Fabian Andsem Harang

Stochastics and Partial Differential Equations: Analysis and Computations Doi: https://doi.org/10.1007/s40072-023-00295-9

We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016). To this end, we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. This concept is then used in order to prove regularization by noise for stochastic equations on the plane. The statement of regularization by noise is formulated in terms of the regularity of the local time associated to the perturbing stochastic field. For this, we provide two quantified example: a fractional Brownian sheet and the sum of two one-parameter fractional Brownian motions. As a further illustration of our regularization results, we also prove well-posedness of a 1D non-linear wave equation with a noisy boundary given by fractional Brownian motions. A discussion of open problems and further investigations is provided.

Probability theory and related fields p. 1-59 Doi: https://doi.org/10.1007/s00440-022-01145-w

Stochastics: An International Journal of Probability and Stochastic Processes 94(8) Doi: https://doi.org/10.1080/17442508.2022.2027948

In this article, we will present a new perspective on the variable-order fractional calculus, which allows for differentiation and integration to a variable order. The concept of multifractional calculus has been a scarcely studied topic within the field of functional analysis in the past 20 years. We develop a multifractional differential operator which acts as the inverse of the multifractional integral operator. This is done by solving the Abel integral equation generalized to a multifractional order. With this new multifractional differential operator, we prove a Girsanov's theorem for multifractional Brownian motions of Riemann–Liouville type. As an application, we show how Girsanov's theorem can then be applied to prove the existence of a unique strong solution to stochastic differential equations where the drift coefficient is merely of linear growth, and the driving noise is given by a non-stationary multifractional Brownian motion with a Hurst parameter as a function of time. The Hurst functions we study will take values in a bounded subset of (0,1/2) . The application of multifractional calculus to SDEs is based on a generalization of the works of D. Nualart and Y. Ouknine [Regularization of differential equations by fractional noise, Stoch Process Appl. 102(1) (2002), pp. 103–116].

The Annals of Applied Probability 32(5) p. 3930-3963 Doi: https://doi.org/10.1214/21-AAP1778

We investigate the regularizing effect of certain additive continuous perturbations on SDEs with multiplicative fractional Brownian motion (fBm). Traditionally, a Lipschitz requirement on the drift and diffusion coefficients is imposed to ensure existence and uniqueness of the SDE. We show that suitable perturbations restore existence, uniqueness and regularity of the flow for the resulting equation, even when both the drift and the diffusion coefficients are distributional, thus extending the program of regularization by noise to the case of multiplicative SDEs. Our method relies on a combination of the nonlinear Young formalism developed by Catellier and Gubinelli (Stochastic Process. Appl. 126 (2016) 2323–2366), and stochastic averaging estimates recently obtained by Hairer and Li (Ann. Probab. 48 (2020) 1826–1860).

Electronic Journal of Probability (EJP) 27 p. 1-38 Doi: https://doi.org/10.1214/22-EJP756

We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [17]. We provide several criteria for existence and uniqueness of solutions which go beyond the classical globally Lipschitz setting. In particular we show well-posedness of the equation, as well as almost sure convergence of the associated particle system, for drifts satisfying either Osgood-continuity, monotonicity, local Lipschitz or Sobolev differentiability type assumptions.

Stochastics and Dynamics Doi: https://doi.org/10.1142/S0219493723500028

We extend the recently developed rough path theory for Volterra equations from [12] to the case of more rough noise and/or more singular Volterra kernels. It was already observed in [12] that the Volterra rough path introduced there did not satisfy any geometric relation, similar to that observed in classical rough path theory. Thus, an extension of the theory to more irregular driving signals requires a deeper understanding of the specific algebraic structure arising in the Volterra rough path. Inspired by the elements of \non-geometric rough paths" developed in [11] and [14] we provide a simple description of the Volterra rough path and the controlled Volterra process in terms of rooted trees, and with this description we are able to solve rough Volterra equations driven by more irregular signals.

arXiv.org

Electronic Journal of Probability (EJP) 26 Doi: https://doi.org/10.1214/21-EJP683

We investigate the probabilistic and analytic properties of Volterra processes constructed as pathwise integrals of deterministic kernels with respect to the Hölder continuous trajectories of Hilbert-valued Gaussian processes. To this end, we extend the Volterra sewing lemma from [18] to the two dimensional case, in order to construct two dimensional operator-valued Volterra integrals of Young type. We prove that the covariance operator associated to infinite dimensional Volterra processes can be represented by such a two dimensional integral, which extends the current notion of representation for such covariance operators. We then discuss a series of applications of these results, including the construction of a rough path associated to a Volterra process driven by Gaussian noise with possibly irregular covariance structures, as well as a description of the irregular covariance structure arising from Gaussian processes time-shifted along irregular trajectories. Furthermore, we consider an infinite dimensional fractional Ornstein-Uhlenbeck process driven by Gaussian noise, which can be seen as an extension of the volatility model proposed by Rosenbaum et al.

Stochastic Processes and their Applications 142 p. 34-78 Doi: https://doi.org/10.1016/j.spa.2021.08.001

This article is devoted to the extension of the theory of rough paths in the context of Volterra equations with possibly singular kernels. We begin to describe a class of two parameter functions defined on the simplex called Volterra paths. These paths are used to construct a so-called Volterra-signature, analogously to the signature used in Lyon’s theory of rough paths. We provide a detailed algebraic and analytic description of this object. Interestingly, the Volterra signature does not have a multiplicative property similar to the classical signature, and we introduce an integral product behaving like a convolution extending the classical tensor product. We show that this convolution product is well defined for a large class of Volterra paths, and we provide an analogue of the extension theorem from the theory of rough paths (which guarantees in particular the existence of a Volterra signature). Moreover the concept of convolution product is essential in the construction of Volterra controlled paths, which is the natural class of processes to be integrated with respect to the driving noise in our situation. This leads to a rough integral given as a functional of the Volterra signature and the Volterra controlled paths, combined through the convolution product. The rough integral is then used in the construction of unique solutions to Volterra equations driven by Hölder noises with singular kernels. An example concerning Brownian noises and a singular kernel is treated.

Journal of theoretical probability Doi: https://doi.org/10.1007/s10959-021-01114-4

We investigate the space-time regularity of the local time associated with Volterra–Lévy processes, including Volterra processes driven by α-stable processes for α∈(0,2]. We show that the spatial regularity of the local time for Volterra–Lévy process is P-a.s. inverse proportional to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturbation of ordinary differential equations by a Volterra–Lévy process which has sufficiently regular local time. Following along the lines of Harang and Perkowski (2020), we show existence, uniqueness and differentiability of the flow associated with such equations.

Journal of Applied Probability 58(1) p. 22-41 Doi: https://doi.org/10.1017/jpr.2020.88

We propose a new multifractional stochastic process which allows for self-exciting behavior, similar to what can be seen for example in earthquakes and other self-organizing phenomena. The process can be seen as an extension of a multifractional Brownian motion, where the Hurst function is dependent on the past of the process. We define this by means of a stochastic Volterra equation, and we prove existence and uniqueness of this equation, as well as giving bounds on the p-order moments, for all p ≥ 1. We show convergence of an Euler–Maruyama scheme for the process, and also give the rate of convergence, which is dependent on the self-exciting dynamics of the process. Moreover, we discuss various applications of this process, and give examples of different functions to model self-exciting behavior.

SIAM Journal on Financial Mathematics 12(3) p. 1257-1284 Doi: https://doi.org/10.1137/20M135902X

We propose a new class of rough stochastic volatility models obtained by modulating the power-law kernel defining the fractional Brownian motion (fBm) by a logarithmic term, such that the kernel retains square integrability even in the limit case of vanishing Hurst index $H$. The so-obtained log-modulated fractional Brownian motion (log-fBm) is a continuous Gaussian process even for $H = 0$. As a consequence, the resulting super-rough stochastic volatility models can be analyzed over the whole range $0 \le H < 1/2$ without the need of further normalization. We obtain skew asymptotics of the form $\log(1/T)^{-p} T^{H-1/2}$ as $T\to 0$, $H \ge 0$, so no flattening of the skew occurs as $H \to 0$.

Stochastics and Dynamics 21(8) Doi: https://doi.org/10.1142/S0219493721400104

We study ordinary differential equations (ODEs) with vector fields given by general Schwartz distributions, and we show that if we perturb such an equation by adding an “infinitely regularizing” path, then it has a unique solution and it induces an infinitely smooth flow of diffeomorphisms. We also introduce a criterion under which the sample paths of a Gaussian process are infinitely regularizing, and we present two processes which satisfy our criterion. The results are based on the path-wise space–time regularity properties of local times, and solutions are constructed using the approach of Catellier–Gubinelli based on nonlinear Young integrals.

Communications in Mathematical Sciences 18(7) p. 1863-1890 Doi: https://doi.org/10.4310/CMS.2020.v18.n7.a3

Stochastic Processes and their Applications 159 p. 499-540 Doi: https://doi.org/10.1016/j.spa.2023.02.005

Stochastics and Partial Differential Equations: Analysis and Computations Doi: https://doi.org/10.1007/s40072-020-00184-5

Stochastics: An International Journal of Probability and Stochastic Processes Doi: https://doi.org/10.1080/17442508.2019.1602130

arXiv.org

Young researchers between geometry and stochastic analysis

Seminar for Probability group at Université Nice-Sophia-Antipolis

Oxford Stochastic Analysis and Mathematical Finance Seminar

DNA seminar, Mathematics department, NTNU Trondheim

Seminar For Berlin Rough Paths Group

Conference in Stochastic Analysis and Applications

Abstract. We propose a new multifractional stochastic process which allows for self-exciting behavior, similar to what can be seen for example in earthquakes and other self-organizing phenomena. The process can be seen as an extension of a multifractional Brownian motion, where the Hurst function is dependent on the past of the process. We define this through a stochastic Volterra equation, and we prove existence and uniqueness of this equation, as well as give bounds on the p-order moments, for all p>=1. We show convergence of an Euler-Maruyama scheme for the process, and also give the rate of convergence, which is depending on the self-exciting dynamics of the process. Moreover, we discuss different applications of this process, and give examples of different functions to model self-exciting behavior.

CSA2019 - Conference in Stochastic Analysis - Risør

Berlin -Oxford meeting, WIAS Berlin

Seminar at Purdue University, Indiana, USA

Seminar at NTNU Trondhiem