arXiv research

Develops information geometry for Lévy processes in finance.

We study CR geometry in arbitrary codimension, and introduce a process, which we call the Levi-Kahler quotient, for constructing Kahler metrics from CR structures with a transverse torus action. Most of the paper is devoted to the study of Levi-Kahler quotients of toric CR manifolds, and in particular, products of odd …

Study prolongations of nilpotent Lie algebras with specific structural subalgebras.

Develops new approach to recover CR structures from their Levi foliations.

Equations of motion of low-energy string effective actions can be conveniently described in terms of generalized geometry and Levi-Civita connections on Courant algebroids. This approach is used to propose and prove a suitable version of the Kaluza-Klein-like reduction. Necessary geometrical tools are recalled.

Enhances option pricing for American-style options using JDOI method.

We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Le…

We discuss Levi-Civita connections on Courant algebroids. We define an appropriate generalization of the curvature tensor and compute the corresponding scalar curvatures in the exact and heterotic case, leading to generalized (bosonic) Einstein-Hilbert type of actions known from supergravity. In particular, we carefull…

A Monte Carlo method for pairs trading on mean-reverting spreads with Lévy processes.

Neural SDEs reduce variance in stochastic simulations.

Let $\mathcal M= (M,\mathcal O_\mathcal M)$ be a smooth supermanifold with connection $\nabla$ and Batchelor model $\mathcal O_\mathcal M\congΓ_{ΛE^\ast}$. From $(\mathcal M,\nabla)$ we construct a connection on the total space of the vector bundle $E\to{M}$. This reduction of $\nabla$ is well-defined independently of …

Generative model for Lévy area improves SDE simulation accuracy.

We develop generic and efficient importance sampling estimators for Monte Carlo evaluation of prices of single- and multi-asset European and path-dependent options in asset price models driven by Lévy processes, extending earlier works which focused on the Black-Scholes and continuous stochastic volatility models. Usin…

Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have a rich geometry determined by their parabolic subgroups and subalgebras, which carry the structure of a building in the sense of J. Tits. We present herein an elementary approach to the geometry of parabolic subalgebras, over an arbitrary fiel…

The paper constructs a star product on a symplectically reduced phase space for a lattice gauge model.

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kähler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connectio…

This paper establishes a correspondence between principal bundles and invariant connections.

This work extends the variance reduction method for the pricing of possibly path-dependent derivatives, which was developed in (Genin and Tankov, 2016) for exponential Lévy models, to affine stochastic volatility models (Keller-Ressel, 2011). We begin by proving a pathwise large deviations principle for affine stochast…

Neural Lévy model improves risk and density forecasting for financial returns.

Study Higgs bundles and their reductions to prove stability and cohomology properties.

Study reveals finite-size effects and sensitivity to random numbers in Levy-Levy-Solomon model.

These lectures notes aim at introducing Lévy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of Lévy processes. We analyze a `toy' example of a Lévy process, viz. a Lévy jump-diffusion, which yet offers significant insight into the distri…

Given a reductive homogeneous space M=G/H endowed with a naturally reductive metric, we study the one-parameter family of connections joining the canonical and the Levi-Civita connection (t=0, 1/2). We show that the Dirac operator D^t corresponding to t=1/3 is the so-called ``cubic'' Dirac operator recently introduced …

Let $(X, T^{1,0}X)$ be a compact connected orientable CR manifold of dimension $2n+1$ with non-degenerate Levi curvature. Assume that $X$ admits a connected compact Lie group action $G$. Under certain natural assumptions about the group action $G$, we show that the $G$-invariant Szegö kernel for $(0,q)$ forms is a comp…

In this paper we study flag curvature of invariant $(α,β)$-metrics of the form $\frac{(α+β)^2}α$ on homogeneous spaces and Lie groups. We give a formula for flag curvature of invariant metrics of the form $F=\frac{(α+β)^2}α$ such that $α$ is induced by an invariant Riemannian metric $g$ on the homogeneous space and the…

The problem of an arbitrary truncated Levy flight description using the method of cumulant approach has been solved. The set of cumulants of the truncated Levy distribution given the assumption of arbitrary truncation has been found. The influence of truncation shape on the truncated Levy flight properties in the Gauss…

Study of Yang-Mills fields on 4-manifolds using modified Lévy Laplacians.

The paper studies invariant Einstein metrics on Lie supergroups.

Analyzes the Levi form on CR manifolds of any dimension.

The pricing of options in exponential Levy models amounts to the computation of expectations of functionals of Levy processes. In many situations, Monte-Carlo methods are used. However, the simulation of a Levy process with infinite Levy measure generally requires either to truncate small jumps or to replace them by a …

Efficient methods for Lévy models using SINH-regular processes.

Paper proves existence of Lévy term structure models.

Study of Lévy flights on Zoll surfaces, revealing geometric information.

Study shows convergence rates for BSDEs approximated by compound Poisson processes.

Levy copulas are the most general concept to capture jump dependence in multivariate Levy processes. They translate the intuition and many features of the copula concept into a time series setting. A challenge faced by both, distributional and Levy copulas, is to find flexible but still applicable models for higher dim…

Motivated by the pricing of lookback options in exponential Lévy models, we study the difference between the continuous and discrete supremum of Lévy processes. In particular, we extend the results of Broadie et al. (1999) to jump-diffusion models. We also derive bounds for general exponential Lévy models.

We introduce an algorithm for the pricing of finite expiry American options driven by Lévy processes. The idea is to tweak Carr's `Canadisation' method, cf. Carr [9] (see also Bouchard et al [5]), in such a way that the adjusted algorithm is viable for any Lévy process whose law at an independent, exponentially distrib…

Study of bandit problem with Poisson decision times and Lévy processes.

The paper constructs CR manifolds with arbitrary Levi nondegeneracy.

Paper calculates perpetual American put option pricing with drawdown event in Lévy market.

In this note we apply the recently established Wiener-Hopf Monte Carlo (WHMC) simulation technique for Levy processes from Kuznetsov et al. [17] to path functionals, in particular first passage times, overshoots, undershoots and the last maximum before the passage time. Such functionals have many applications, for inst…

Analyzes Lévy flights on manifolds for finding small targets.

Optimizes liquidation strategies for assets with Levy process price dynamics.

The Levy Laplacian is studied on manifolds, with heat flow solutions tending to constant functionals over time.

Paper extends Lévy models with memory to better price FX double barrier options.

Efficient algorithm for CMDPs reduces to offline density estimation.

Sequences of Levy transformations for the Darboux system of conjugates nets in multidimensions are studied. We show that after a suitable number of Levy transformations, with at least a Levy transformation in each direction, we get closed formulae in terms of multi-Wroński determinants. These formulae are for the tange…

We construct a complete convergent normal form for a real hypersurface in $\CC{N},\,N\geq 2$ at generic Levi degeneracy. This seems to be the first convergent normal form for a Levi-degenerate hypersurface. In particular, we obtain, in the spirit of the work of Chern and Moser \cite{chern}, distinguished curves in the …