I am working as a professor in mathematics at BI Norwegian Business School. I hold a Ph.D in Mathematics from the University of Oslo, where I specialized in stochastic analysis and financial mathematics.
My research is mostly focused around various topics in the field of stochastic analysis, rough path theory, and applications of these theories towards economic modelling and financial markets. I am currently very interested in mean field equations and mean field games, and applications in modelling of economic growth. I am also actively working on projects related to Volterra rough paths, rough volatiltiy modelling, and generally the application of stochastic processes with memory in finance.
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.
Harang, Fabian Andsem & Benth, Fred Espen (2021)
Infinite Dimensional Pathwise Volterra Processes Driven by Gaussian Noise - Probabilistic Properties and Applications