies a set of polynomial positivity constraints. This characterization finds an application in mathematical finance, in a recent study on price impact models.

math.CA↗

arXiv2025

Self-similar Gaussian Markov processes

We characterize all multi-dimensional real self-similar Gaussian Markov processes. Three types of covariance matrix functions occur: white-noise type functions, covariances that can be expressed by continuous matrix semigroups, and covariances based on non-continuous solutions of Cauchy's functional equation. Characterizing the latter requires us to develop some results on the representation theory of non-continuous matrix semigroups, which are presented in a companion paper. In dimension one, besides white noise, the self-similar Gaussian Markov processes reduce to a two-parameter family of time-changed Brownian motions. This observation simplifies several proofs of non-Markovianity of concrete processes found in the literature.

math.PR↗

arXiv2025

On some inequalities for the two-parameter Mittag-Leffler function in the complex plane

For the two-parameter Mittag-Leffler function $E_{α,β}$ with $α> 0$ and $β\ge 0,$ we consider the question whether $|E_{α,β}(z)|$ and $E_{α,β}(\Re z)$ are comparable on the whole complex plane. We show that the inequality $|E_{α,β}(z)|\le E_{α,β}(\Re z)$ holds globally if and only if $E_{α,β}(-x)$ is completely monotone on $(0,\infty)$. For $α\in [1,2)$ we prove that the complete monotonicity of $1/E_{α,β}(x)$ on $(0,\infty)$ is necessary for the global inequality $|E_{α,β}(z)|\ge E_{α,β}(\Re z),$ and also sufficient for $α=1.$ For $α\ge 2$ we show that the absence of non-real zeros for $E_{α,β}$ is sufficient for the global inequality $|E_{α,β}(z)|\ge E_{α,β}(\Re z),$ and also necessary for $α=2.$ All these results have an explicit description in terms of the values of the parameters $α,β.$ Along the way, several inequalities for $E_{α,β}$ on the half-plane $\{\Re z \ge 0\}$ are established, and a characterization of its log-convexity and log-concavity on the positive half-line is obtained.

math.CV↗

arXiv2025

Unparalleled instances of prolifickness, random walks, and square root boundaries

We revisit the problem of influencing the sex ratio of a population by subjecting reproduction of each family to some stopping rule. As an easy consequence of the strong law of large numbers, no such modification is possible in the sense that the ratio converges to 1 almost surely, for any stopping rule that is finite almost surely. We proceed to quantify the effects and provide limit distributions for the properly rescaled sex ratio. Besides the total ratio, which is predominantly considered in the pertinent literature, we also analyze the average sex ratio, which may converge to values different from 1. The first part of this note is largely expository, applying classical results and standard methods from the fluctuation theory of random walks. In the second part we apply tail asymptotics for the time at which a random walk hits a one-sided square root boundary, exhibit the differences to the corresponding two-sided problem, and use a limit law related to the empirical dispersion coefficient of a heavy-tailed distribution. Finally, we derive a large deviations result for a special stopping strategy, using saddle point asymptotics.

math.PR↗

arXiv2024

Small-time central limit theorems for stochastic Volterra integral equations and their Markovian lifts

We study small-time central limit theorems for stochastic Volterra integral equations with Hölder continuous coefficients and general locally square integrable Volterra kernels. We prove the convergence of the finite-dimensional distributions, a functional CLT, and limit theorems for smooth transformations of the process, which covers a large class of Volterra kernels that includes rough models based on Riemann-Liouville kernels with short- and long-range dependencies. To illustrate our results, we derive asymptotic pricing formulae for digital calls on the realized variance in three different regimes. The latter provides a robust and model-independent pricing method for small maturities in rough volatility models. Finally, for the case of completely monotone kernels, we introduce a flexible framework of Hilbert space-valued Markovian lifts and derive analogous limit theorems for such lifts.

math.PR↗

arXiv2023

A characterization of real matrix semigroups

We characterize all real matrix semigroups satisfying a mild boundedness assumption, without assuming continuity. Besides the continuous solutions of the semigroup functional equation, we give a description of solutions arising from non-measurable solutions of Cauchy's functional equation. To do so, we discuss the primary decomposition and the Jordan-Chevalley decomposition of a matrix semigroup. Our motivation stems from a characterization of all multi-dimensional self-similar Gaussian Markov processes, which is given in a companion paper.

math.RA↗

arXiv2023

The Fan-Taussky-Todd inequalities and the Lumer-Phillips theorem

We argue that a classical inequality due to Fan, Taussky and Todd (1955) is equivalent to the dissipativity of a Jordan block. As the latter can be characterised via the zeros of Chebyshev polynomials, we obtain a short new proof of the inequality. Three other inequalities of Fan-Taussky-Todd are reproven similarly. By the Lumer-Phillips theorem, the semigroup defined by the Jordan block is contractive. This yields new extensions of the classical Fan-Taussky-Todd inequalities. As an application, we give an estimate for the partial sums of a Bessel function.

math.CA↗

arXiv2023

Small ball probabilities and large deviations for grey Brownian motion

We show that the uniform norm of generalized grey Brownian motion over the unit interval has an analytic density, excluding the special case of fractional Brownian motion. Our main result is an asymptotic expansion for the small ball probability of generalized grey Brownian motion, which extends to other norms on path space. The decay rate is not exponential but polynomial, of degree two. For the uniform norm and the Hölder norm, we also prove a large deviations estimate.

math.PR↗

arXiv2021

A converse to the neo-classical inequality with an application to the Mittag-Leffler function

We prove two inequalities for the Mittag-Leffler function, namely that the function $\log E_α(x^α)$ is sub-additive for $0<α<1,$ and super-additive for $α>1.$ These assertions follow from two new binomial inequalities, one of which is a converse to the neo-classical inequality. The proofs use a generalization of the binomial theorem due to Hara and Hino (Bull. London Math. Soc. 2010). For $0<α<2,$ we also show that $E_α(x^α)$ is log-concave resp. log-convex, using analytic as well as probabilistic arguments.

math.CA↗

arXiv2021

Geometric properties of some generalized Mathieu power series inside the unit disk

We consider two parametric families of special functions: One is defined by a power series generalizing the classical Mathieu series, and the other one is a generalized Mathieu type power series involving factorials in its coefficients. Using criteria due to Fejer and Ozaki, we provide sufficient conditions for these functions to be close-to-convex or starlike inside the unit disk, and thus univalent.

math.CV↗

arXiv2020

Small-time, large-time and $H\to 0$ asymptotics for the Rough Heston model

We characterize the behaviour of the Rough Heston model introduced by Jaisson\&Rosenbaum \cite{JR16} in the small-time, large-time and $α\to 1/2$ (i.e. $H\to 0$) limits. We show that the short-maturity smile scales in qualitatively the same way as a general rough stochastic volatility model (cf.\ \cite{FZ17}, \cite{FGP18a} et al.), and the rate function is equal to the Fenchel-Legendre transform of a simple transformation of the solution to the same Volterra integral equation (VIE) that appears in \cite{ER19}, but with the drift and mean reversion terms removed. The solution to this VIE satisfies a space-time scaling property which means we only need to solve this equation for the moment values of $p=1$ and $p=-1$ so the rate function can be efficiently computed using an Adams scheme or a power series, and we compute a power series in the log-moneyness variable for the asymptotic implied volatility which yields tractable expressions for the implied vol skew and convexity. The limiting asymptotic smile in the large-maturity regime is obtained via a stability analysis of the fixed points of the VIE, and is the same as for the standard Heston model in \cite{FJ11}. Finally, using Lévy's convergence theorem, we show that the log stock price $X_t$ tends weakly to a non-symmetric random variable $X^{(1/2)}_t$ as $α\to 1/2$ (i.e. $H\to 0$) whose mgf is also the solution to the Rough Heston VIE with $α=1/2$, and we show that $X^{(1/2)}_t/\sqrt{t}$ tends weakly to a non-symmetric random variable as $t\to 0$, which leads to a non-flat non-symmetric asymptotic smile in the Edgeworth regime. We also show that the third moment of the log stock price tends to a finite constant as $H\to 0$ (in contrast to the Rough Bergomi model discussed in \cite{FFGS20} where the skew flattens or blows up) and the $V$ process converges on pathspace to a random tempered distribution.

q-fin.PR↗

arXiv2020

A note on large deviations in life insurance

We study large and moderate deviations for a life insurance portfolio, without assuming identically distributed losses. The crucial assumption is that losses are bounded, and that variances are bounded below. From a standard large deviations upper bound, we get an exponential bound for the probability of the average loss exceeding a threshold. A counterexample shows that a full large deviation principle does not follow from our assumptions.

math.PR↗

arXiv2020

The running maximum of the Cox-Ingersoll-Ross process with some properties of the Kummer function

We derive tail asymptotics for the running maximum of the Cox-Ingersoll-Ross process. The main result is proved by the saddle point method, where the tail estimate uses a new monotonicity property of the Kummer function. This auxiliary result is established by a computer algebra assisted proof. Moreover, we analyse the coefficients of the eigenfunction expansion of the running maximum distribution asymptotically.

math.PR↗

arXiv2020

Large deviations for fractional volatility models with non-Gaussian volatility driver

We study stochastic volatility models in which the volatility process is a function of a continuous fractional stochastic process, which is an integral transform of the solution of an SDE satisfying the Yamada-Watanabe condition. We establish a small-noise large deviation principle for the log-price, and, for a special case of our setup, obtain logarithmic call price asymptotics for large strikes.

math.PR↗