arXiv subjects

Martin Friesen

Publications and source records attributed to Martin Friesen.

At least 19 recordsLinked to original sources

arXiv2026

Pricing and Hedging of Discretely Monitored Asian Options in the Volterra-Heston Model

We develop semi-closed pricing formulas and lifted-model hedging methods for discretely monitored geometric and arithmetic Asian options in the Volterra-Heston stochastic volatility model. Exploiting the affine Volterra structure, we derive a tractable transform for the joint law of the terminal log-price and the discretely monitored geometric average. This transform yields semi-closed pricing formulas for geometric Asian options, which in turn provide effective control variates for Monte Carlo valuation of arithmetic Asian options. Under the stated real-moment and affine-transform hypotheses, we also derive the Galtchouk-Kunita-Watanabe decomposition for Fourier-representable payoffs and obtain a variance-optimal hedge in terms of the Riccati-Volterra equation and the forward-variance curve. Using N-factor Markovian approximations, we obtain a finite-dimensional numerical implementation for hedging Asian options. Our numerical experiments document factor convergence for a regular non-Markovian kernel and the effect of rebalancing frequency on hedging error. In the Heston benchmark, geometric Asian controls substantially reduce the variance of arithmetic-Asian price estimates and improve the finite-sample stability of regression-based hedging relative to direct regression.

q-fin.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↗

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 H\"older-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

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 H\"older 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↗

arXiv2025

Limit theorems for stochastic Volterra processes

We introduce an abstract Hilbert space-valued framework of Markovian lifts for stochastic Volterra equations with operator-valued Volterra kernels. Our main results address the existence and characterisation of possibly multiple limit distributions and stationary processes, a law of large numbers including a convergence rate, and the central limit theorem for time averages of the process within the Gaussian domain of attraction. As particular examples, we study Markovian lifts based on Laplace transforms in a weighted Hilbert space of densities and Markovian lifts based on the shift semigroup on the Filipovi\'c space. We illustrate our results for the case of fractional stochastic Volterra equations with additive or multiplicative Gaussian noise.

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 H\"older 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↗

arXiv2024

Ergodicity and Law-of-large numbers for the Volterra Cox-Ingersoll-Ross process

We study the Volterra Volterra Cox-Ingersoll-Ross process on $\mathbb{R}_+$ and its stationary version. Based on a fine asymptotic analysis of the corresponding Volterra Riccati equation combined with the affine transformation formula, we first show that the finite-dimensional distributions of this process are asymptotically independent. Afterwards, we prove a law-of-large numbers in $L^p$(\Omega)$ with $p \geq 2$ and show that the stationary process is ergodic. As an application, we prove the consistency of the method of moments and study the maximum-likelihood estimation for continuous and discrete high-frequency observations.

math.PR↗

arXiv2024

Regular occupation measures of Volterra processes

We introduce a local non-determinism condition for Volterra It\^{o} processes that captures smoothing properties of possibly degenerate noise. By combining the stochastic sewing lemma with one-step Euler approximations, we first prove the joint space-time regularity for their occupation measure, self-intersection measure, and time marginals for such Volterra It\^{o} processes. As an application, we obtain the space-time regularity of local times and self-intersection times for rough perturbations of Gaussian Volterra processes, and construct a class of non-Gaussian Volterra I\^{o} processes that are $C^{\infty}$-regularising. Secondly, for the particular class of stochastic Volterra equations with H\"older continuous coefficients, using disintegration of measures for their Markovian lifts, we further establish the absolute continuity of finite-dimensional distributions. Finally, we prove the existence, uniqueness, and stability for self-interacting stochastic equations with distributional drifts.

math.PR↗

arXiv2024

Maximum likelihood estimation in the ergodic Volterra Ornstein-Uhlenbeck process

We study statistical inference of the drift parameters for the Volterra Ornstein-Uhlenbeck process on R in the ergodic regime. For continuous-time observations, we derive the corresponding maximum likelihood estimators and show that they are strongly consistent and asymptotically normal locally uniformly in the parameters. For the case of discrete high-frequency observations, we prove similar results by discretization of the continuous-time maximum likelihood estimator. Finally, for discrete low-frequency observations, we show that the method of moments is consistent. Our proofs are crucially based on the law of large numbers.

math.ST↗

arXiv2024

Comparison principles for stochastic Volterra equations

In this work, we establish a comparison principle for stochastic Volterra equations with respect to the initial condition and the drift $b$ applicable to a wide class of Volterra kernels and input curves $g$. Such input curves are allowed to be singular in zero, and appear, e.g., in Markovian lifts for Volterra equations. For completely monotone kernels, our result holds without any further restrictions, while for regular kernels we give a characterisation of the comparison principle. Finally, we show that for not completely monotone kernels such a principle fails unless the drift is monotone. As a side-product of our results, we also complement the literature on the weak existence of continuous nonnegative solutions, which covers the rough Cox-Ingersoll-Ross process with singular initial conditions.

math.PR↗

arXiv2023

Limits of stochastic Volterra equations driven by Gaussian noise

We study stochastic Volterra equations in Hilbert spaces driven by cylindrical Gaussian noise. We derive a mild formulation for the stochastic Volterra equation, prove the equivalence of mild and strong solutions, the existence and uniqueness of mild solutions, and study space-time regularity. Furthermore, we establish the stability of mild solutions in $L^q(\R_+)$, prove the existence of limit distributions in the Wasserstein $p$-distance with $p \in [1,\infty)$, and characterise when these limit distributions are independent of the initial state of the process despite the presence of memory. While our techniques allow for a general class of Volterra kernels, they are particularly suited for completely monotone kernels and fractional Riemann-Liouville kernels in the full range $\alpha \in (0,2)$.

math.PR↗

arXiv2023

Interacting particle systems with continuous spins

We study a general class of interacting particle systems over a countable state space $V$ where on each site $x \in V$ the particle mass $\eta(x) \geq 0$ follows a stochastic differential equation. We construct the corresponding Markovian dynamics in terms of strong solutions to an infinite coupled system of stochastic differential equations and prove a comparison principle with respect to the initial configuration as well as the drift of the process. Using this comparison principle, we provide sufficient conditions for the existence and uniqueness of an invariant measure in the subcritical regime and prove convergence of the transition probabilities in the Wasserstein-1-distance. Finally, for sublinear drifts, we establish a linear growth theorem showing that the spatial spread is at most linear in time. Our results cover a large class of finite and infinite branching particle systems with interactions among different sites.

math.PR↗

arXiv2022

Limit theorems for time averages of continuous-state branching processes with immigration

In this work we investigate limit theorems for the time-averaged process $\left(\frac{1}{t}\int_0^t X_s^x ds\right)_{t\geq 0}$ where $X^x$ is a subcritical continuous-state branching processes with immigration (CBI processes) starting in $x \geq 0$. Under a second moment condition on the branching and immigration measures we first prove the law of large numbers in $L^2$ and afterward establish the central limit theorem. Assuming additionally that the big jumps of the branching and immigration measures have finite exponential moments of some order, we prove in our main result the large deviation principle and provide a semi-explicit expression for the good rate function in terms of the branching and immigration mechanisms. Our methods are deeply based on a detailed study of the corresponding generalized Riccati equation and related exponential moments of the time-averaged process.

math.PR↗

arXiv2022

Volterra square-root process: Stationarity and regularity of the law

The Volterra square-root process on $\mathbb{R}_+^m$ is an affine Volterra process with continuous sample paths. Under a suitable integrability condition on the resolvent of the second kind associated with the Volterra convolution kernel, we establish the existence of limiting distributions. In contrast to the classical square-root diffusion process, here the limiting distributions may depend on the initial state of the process. Our result shows that the non-uniqueness of limiting distributions is closely related to the integrability of the Volterra convolution kernel. Using an extension of the exponential-affine transformation formula we also give the construction of stationary processes associated with the limiting distributions. Finally, we prove that the time marginals as well as the limiting distributions, when restricted to the interior of the state space $\mathbb{R}_{+}^m$, are absolutely continuous with respect to the Lebesgue measure and their densities belong to some weighted Besov space of type $B_{1,\infty}^λ$.

math.PR↗

arXiv2022

Stationary Covariance Regime for Affine Stochastic Covariance Models in Hilbert Spaces

We study the long-time behavior of affine processes on positive self-adjoiont Hilbert-Schmidt operators which are of pure-jump type, conservative and have finite second moment. For subcritical processes we prove the existence of a unique limit distribution and construct the corresponding stationary affine process. Moreover, we obtain an explicit convergence rate of the underlying transition kernels to the limit distribution in the Wasserstein distance of order $p\in [1, 2]$ and provide explicit formulas for the first two moments of the limit distribution. We apply our results to the study of infinite-dimensional affine stochastic covariance models in the stationary covariance regime, where the stationary affine process models the instantaneous covariance process. In this context we investigate the behavior of the implied forward volatility smile for large forward dates in a geometric affine forward curve model used for the modeling of forward curve dynamics in fixed income or commodity markets formulated in the Heath-Jarrow-Morton-Musiela framework.

math.PR↗

arXiv2021

Spontaneous wave formation in stochastic self-driven particle systems

Waves and oscillations are commonly observed in the dynamics of self-driven agents such as pedestrians or vehicles. Interestingly, many factors may perturb the stability of space homogeneous streaming, leading to the spontaneous formation of collective oscillations of the agents related to stop-and-go waves, jamiton, or phantom jam in the literature. In this article, we demonstrate that even a minimal additive stochastic noise in stable first-order dynamics can initiate stop-and-go phenomena. The noise is not a classic white one, but a colored noise described by a Gaussian Ornstein-Uhlenbeck process. It turns out that the joint dynamics of particles and noises forms again a (Gaussian) Ornstein-Uhlenbeck process whose characteristics can be explicitly expressed in terms of parameters of the model. We analyze its stability and characterize the presence of waves through oscillation patterns in the correlation and autocorrelation of the distance spacing between the particles. We determine exact solutions for the correlation functions for the finite system with periodic boundaries and in the continuum limit when the system size is infinite. Finally, we compare experimental trajectories of single-file pedestrian motions to simulation results.

physics.soc-ph↗

arXiv2020

Regularity of transition densities and ergodicity for affine jump-diffusion processes

In this paper we study the transition density and exponential ergodicity in total variation for an affine process on the canonical state space $\mathbb{R}_{\geq0}^{m}\times\mathbb{R}^{n}$. Under a Hörmander-type condition for diffusion components as well as a boundary non-attainment assumption, we derive the existence and regularity of the transition density for the affine process and then prove the strong Feller property. Moreover, we also show that under these and the additional subcritical conditions the corresponding affine process on the canonical state space is exponentially ergodic in the total variation distance. To prove existence and regularity of the transition density we derive some precise estimates for the real part of the characteristic function of the process. Our ergodicity result is a consequence of a suitable application of a Harris-type theorem based on a local Dobrushin condition combined with the regularity of the transition densities.

math.PR↗

arXiv2020

On a class of stochastic partial differential equations with multiple invariant measures

In this work we investigate the long-time behavior, that is the existence and characterization of invariant measures as well as convergence of transition probabilities, for Markov processes obtained as the unique mild solution to stochastic partial differential equations in a Hilbert space. Contrary to the existing literature where typically uniqueness of invariant measures is studied, we focus on the case where the uniqueness of invariant measures fails to hold. Namely, using a \textit{generalized dissipativity condition} combined with a decomposition of the Hilbert space, we prove the existence of multiple limiting distributions in dependence of the initial state of the process and study the convergence of transition probabilities in the Wasserstein 2-distance. Finally, we show that these results contain Lévy driven Ornstein-Uhlenbeck processes, the Heath-Jarrow-Morton-Musiela equation as well as stochastic partial differential equations with delay as a particular case.

math.PR↗