Parabolic Partial Integro-differential Equations Research Articles

Numerical Solution of Partial Integrodifferential Equations of Diffusion Type

A collocation method based on linear Legendre multiwavelets is developed for numerical solution of one-dimensional parabolic partial integrodifferential equations of diffusion type. Such equations have numerous applications in many problems in the applied sciences to model dynamical systems. The proposed numerical method is validated by applying it to various benchmark problems from the existing literature. The numerical results confirm the accuracy, efficiency, and robustness of the proposed method.

Chebyshev Collocation Method for Parabolic Partial Integrodifferential Equations

An efficient technique for solving parabolic partial integrodifferential equation is presented. This technique is based on Chebyshev polynomials and finite difference method.A priorierror estimate for the proposed technique is deduced. Some examples are presented to illustrate the validity and efficiency of the presented method.

Numerical schemes for pricing Asian options under state-dependent regime-switching jump–diffusion models

We study the pricing problem of Asian options when the underlying asset price follows a very general state-dependent regime-switching jump–diffusion process via a partial differential equation approach. Under this model, the price of the option can be obtained by solving a highly complex system of coupled two-dimensional parabolic partial integro-differential equations (PIDEs). We prove existence of the solution to this system of PIDEs by the method of upper and lower solutions via constructing a monotonic sequence of approximating solutions whose limit is a strong solution of the PIDE system. We then propose several numerical schemes for solving the system of PIDEs. One of the proposed schemes is built upon the constructive proof, hence its results are provably convergent to the solution of the system of PIDEs. We illustrate the accuracy of the proposed methods by several numerical examples.

Numerical methods for the quadratic hedging problem in Markov models with jumps

We develop algorithms for the numerical computation of the quadratic hedging strategy in incomplete markets modeled by a pure jump Markov process. Using the Hamilton-Jacobi-Bellman approach, the value function of the quadratic hedging problem can be related to a triangular system of parabolic partial integrodifferential equations (PIDEs), which can be shown to possess unique smooth solutions in our setting. The first equation is nonlinear, but does not depend on the payoff of the option to hedge (the pure investment problem), while the other two equations are linear. We propose convergent finite-difference schemes for the numerical solution of these PIDEs and illustrate our results with an application to electricity markets, where time-inhomogeneous pure jump Markov processes naturally occur.

Numerical Schemes for Pricing Asian Options Under State-Dependent Regime-Switching Jump-Diffusion Models

We propose numerical schemes for pricing Asian options when the underlying asset price follows a very general state-dependent regime-switching jump-diffusion process. Under this model, the price of the option can be obtained by solving a highly complex system of coupled two-dimensional parabolic partial integro-differential equations (PIDEs) via iterative techniques. One of the proposed schemes is provably convergent to the solution of the system of PIDEs. In addition, by treating the coupling and integral terms explicitly, over each iteration of the scheme, the pricing problem under this scheme can be partitioned into independent pricing sub-problem, with communication at the end of the iteration. Hence, this method allows for a very natural and easy-to-implement, yet efficient, parallelization of the solution process on multi-core architectures. We illustrate the accuracy and efficiency of the proposed methods by several numerical examples.

Second Order Accurate IMEX Methods for Option Pricing Under Merton and Kou Jump-Diffusion Models

In this paper three implicit-explicit (IMEX) time semi-discrete methods, namely IMEX-BDF1, IMEX-BDF2 and CN-LF, are developed for solving parabolic partial integro-differential equations which arise in option pricing theory when the underlying asset follows a jump diffusion process. It is shown that IMEX-BDF2 and CN-LF are stable and second order accurate, whereas IMEX-BDF1 is stable but only first order accurate. After time semi-discretization, the resulting linear differential equations are solved by using a cubic B-spline collocation method. The methods so developed have computational complexity of $$O(MNlog_{2}(M))$$O(MNlog2(M)) for Merton model and of $$O(MN)$$O(MN) for Kou model, where $$N$$N denotes the number of time steps and $$M$$M the number of collocation points. Some numerical examples, for pricing European options under Merton and Kou jump-diffusion models with constant as well as variable volatility, are presented to demonstrate the stability, convergence and computational complexity of the methods.

A New Chebyshev Spectral-collocation Method for Solving a Class of One-dimensional Linear Parabolic Partial Integro-differential Equations

In this work, the Chebyshev spectral-collocation method is applied to obtain approximate solution for some types of linear parabolic partial integro–differential equations (PPIDEs). In the first approach, we convert our equation into two coupled Volterra integral equations of the second kind by using a proper transformation. In the second approach, the integration in the resulting equations are approximated by replacing the integrand by its interpolating polynomials in terms of the Chebyshev polynomials instead of using the approximation by Gauss quadrature rules. After approximation a linear algebraic system were raised, then it tested by the conditional number. Finally, some numerical examples are included to illustrate the validity and applicability of the proposed technique.

The obstacle problem for semilinear parabolic partial integro-differential equations

This paper presents a probabilistic interpretation for the weak Sobolev solution of the obstacle problem for semilinear parabolic partial integro-differential equations (PIDEs). The results of Léandre [32] concerning the homeomorphic property for the solution of SDEs with jumps are used to construct random test functions for the variational equation for such PIDEs. This results in the natural connection with the associated Reflected Backward Stochastic Differential Equations with jumps (RBSDEs), namely Feynman–Kac's formula for the solution of the PIDEs. Moreover, it gives an application to the pricing and hedging of contingent claims with constraints in the wealth or portfolio processes in financial markets including jumps.

Efficient pricing of swing options in Lévy-driven models

We consider the problem of pricing swing options with multiple exercise rights in Lévy-driven models. We propose an efficient Wiener–Hopf factorization method that solves multiple parabolic partial integro-differential equations associated with the pricing problem. We compare the proposed method with a finite difference algorithm. Both proposed deterministic methods are related to the dynamic programming principle and lead to the solution of a multiple optimal stopping problem. Numerical examples illustrate the efficiency and the precision of the proposed methods.

A New Positive Definite Expanded Mixed Finite Element Method for Parabolic Integrodifferential Equations

A new positive definite expanded mixed finite element method is proposed for parabolic partial integrodifferential equations. Compared to expanded mixed scheme, the new expanded mixed element system is symmetric positive definite and both the gradient equation and the flux equation are separated from its scalar unknown equation. The existence and uniqueness for semidiscrete scheme are proved and error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are provided to confirm our theoretical analysis.

ADI orthogonal spline collocation methods for parabolic partial integro-differential equations

Alternating direction implicit (ADI) orthogonal spline collocation schemes are formulated and analysed for a class of partial integro-differential equations of parabolic type. These techniques are based on the θ-method, for θ ∈ [1/2, 1], (where θ = 1 yields the backward Euler method and θ = 1/2 yields the Crank-Nicolson method) and the second-order backward differentiation formula (BDF) method. For each method, optimal estimates in various norms at each time step are derived and confirmed by results of numerical experiments.

Weak Solutions of Abstract Evolutionary Integro-Differential Equations in Hilbert Spaces

We prove existence and uniqueness of weak solutions to certain abstract evolutionary integro-differential equations in Hilbert spaces, including evolution equations of fractional order less than 1. Our results apply, e.g., to parabolic partial integro-differential equations in divergence form with merely bounded and measurable coefficients. AMS subject classification: 45N05, 45K05

Quasilinear parabolic integro-differential equations with nonlinear boundary conditions

We study the $L_p$-theory of a class of quasilinear parabolic partial integro-differential equations with nonlinear boundary conditions. The main objective here is to prove existence and uniqueness of local (in time) strong solutions of these problems. Our approach relies on linearization and the contraction mapping principle. To make this work we establish optimal regularity estimates of $L_p$ type for associated linear problems with inhomogeneous boundary data, using here recent results on maximal $L_p$-regularity for abstract parabolic Volterra equations.

A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models

We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicit-implicit finite difference scheme which can be used to price European and barrier options in such models. We study stability and convergence of the scheme proposed and, under additional conditions, provide estimates on the rate of convergence. Numerical tests are performed with smooth and nonsmooth initial conditions.

Backstepping observers for a class of parabolic PDEs

In this paper we design exponentially convergent observers for a class of parabolic partial integro-differential equations (P(I)DEs) with only boundary sensing available. The problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system. Observer gain (output injection function) is shown to satisfy a well-posed hyperbolic PDE that is closely related to the hyperbolic PDE governing backstepping control gain for the state-feedback problem. For several physically relevant problems the observer gains are obtained in closed form. The observer gains are then used for an output-feedback design in both collocated and anti-collocated setting of sensor and actuator. The order of the resulting compensator can be substantially lowered without affecting stability. Explicit solutions of a closed loop system are found in particular cases.

The numerical solution of parabolic Volterra integro-differential equations on unbounded spatial domains

In this paper we study the numerical approximation of solutions to parabolic partial Volterra integro-differential equations in unbounded (one-dimensional) spatial domains. The key to these methods is the derivation of fully transparent artificial boundary conditions for some feasible, prescribed computational domain. Two numerical examples illustrate the application and the effectiveness of the artificial boundary method.

Closed-Form Boundary State Feedbacks for a Class of 1-D Partial Integro-Differential Equations

In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equation's well posedness and the kernel's smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.

A Finite Difference Scheme for Option Pricing in Jump-diffusion and Exponential Levy Models

We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicit-implicit time-stepping scheme to solve the equation and study stability and convergence of the schemes proposed, using the notion of viscosity solution. Numerical tests are performed for the Merton jump-diffusion model and for the Variance Gamma model with smooth and non-smooth payoff functions. Our scheme can be used for European and barrier options, applies in the case of pure-jump models or degenerate diffusion coefficients, and extends to time-dependent coefficients.

H1-Galerkin mixed finite element methods for parabolic partial integro-differential equations

H1‐Galerkin mixed finite element methods are analysed for parabolic partial integro‐differential equations which arise in mathematical models of reactive flows in porous media and of materials with memory effects. Depending on the physical quantities of interest, two methods are discussed. Optimal error estimates are derived for both semidiscrete and fully discrete schemes for problems in one space dimension. An extension to problems in two and three space variables is also discussed and it is shown that the H1‐Galerkin mixed finite element approximations have the same rate of convergence as in the classical methods without requiring the LBB consistency condition.

Orthogonal spline collocation methods for partial differential equations

This paper provides an overview of the formulation, analysis and implementation of orthogonal spline collocation (OSC), also known as spline collocation at Gauss points, for the numerical solution of partial differential equations in two space variables. Advances in the OSC theory for elliptic boundary value problems are discussed, and direct and iterative methods for the solution of the OSC equations examined. The use of OSC methods in the solution of initial–boundary value problems for parabolic, hyperbolic and Schrödinger-type systems is described, with emphasis on alternating direction implicit methods. The OSC solution of parabolic and hyperbolic partial integro-differential equations is also mentioned. Finally, recent applications of a second spline collocation method, modified spline collocation, are outlined.