Efficient Numerical Algorithms For Spatially Nonlocal Partial Integro-differential Equations And Their Applications In Financ

The pricing of financial derivatives is one of the most important research areas in financial mathematics.Options as one of the financial derivatives,its pricing has been widely concerned from academics and financial practitioners.The classical Black-Scholes model is unable to accurately portray the characteristics of the actual financial market due to its strong assumptions.As a result,many scholars improve Black-Scholes model by introducing a large number of models based on the Lévy process,such as jump-diffusion model.The corresponding option pricing problems transform into a spatially non-local partial integro-differential equation with a nonsmooth initial functions.When solving such equations numerically,non-uniform grids are essential to recover the convergence order and improve the computational accuracy.Therefore,this paper focuses on the Euro-pean and American option pricing problems under the jump-diffusion model by studying several types of variable step-size numerical methods for solving non-local partial integro-differential equations.The Bates model of stochastic volatility with jumps under European options can be transformed into a partial integro-differential equations with spatially mixed deriva-tive terms.In view of the initial singularity of the equations and the non-local prop-erty of the integral term,we use the variable step-size implicit explicit Crank-Nicolson Adams-Bashforth method(IMEX CNAB)combined with the second-order finite differ-ence method to solve the equations,which can not only improve the computational accu-racy,but also reduce the computational complexity.Firstly,the stability of the variable step-size IMEX CNAB semi-discrete scheme is proved by the energy method,and based on the regularity assumption of the model,the global error bounds of the numerical scheme are derived in the L~2norm.The stability and error estimates of the fully discrete scheme are then given by estimating the difference operator.Finally,numerical experiments ver-ify the validity of the variable step-size IMEX CNAB method.The diffusion model with jumps under American options can be formulated as an op-timal control problem which can be written in the form of the Hamilton-Jacobi-Bellman(HJB)partial integro-differential equations.We use the variable step-size implicit ex-plicit two step-size backward differentiation method(IMEX BDF2)combined with a fi-nite difference method to obtain the full discrete scheme.We firstly prove the stability with respect to perturbations in the discrete L~2norm for linear and semi-linear equations,and in the discrete H~1norm for nonlinear equations.These results are then extended to possibly degenerate linear equations.On the basis of these stability results,we de-rive error estimates for classical solutions of semi-linear uniformly parabolic HJB partial integro-differential equations in the discrete L~2norm,whose convergence order depend on their H?lder regularity,recovering second-order convergence in the case of sufficiently smooth.Numerical experiments verify the theoretical results.Adaptive algorithm provide effective error control for numerically solving equations with singularities,and the a posteriori error estimates is the basis for designing adaptive algorithms.A commonly used spatial discretization method for option pricing models is the finite difference method.Due to the relative difficulty in deriving objective-oriented a posteriori error estimates,we firstly study the a posteriori error estimates of the fully dis-crete methods for one-dimensional linear parabolic equations,the backward Euler method and Crank-Nicolson(CN)method are used for the time discretization,and the second-order finite difference method is used for the spatial discretization.The a posteriori error quantities corresponding to the spatial discretization are obtained by linear interpolation and its estimates.For the backward Euler and CN methods,the temporal discretization error is obtained by exploring the linear continuous approximation and continuous,piece-wise quadratic time reconstructions,respectively.Upper and lower bounds of the a poste-riori error estimates for the fully discrete finite difference methods are eventually obtained,which depend only on the discretization parameters and the data of the model problem.These techniques are then extended to two-dimensional linear parabolic equations.Nu-merical experiments verify the theoretical results.Based on the above,we study the a posteriori error estimates of IMEX numerical methods for partial integro-differential equations in finance.The IMEX Euler method and the IMEX CNAB method are used for the time discretization,and the second-order finite difference method based on non-uniform grids is used for the spatial discretization.Upper and lower bounds of the a posteriori error estimates for the fully discrete finite difference method are derived,again through the use of continuous,piecewise time re-constructions.Based on these a posteriori error estimates,we further design a time-space adaptive algorithm.Numerical experiments are carried out under uniform partitioning and time-space adaptive algorithm,and the numerical results validate the effectiveness of the adaptive algorithm.