I am a Professor of Mathematics at BI Norwegian Business School, where I lead the AMOR – Center for Applied Mathematics and Operations Research. My research lies at the intersection of stochastic analysis, dynamical systems, functional analysis, and machine learning, with a strong focus on applications in operations research, economics, and finance.
Currently, my work focuses on mathematical and statistical problems in operations research related to the retail industry. Through the "Smart Matflyt" (Smart Food Flow)project, a part of AMOR Retail, I lead an interdisciplinary initiative to develop advanced models for forecasting, optimal pricing, and optimizing supply chains and logistics in the food retail sector.
Previously, I co-led the Signatures for Images project together with Prof. Kurusch Ebrahimi Fard at the Centre for Advanced Study (CAS) in Oslo, an international initiative aimed at advancing the mathematical understanding of image-type data through iterated integral signatures. This project brought together experts in stochastic analysis, combinatoric algebra, geometry, and applied mathematics to develop novel frameworks for studying complex visual and structured data.
At BI, I work to advance interdisciplinary research and the role of applied mathematics in solving real-world challenges. As the director of AMOR, I am particularly interested in understanding research challenges faced by industry and exploring how mathematical modeling, optimization, and AI-driven decision-making can help address them.
Beyond research, I teach courses in the quantitative disciplines at BI. I am also interested in supervising master’s students, so please feel free to contact me if you are interested in writing a thesis with me.
See my personal homepage for more information.
Over the past decade, the importance of the 1D signature which can be seen as a functional defined over a path, has been pivotal in both path-wise stochastic calculus and the analysis of time series data. By considering an image as a two-parameter function that takes values in a -dimensional space, we introduce an extension of the path signature to images. We address numerous challenges associated with this extension and demonstrate that the 2D signature satisfies a version of Chen’s relation in addition to a shuffle-type product. Furthermore, we show that specific variations of the 2D signature can be recursively defined, thereby satisfying an integral-type equation. We analyze the properties of the proposed signature, such as continuity, invariance to stretching, translation and rotation of the underlying image. Additionally, we establish that the proposed 2D signature over an image satisfies a universal approximation property.
Bechtold, Florian & Harang, Fabian Andsem (2025)
Pathwise regularization by noise for semilinear SPDEs driven by a multiplicative cylindrical Brownian motion
We explore the implications of a preference ordering for an investor-consumer with a strong preference for keeping consumption above an exogenous social norm, but who is willing to tolerate occasional dips below it. We do this by splicing two CRRA preference orderings, one with high curvature below the norm and the other with low curvature at or above it. We find this formulation appealing for many endowment funds and sovereign wealth funds, including the Norwegian Government Pension Fund Global, which inspired our research. We derive an analytical solution, which we use to describe key properties of the policy functions for consumption and portfolio allocation. We find that annual spending should not only be significantly lower than the expected financial return, but mostly also procyclical. In particular, financial losses should, as a rule, be followed by larger than proportional spending cuts, except when some smoothing is needed to keep spending from falling too far below the social norm. Yet, at very low wealth levels, spending should be kept particularly low in order to build sufficient wealth to raise consumption above the social norm. Financial risk taking should also be modest and procyclical, so that the investor sometimes may want to “buy at the top” and “sell at the bottom.” Many of these features are shared by habit-formation models and other models with some lower bound for consumption. However, our specification is more flexible and thus more easily adaptable to actual fund management. The nonlinearity of the policy functions may present challenges regarding delegation to professional managers. However, simpler rules of thumb with constant or slowly moving equity share and consumption-wealth ratio can reach almost the same expected discounted utility. Nevertheless, the constant levels will then look very different from the implications of expected CRRA utility or Epstein–Zin preferences in that consumption is much lower.
Based on the recent development of the framework of Volterra rough paths (Harang
and Tindel in Stoch Process Appl 142:34–78, 2021), we consider here the probabilistic
construction of the Volterra rough path associated to the fractional Brownian motion
with H > 1
2 and for the standard Brownian motion. The Volterra kernel k(t,s) is
allowed to be singular, and behaving similar to |t − s|
−γ for some γ ≥ 0. The
construction is done in both the Stratonovich and Itô senses. It is based on a modified
Garsia–Rodemich–Romsey lemma which is of interest in its own right, as well as
tools from Malliavin calculus. A discussion of challenges and potential extensions is
provided.
Catellier, Rémi & Harang, Fabian Andsem (2023)
Pathwise regularization of the stochastic heat equation with multiplicative noise through irregular perturbation
Existence and uniqueness of solutions to the stochastic heat equation with multiplicative spatial noise is studied. In the spirit of pathwise regularization by noise, we show that a perturbation by a sufficiently irregular continuous path establish wellposedness of such equations, even when the drift and diffusion coefficients are given as generalized functions or distributions. In addition we prove regularity of the averaged field associated to a Lévy fractional stable motion, and use this as an example of a perturbation regularizing the multiplicative stochastic heat equation.
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.
Galeati, Lucio; Harang, Fabian Andsem & Mayorcas, Avi (2022)
Distribution dependent SDEs driven by additive fractional Brownian motion
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].
Galeati, Lucio & Harang, Fabian Andsem (2022)
Regularization of multiplicative SDEs through additive noise
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 & Ling, Chengcheng (2021)
Regularity of Local Times Associated with Volterra–Lévy Processes and Path-Wise Regularization of Stochastic Differential Equations
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.
Harang, Fabian Andsem; Lagunas, Marc & Ortiz-Latorre, Salvador (2021)
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.
