Organized by MAM (Mathematics and Applied Mathematics research environment) https://mammath.wordpress.com/ External link.

Division of Mathematics and Physics, The School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden, with financial support from Nordplus project “Network FinEng2024”, project ID NPHE-2024/10408

Organizing committee: Associate Professor Ying Ni, Professor Sergei Silvestrov

Program

Each presentation comprises a lecture of approximately 40 minutes, followed by around 5 minutes allocated to questions and discussion. The precise timing is indicated in the program schedule below.

September 15

10:00-10:15 Registration and welcome from organizers, room U2-129

Session 1: Stochastic Processes and Financial Engineering

Chair: Assoc. Prof. Ying Ni

10:15-11:00 Assoc. Prof. Achref Bachouch, Mälardalen University, Sweden

Numerical probabilistic method for Semi-linear Stochastic PDEs using Backward Doubly SDEs

Abstract. In the first part of this talk, I will present a numerical probabilistic method to approximate the solution of a class of semi-linear stochastic partial differential equations (SPDEs in short). Pardoux and Peng [5] related Semi-linear SPDEs to backward doubly stochastic differential equations (BDSDEs in short). Our numerical scheme is based on discrete time approximation for solutions of systems of decoupled Forward-BDSDEs, generalizing numerical schemes for standard backward stochastic differential equations studied in [3] and [6]. Under standard assumptions on the parameters, we prove the convergence in time and the rate of convergence of our numerical scheme.

In the second part of the talk, I will present the resolution of the discrete dynamic programming equation arising from the time discretization by using nested empirical regression problems, following [4]. Error estimates are derived conditionally to the exterior noise.

The first part of this talk is based on [1] and the second part is based on [2].

[1] A. Bachouch, M.A. Ben Lasmer, A. Matoussi, and M. Mnif. Euler time discretization of Backward Doubly SDEs and application to Semilinear SPDEs. Stochastics and Partial Differential Equations: Analysis and Computations, 4(3), 592-634, 2016.

[2] A. Bachouch, E. Gobet, and A. Matoussi. Empirical regression method for Backward Doubly Stochastic Differential Equations. SIAM/ASA J. Uncertainty Quantification, 4(1), 358-379, 2016.

[3] B. Bouchard and N. Touzi. Discrete time approximation and Monte-Carlo Simulation of Backward Stochastic differential equations. Stochastic Processes and Their applications, 111, 175-206, 2004.

[4] [4] E. Gobet and P. Turkedjiev. Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions. Mathematics of Computation, 85(299), 1359-1391, 2016.

[5] E. Pardoux and S.G. Peng. Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probability Theory and Related Fields, 98(2), 209–227, 1994.

[6] J. Zhang. A numerical scheme for BSDEs. The Annals of Applied Probability,14(1), 459–488, 2004

11:00- 11:35 Doctoral student Mara Kalicanin Dimitrov, Mälardalen University, Sweden

Mixed LSMC-PDE method applied to Gatheral double mean reverting model

Abstract. This talk is based on a joint working paper that we are currently developing. We study Bermudan option pricing under the Gatheral double mean-reverting (DMR) stochastic volatility model. Our focus is on the mixed least-squares Monte Carlo–partial differential equation (LSMC–PDE) method, introduced by Farahany, Jackson, and Jaimungal (2020) for Bermudan options with stochastic volatility.

The DMR stochastic volatility model is characterized by two coupled variance factors, with one mean-reverting to a stochastic variance level. This structure provides greater flexibility than the classical Heston stochastic volatility model, while also introducing additional analytical challenges. Since the DMR model belongs to the separable class of one-way coupled stochastic volatility models, the hybrid LSMC–PDE algorithm and its almost-sure convergence results apply. In this study, we propose two variants that adapt the LSMC–PDE method to the DMR framework: a standard adaptation, and an enhanced version that employs the mixed almost-exact scheme developed in earlier work.

11:35-12:20 Assoc. Prof. Ying Ni, Mälardalen University, Sweden

Financial Option Pricing and Hedging using Modern Machine Learning Tools

Abstract. In this talk, I will briefly review the fundamentals of option pricing and hedging and present results from two collaborative papers with colleagues at the Norwegian University of Science and Technology.

The first paper addresses option pricing, specifically the pricing of Bermudan basket options with maximum and minimum payoffs on many underlying assets. The novelty of our work lies in implementing neural network architecture tailored for high-dimensional max- and min-basket options under the high-dimensional Heston stochastic volatility model — a setting that has received little attention so far. As a by-product, we also propose a simple synthetic regression method and compare its performance to that of the neural network approach.

The second paper focuses on hedging options, where we introduce a novel framework called X-hedging. While deep neural networks have proven effective in capturing complex nonlinear relationships, their lack of explainability, as famous black box models, remains a significant drawback. Our X-hedging framework uses gradient boosted decision trees (GBDTs), specifically LightGBM models, to achieve high hedging performance alongside enhanced model transparency. Experimental results demonstrate that X-hedging not only outperforms the current state-of-the-art deep neural network approaches but also provides improved user-centered explainability. X-hedging framework is flexible and can be readily adapted to other market models incorporating different types of market frictions.

12:30- 14:00 Refreshments and scientific discussion in UKK coffee room

Session 2. Stochastic Processes and Financial Engineering

Chair: Assoc. Prof. Ying Ni

14:00- 14:50 Prof. em. Dmitrii Silvestrov, Stockholm University, Mälardalen University (Sweden)

Limit and Ergodic Theorems for Perturbed Semi-Markov-Type Processes

Abstract. The lecture aims to present and comment on the results of the recent books on perturbed semi-Markov-type processes and their applications:

[1] Silvestrov, D. (2025). Coupling and Ergodic Theorems for Semi-Markov-Type Processes II: Semi-Markov Processes and Multi-Alternating Regenerative Processes with Semi-Markov Modulation. Springer, Cham, xix+606 pp.

[2] Silvestrov, D. (2025). Coupling and Ergodic Theorems for Semi-Markov-Type Processes I: Markov Chains, Renewal and Regenerative Processes. Springer, Cham, xix+611 pp.

[3] Silvestrov, D. (2022). Perturbed Semi-Markov Type Processes II: Ergodic Theorems for Multi-Alternating Regenerative Processes. Springer, Cham, xvii+413 pp.

[4] Silvestrov, D. (2022). Perturbed Semi-Markov Type Processes I: Limit Theorems for Rare-Event Times and Processes. Springer, Cham, xvii+401 pp.

[5] Silvestrov, D., Silvestrov, S. (2017). Nonlinearly Perturbed Semi-Markov Processes. Springer Briefs in Probability and Mathematical Statistics, Springer, Cham, xiv+143 pp.

Session 3. Numerical Methods and Engineering Mathematics

Chair: Prof. Sergei Silvestrov

14:50-15:40 Dr. Lars Hellström, Mälardalen University

The Lanczos Moments of a Random Variable

Abstract. The Lanczos algorithm in linear algebra involves computing a sequence of scalar quantities that become elements of a tridiagonal matrix. These scalars may also be interpreted as moments of a probability distribution (or more general measure), which provides an alternative characterisation of that distribution. In particular, it is straightforward to construct finite discrete approximations from these moments. Moreover, there exists an associated sequence of orthogonal polynomials which may be constructed as characteristic polynomials of truncations of the tridiagonal matrices; the standard families of orthogonal polynomials turn out to be such sequences for certain standard probability distributions.

From the descriptive statistics point of view, it may be mentioned that the first and second Lanczos moments natively become the mean and standard deviation, whereas the third and fourth provide interpretations for skewness and kurtosis. An interesting feature of the Lanczos moments is that they are all in the same unit as the underlying random variable and transform affinely under affine transformations of that variable. The sequence of Lanczos moments is bounded if and only if the random variable is bounded.

15:40 -16:00 Coffee break

16:00 -16:50 Prof. Sergei Korotov, Mälardalen University, Sweden

Monotone matrices and discrete maximum principles

Abstract. In this talk we will demonstrate how the results on monotone matrices can be used to guarantee the validity of the so-called discrete maximum principles in finite element simulations. Several associated geometric problems on the finite element meshes will be discussed as well.

16:50-17:30 Dr. Jean-Paul Murara, Mälardalen University,Sweden

Stochastic Optimal Control Problem with a Multiscale Volatility Model

Abstract. In this talk, using generalized reference probability space, we start by giving an introduction of a strong and a weak formulation of an optimal control problem. We extend the general problem to a particular stochastic optimal control problem. After, we investigate the application of the above in the case of a portfolio optimization problem that follows a multi-scale stochastic volatility model.

September 16

Room U2-129 (afternoon)

Session 4: Algebraic Structures

Chair: Prof. Sergei Silvestrov

13.00-13.50 Assoc. Prof. Olga Liivapuu, Estonian University of Life Sciences, Estonia

Odd and even derivations, transposed Poisson superalgebra and 3-Lie superalgebra

Abstract. One important example of a transposed Poisson algebra can be constructed by means of a commutative algebra and its derivation. This approach can be extended to superalgebras, that is, one can construct a transposed Poisson superalgebra given a commutative superalgebra and its even derivation. In this paper we show that including odd derivations in the framework of this approach requires introducing a new notion. It is a super vector space with two operations that satisfy the compatibility condition of transposed Poisson superalgebra. The first operation is determined by a left supermodule over commutative superalgebra and the second is a Jordan bracket. Then it is proved that the super vector space generated by an odd derivation of a commutative superalgebra satisfies all the requirements of introduced notion. We also show how to construct a 3-Lie superalgebra if we are given a transposed Poisson superalgebra and its even derivation.

13.50-14.40 Prof. Viktor Abramov, University of Tartu, Estonia

Ternary Lie algebras at cube roots of unity

Abstract. We extend the concepts of the associator and commutator from algebras with a binary multiplication law to algebras with a ternary multiplication law using cube roots of unity. By analogy with the Jacobi identity for the binary commutator, we derive an identity for the proposed ternary commutator. While the Jacobi identity is based on the cyclic permutation group of three elements Z_3, the identity we establish for the ternary commutator is based on the general affine group GA(1,5). We introduce the notion of a ternary Lie algebra at cube roots of unity. A broad class of such algebras is constructed using associative ternary multiplications of rectangular and cubic matrices.

14:40-15.30 Doctoral student Anti Maria Aader, University of Tartu, Estonia

On the classification of ternary Lie algebras at cube roots of unity

Abstract. We study the classification of low-dimensional ternary Lie algebras at cube roots of unity by means of structure constants and their tensorial transformation under a change of basis. We propose the complete classification of non-isomorphic 2-dimensional ternary Lie algebras at cube roots of unity and show that two of them are simple algebras. We also consider the case of 3-dimensional ternary Lie algebras and show that this case is more complicated compared to the 2-dimensional case. We show how we have studied its structure using Python.

15:30-16:00 Coffee break

16:00-16:40 Doctoral student German Garcia, Mälardalen University, Sweden

Hom-associative structures in one-sided unital algebras

Abstract. It is an inherently interesting question whether a given algebra can be seen as an algebra of a different type. This talk explores whether it is possible to see one-sided unital algebras as hom-associative. I will discuss several interactions between both axioms and the twisting map and prove that maps yielding hom-associativity into the algebra are multiplications by elements of a certain subalgebra. An important discussion in hom-algebra theory is multiplicativity of the twisting map, which I will prove to be tied to the idempotents of the algebra.

16:40-17:20 Doctoral student Stephen Mboya, School of Mathematics, University of Nairobi, Kenya

Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras of generalized sl(2)-type

Abstract. In this talk, we give some algebraic properties of skew-symmetric ternary Hom-Nambu algebras. We construct ternary Hom-Nambu-Lie algebras from Hom-Lie algebras of generalized sl(2)-type using an induction procedure. We present isomorphism criteria for this class of ternary Hom-Nambu-Lie algebras in terms of the Hom-Lie algebras inducing them. We also discuss the notion of solvability and nilpotency for this class of skew-symmetric algebras. Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras of generalized sl(2)-type.

September 17

Scientific discussions and FinEng2025 project planningx