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 s. 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) s. 3930-3963 Doi: https://doi.org/10.1214/21-AAP1778

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

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

arXiv.org

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

Stochastic Processes and their Applications 142 s. 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) s. 22-41 Doi: https://doi.org/10.1017/jpr.2020.88

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

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

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

Stochastic Processes and their Applications 159 s. 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