arXiv research

Novel approach ensures stability of compact schemes for variable PDEs.

We extend the scheme developed in B. Düring, A. Pitkin, "High-order compact finite difference scheme for option pricing in stochastic volatility jump models", 2019, to the so-called stochastic volatility with contemporaneous jumps (SVCJ) model, derived by Duffie, Pan and Singleton. The performance of the scheme is asse…

It was proved in 1998 by Ben-David and Litman that a concept space has a sample compression scheme of size d if and only if every finite subspace has a sample compression scheme of size d. In the compactness theorem, measurability of the hypotheses of the created sample compression scheme is not guaranteed; at the same…

We present a new high-order compact scheme for the multi-dimensional Black-Scholes model with application to European Put options on a basket of two underlying assets. The scheme is second-order accurate in time and fourth-order accurate in space. Numerical examples confirm that a standard second-order finite differenc…

A fast, accurate method for pricing American options with free boundaries.

In this article, a three-time levels compact scheme is proposed to solve the partial integro-differential equation governing the option prices under jump-diffusion models. In the proposed compact scheme, the second derivative approximation of unknowns is approximated by the value of unknowns and their first derivative …

The stability and robustness of compact schemes for parabolic PDEs are analyzed.

In this thesis, we study deformations of compact holomorphic Poisson manifolds and algebraic Poisson schemes in the framework of Kodaira-Spencer's analytic deformation theory and Grothendieck's algebraic deformation theory.

We evaluate the hedging performance of a high-order compact finite difference scheme from [4] for option pricing in Bates model. We compare the scheme's hedging performance to standard finite difference methods in different examples. We observe that the new scheme outperforms a standard, second-order central finite dif…

We derive high-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In our numerical study we obtain high-order numerical convergence also for non-zero …

New method for pricing options in stochastic volatility models.

We present high-order compact schemes for a linear second-order parabolic partial differential equation (PDE) with mixed second-order derivative terms in two spatial dimensions. The schemes are applied to option pricing PDE for a family of stochastic volatility models. We use a non-uniform grid with more grid-points ar…

We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility models. The scheme is fourth-order accurate in space and second-order accurate in time. Under some restrictions, theoretical results like unconditional stability in the sense of von Neumann are presented. Where the a…

We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility jump models, e.g. in Bates model. In such models the option price is determined as the solution of a partial integro-differential equation. The scheme is fourth order accurate in space and second order accurate in ti…

Study of Tannakian categories for integrable connections on Kaehler manifolds.

Study on solutions to spinorial Yamabe equation on manifolds with boundary.

Enhances CEV model pricing with high-order scheme and adaptive time stepping.

Improved fourth-order compact scheme for option valuation with Robin boundary condition.

In this paper, a standard PDE for the pricing of arithmetic average strike Asian call option is presented. A Crank-Nicolson Implicit Method and a Higher Order Compact finite difference scheme for this pricing problem is derived. Both these schemes were implemented for various values of risk free rate and volatility. Th…

An unobstructedness theorem is proved for deformations of compact holomorphic Poisson manifolds and applied to a class of examples. These include certain rational surfaces and Hilbert schemes of points on Poisson surfaces. We study in particular the Hilbert schemes of the projective plane and show that a generic deform…

We consider a system of coupled free boundary problems for pricing American put options with regime-switching. To solve this system, we first employ the logarithmic transformation to map the free boundary for each regime to multi-fixed intervals and then eliminate the first-order derivative in the transformed model by …

We construct a three-point compact finite difference scheme on a non-uniform mesh for the time-fractional Black-Scholes equation. We show that for special graded meshes used in finance, the Tavella-Randall and the quadratic meshes the numerical solution has a fourth-order accuracy in space. Numerical experiments are di…

We construct harmonic morphisms on the compact simple Lie group G2. The construction uses eigenfamilies in a representation theoretic scheme.

We prove unobstructed deformations for compact Kaehlerian even-dimensional Poisson manifolds whose Poisson tensor degenerates along a divisor with mild singularities. Examples include Hilbert schemes of del Pezzo surfaces.

Let $X$ be a compact connected Riemann surface of genus $g$, with $g \geq 2$, and let ${\mathcal O}_X$ denote the sheaf of holomorphic functions on $X$. Fix positive integers $r$ and $d$ and let ${\mathcal Q}(r,d)$ be the Quot scheme parametrizing all torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of degree …

Algorithm solves American options with regime-switching using multigrid and compact finite difference.

We study two quantization schemes for compact symplectic manifolds with almost complex structures. The first of these is the Spin-c quantization. We prove the analog of Kodaira vanishing for the Spin-c Dirac operator, which shows that the index space of this operator provides an honest (not virtual) vector space semicl…

Let X be a compact connected Riemann surface of genus at least two. Fix positive integers r and d. Let Q denote the Quot scheme that parametrizes the torsion quotients of {\mathcal O}^{\oplus r}_X of degree d. This Q is also the moduli space of vortices for the standard action of U(r) on {\mathbb C}^r. The group \text{…

A new method for pricing options with stochastic volatility and jumps.

A family of holomorphic vector bundles is constructed on a complex manifold $X$. The space of the holomorphic sections of these bundles are calculated in certain cases. As an application, if $X$ is an $N$-dimensional compact Kähler manifold with holonomy group $SU(N)$, the space of holomorphic vector fields on its jet …

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${\mathcal Q}_X(r,d)$ be the quot scheme that parametrizes all the torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of degree $d$. This ${\mathcal Q}_X(r,d)$ is also a moduli space of vortices on $X$. Its geometric properties have be…

We develope a new scheme for the construction of explicit complex-valued proper biharmonic functions on Riemannian Lie groups. We exploit this and manufacture many infinite series of uncountable families of new solutions on the special unitary group $SU(n)$. We then show that the special orthogonal group $SO(n)$ and th…

This paper generalizes the Maurer--Pontil framework of finite-dimensional lossy coding schemes to the setting where a high-dimensional random vector is mapped to an element of a compact set of latent representations in a lower-dimensional Euclidean space, and the reconstruction map belongs to a given class of nonlinear…

Let $S^n(X)$ be the $n$-fold symmetric product of a compact connected Riemann surface $X$ of genus $g$ and gonality $d$. We prove that $S^n(X)$ admits a Kähler structure such that all the holomorphic bisectional curvatures are nonpositive if and only if $n < d$. Let ${\mathcal Q}_X(r,n)$ be the Quot scheme parametrizin…

The paper proves existence of solutions for mean field equations on compact Riemann surfaces.

Recently deep learning-based methods have been applied in image compression and achieved many promising results. In this paper, we propose an improved hybrid layered image compression framework by combining deep learning and the traditional image codecs. At the encoder, we first use a convolutional neural network (CNN)…

New method solves complex financial option pricing with varying time steps.

The article classifies curvature functions on compact manifolds with boundaries.

Formula for sections on complex manifolds with non-isolated components.

Extends optimal regularity and Uhlenbeck compactness to non-Riemannian manifolds.

Assume that $(X, g^+)$ is an asymptotically hyperbolic manifold, $(M, [\bar{h}])$ is its conformal infinity, $ρ$ is the geodesic boundary defining function associated to $\bar{h}$ and $\bar{g} = ρ^2 g^+$. For any $γ\in (0,1)$, we prove that the solution set of the $γ$-Yamabe problem on $M$ is compact in $C^2(M)$ provid…

Solves a complex mathematical problem on curved surfaces.

Unified framework for complex-valued eigenfunctions on Riemannian symmetric spaces.

We define parahoric $\cG$--torsors for certain Bruhat--Tits group scheme $\cG$ on a smooth complex projective curve $X$ when the weights are real, and also define connections on them. We prove that a $\cG$--torsor is given by a homomorphism from $π_1(X\setminus D)$ to a maximal compact subgroup of $G$, where $D\, \subs…

In this article, a compact finite difference method is proposed for pricing European and American options under jump-diffusion models. Partial integro-differential equation and linear complementary problem governing European and American options respectively are discretized using Crank-Nicolson Leap-Frog scheme. In pro…

In this article, we prove the Riemannian Penrose inequality for asymptotically flat manifolds with non-compact boundary whose asymptotic region is modelled on a half-space. Such spaces were initially considered by Almaraz, Barbosa and de Lima in 2014. In order to prove the inequality, we develop a new approximation sch…

Paper generalizes Andreev's theorem with obtuse angles.

Utilizing a splitting of geometric flows on surfaces introduced by Buzano and Rupflin, we present a general scheme to prove blow up criteria for such geometric flows. A vital ingredient is a new compactness theorem for families of metrics on surfaces with a uniform bound on their volumes, square integrals of their curv…