High‐order ADI orthogonal spline collocation method for a new 2D fractional integro‐differential problem

This is the peer reviewed version of the following article: Qiao L, Xu D, Yan Y. (2020). High-order ADI orthogonal spline collocation method for a new 2D fractional integro-differential problem. Mathematical Methods in the Applied Sciences, 1-17., which has been published in final form at https://doi.org/10.1002/mma.6258. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.We use the generalized L1 approximation for the Caputo fractional deriva-tive, the second-order fractional quadrature rule approximation for the inte-gral term, and a classical Crank-Nicolson alternating direction implicit (ADI)scheme for the time discretization of a new two-dimensional (2D) fractionalintegro-differential equation, in combination with a space discretization by anarbitrary-order orthogonal spline collocation (OSC) method. The stability of aCrank-Nicolson ADI OSC scheme is rigourously established, and error estimateis also derived. Finally, some numerical tests are give

Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations

We consider an initial-boundary value problem for $\partial_tu-\partial_t^{-\alpha}\nabla^2u=f(t)$, that is, for a fractional diffusion ($-1<\alpha<0$) or wave ($0<\alpha<1$) equation. A numerical solution is found by applying a piecewise-linear, discontinuous Galerkin method in time combined with a piecewise-linear, conforming finite element method in space. The time mesh is graded appropriately near $t=0$, but the spatial mesh is quasiuniform. Previously, we proved that the error, measured in the spatial $L_2$-norm, is of order $k^{2+\alpha_-}+h^2\ell(k)$, uniformly in $t$, where $k$ is the maximum time step, $h$ is the maximum diameter of the spatial finite elements, $\alpha_-=\min(\alpha,0)\le0$ and $\ell(k)=\max(1,|\log k|)$. Here, we generalize a known result for the classical heat equation (i.e., the case $\alpha=0$) by showing that at each time level $t_n$ the solution is superconvergent with respect to $k$: the error is of order $(k^{3+2\alpha_-}+h^2)\ell(k)$. Moreover, a simple postprocessing step employing Lagrange interpolation yields a superconvergent approximation for any $t$. Numerical experiments indicate that our theoretical error bound is pessimistic if $\alpha<0$. Ignoring logarithmic factors, we observe that the error in the DG solution at $t=t_n$, and after postprocessing at all $t$, is of order $k^{3+\alpha_-}+h^2$.Comment: 24 pages, 2 figure

Numerical and optimal control methods for partial differential equations arising in computational finance

The chosen title for my PhD thesis is "Numerical and optimal control methods for partial differential equations arising in computational finance". The body of my research is divided into two parts. The first part is devoted to the application of an alternating direction implicit numerical method for solving stochastic volatility option pricing models. The second part focuses on a partial-integro differential equation constrained optimal control approach to parameter estimation for the forward jump-diffusion option pricing model. The body of the thesis is preceded by an extensive introduction, which seeks to contextualize my work with respect to the field of computational finance, this is followed by a brief conclusion. Finally, the thesis is completed by a list of refer ences. The first project proposes a new high-order alternating direction implicit (ADI) finite difference scheme for the solution of initial-boundary value problems of convection-diffusion type with mixed derivatives and non-constant coefficients, as they arise from stochastic volatility models in option pricing. The approach combines different high-order spatial discretisations with Hundsdorfer and Verwer's ADI time-stepping method, to obtain an efficient method which is fourth-order accurate in space and second-order accurate in time. Numerical experiments for the European put option pricing problem using Heston's stochastic volatility model confirm the high-order convergence. The second project proposes to solve a parameter calibration problem for the forward jump-diffusion option pricing model proposed by Andersen and Andreasen. A distributed optimal control approach is employed, with a partial-integro differential equation as our state equation. By approaching the problem from a functional analysis perspective, I investigate the necessary regularity conditions for our parameters of interest. Following this, the existence of optimal solutions is proven under certain analytical conditions. Furthermore, the first-order necessary conditions for optimality are also established. Finally, a projected-gradient optimization method is applied numerically to empirical market data and results are given

Error analysis of truncated expansion solutions to high-dimensional parabolic PDEs

We study an expansion method for high-dimensional parabolic PDEs which constructs accurate approximate solutions by decomposition into solutions to lower-dimensional PDEs, and which is particularly effective if there are a low number of dominant principal components. The focus of the present article is the derivation of sharp error bounds for the constant coefficient case and a first and second order approximation. We give a precise characterisation when these bounds hold for (non-smooth) option pricing applications and provide numerical results demonstrating that the practically observed convergence speed is in agreement with the theoretical predictions

Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$\partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T),$$ where $\mathfrak{L}^{\sigma,\mu}$ is a general symmetric diffusion operator of L\'evy type and $\varphi$ is merely continuous and non-decreasing. We then use this theory to prove convergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion operators $\mathfrak{L}^{\sigma,\mu}$ are the (fractional) Laplacians $\Delta$ and $-(-\Delta)^{\frac\alpha2}$ for $\alpha\in(0,2)$, discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal L\'evy operators, allows us to give a unified and compact {\em nonlocal} theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and convergence of the methods under minimal assumptions -- including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in \cite{DTEnJa17b}. We also present some numerical tests, but extensive testing is deferred to the companion paper \cite{DTEnJa18b} along with a more detailed discussion of the numerical methods included in our theory.Comment: 34 pages, 3 figures. To appear in SIAM Journal on Numerical Analysi

Probability of Default modelling with L\'evy-driven Ornstein-Uhlenbeck processes and applications in credit risk under the IFRS 9

In this paper we develop a framework for estimating Probability of Default (PD) based on stochastic models governing an appropriate asset value processes. In particular, we build upon a L\'evy-driven Ornstein-Uhlenbeck process and consider a generalized model that incorporates multiple latent variables affecting the evolution of the process. We obtain an Integral Equation (IE) formulation for the corresponding PD as a function of the initial position of the asset value process and the time until maturity, from which we then prove that the PD function satisfies an appropriate Partial Integro-Differential Equation (PIDE). These representations allow us to show that appropriate weak (viscosity) as well as strong solutions exist, and develop subsequent numerical schemes for the estimation of the PD function. Such a framework is necessary under the newly introduced International Financial Reporting Standards (IFRS) 9 regulation, which has imposed further requirements on the sophistication and rigor underlying credit modelling methodologies. We consider special cases of the generalized model that can be used for applications to credit risk modelling and provide examples specific to provisioning under IFRS 9, and more