Nonlinear Parabolic Equations arising in Mathematical Finance

This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models of stochastic dynamic portfolio optimization leading to the Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both problems can be represented by solutions to nonlinear parabolic equations. Qualitative analysis will be focused on issues concerning the existence and uniqueness of solutions. In the numerical part we discuss a stable finite-volume and finite difference schemes for solving fully nonlinear parabolic equations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0387

High-resolution alternating evolution schemes for hyperbolic conservation laws and Hamilton-Jacobi equations

The novel approximation system introduced by Liu is an accurate approximation to systems of hyperbolic conservation laws. We develop a class of global and local alternating evolution (AE) schemes for one- and two-dimensional hyperbolic conservation law and one-dimensional Hamilton-Jacobi equations, where we take advantage of the high accuracy of the AE approximation. The nature of solutions having singularities, which is generic to these equations in handled using the AE methodology. The numerical scheme is constructed from the AE system by sampling over alternating computational grid points. Higher order accuracy is achieved by a combination of high-order polynomial reconstruction and a stable Runge-Kutta discretization in time. Local AE schemes are made possible by letting the scale parameter [epsilon] reflect the local distribution of nonlinear waves. The AE schemes have the advantage of easier formulation and implementation, and efficient computation of the solution. Theoretical numerical stability is proved mainly for the first and second order schemes of hyperbolic conservation law and Hamilton-Jacobi equations. In the case of hyperbolic conservation law, we have also shown that the numerical solutions converge to the weak solution. The designed methods have the advantage of being Riemann solver free, and the performs comparably to the finite volume/difference methods currently used. A series of numerical tests illustrates the capacity and accuracy of our method in describing the solutions

RIACS

Topics considered include: high-performance computing; cognitive and perceptual prostheses (computational aids designed to leverage human abilities); autonomous systems. Also included: development of a 3D unstructured grid code based on a finite volume formulation and applied to the Navier-stokes equations; Cartesian grid methods for complex geometry; multigrid methods for solving elliptic problems on unstructured grids; algebraic non-overlapping domain decomposition methods for compressible fluid flow problems on unstructured meshes; numerical methods for the compressible navier-stokes equations with application to aerodynamic flows; research in aerodynamic shape optimization; S-HARP: a parallel dynamic spectral partitioner; numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains; application of high-order shock capturing schemes to direct simulation of turbulence; multicast technology; network testbeds; supercomputer consolidation project

Some non monotone schemes for Hamilton-Jacobi-Bellman equations

We extend the theory of Barles Jakobsen to develop numerical schemes for Hamilton Jacobi Bellman equations. We show that the monotonicity of the schemes can be relaxed still leading to the convergence to the viscosity solution of the equation. We give some examples of such numerical schemes and show that the bounds obtained by the framework developed are not tight. At last we test some numerical schemes.Comment: 24 page

Error estimates for finite difference schemes associated with Hamilton-Jacobi equations on a junction

This paper is concerned with monotone (time-explicit) finite difference schemes associated with first order Hamilton-Jacobi equations posed on a junction. They extend the schemes recently introduced by Costeseque, Lebacque and Monneau (2013) to general junction conditions. On the one hand, we prove the convergence of the numerical solution towards the viscosity solution of the Hamilton-Jacobi equation as the mesh size tends to zero for general junction conditions. On the other hand, we derive optimal error estimates of order $($\Delta$x)^{\frac{1}{2}}$ in $L\_{loc}^{\infty}$ for junction conditions of optimal-control type at least if the flux is "strictly limited".Comment: 39 pages. In the initial version, the proof of the error estimate only works for Hamiltonians with the same minimum with no flux limiter. In the revised version, we can handle general quasi-convex Hamiltonians and flux limiters. We also provide numerical simulation

High-order filtered schemes for the Hamilton-Jacobi continuum limit of nondominated sorting

We investigate high-order finite difference schemes for the Hamilton-Jacobi equation continuum limit of nondominated sorting. Nondominated sorting is an algorithm for sorting points in Euclidean space into layers by repeatedly removing minimal elements. It is widely used in multi-objective optimization, which finds applications in many scientific and engineering contexts, including machine learning. In this paper, we show how to construct filtered schemes, which combine high order possibly unstable schemes with first order monotone schemes in a way that guarantees stability and convergence while enjoying the additional accuracy of the higher order scheme in regions where the solution is smooth. We prove that our filtered schemes are stable and converge to the viscosity solution of the Hamilton-Jacobi equation, and we provide numerical simulations to investigate the rate of convergence of the new schemes

A New Discontinuous Galerkin Finite Element Method for Directly Solving the Hamilton-Jacobi Equations

In this paper, we improve upon the discontinuous Galerkin (DG) method for Hamilton-Jacobi (HJ) equation with convex Hamiltonians in (Y. Cheng and C.-W. Shu, J. Comput. Phys. 223:398-415,2007) and develop a new DG method for directly solving the general HJ equations. The new method avoids the reconstruction of the solution across elements by utilizing the Roe speed at the cell interface. Besides, we propose an entropy fix by adding penalty terms proportional to the jump of the normal derivative of the numerical solution. The particular form of the entropy fix was inspired by the Harten and Hyman's entropy fix (A. Harten and J. M. Hyman. J. Comput. Phys. 50(2):235-269, 1983) for Roe scheme for the conservation laws. The resulting scheme is compact, simple to implement even on unstructured meshes, and is demonstrated to work for nonconvex Hamiltonians. Benchmark numerical experiments in one dimension and two dimensions are provided to validate the performance of the method