Resolvent conditions for discretizations of diffusion-convection-reaction equations in several space dimensions

Straetemans, F. A. J.

doi:10.1016/s0168-9274(98)00017-8

Cited by 6 publications

(3 citation statements)

References 35 publications

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“…For finite dimensional problems analysis of the spectrum can lead to sufficient conditions but in the PDE context, the size of the matrix becomes unbounded as the grid is refined. In our case the matrix is non-symmetric and non-normal, and for non-normal matrices, counterexamples can be given where, even if the eigenvalues are less than one in magnitude, instability results as the dimension of the matrix becomes large (see [11,7,9,17]). For some values of the market parameters we are able to show that sufficient conditions for stability are satisfied using numerical range arguments [7,8,16,3].…”

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confidence: 96%

Analysis of the stability of the linear boundary condition for the Black–Scholes equation

Windcliff¹,

Forsyth²,

Vetzal³

2004

JCF

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Abstract. The linear asymptotic boundary condition, i.e. assuming that the second derivative of the value of the derivative security vanishes as the asset price becomes large, is commonly used in practice. To our knowledge, there have been no rigorous studies of the stability of these methods, despite the fact that the discrete matrix equations obtained using this boundary condition loses diagonal dominance for large timesteps. In this paper, we demonstrate that the discrete equations obtained using this boundary condition satisfy necessary conditions for stability for a finite difference discretization. Computational experiments also show that this boundary condition satisfies sufficient conditions for stability as well. [19,18,10] have recommended a linear asymptotic boundary condition (that the second derivative of the option value with respect to the underlying asset value be zero) as the asset price becomes large. Although this boundary specification is often applied, to our knowledge there have been no rigorous studies of the stability of this technique. Looking at the form of the discrete matrix equations obtained using this boundary condition, the resulting discrete equations lose diagonal dominance for large timesteps and the usual arguments cannot be applied to guarantee unconditional stability.In order to determine the ranges of parameters (risk-free rate, volatility, etc.) for which this asymptotic boundary condition could cause instability, we derive necessary conditions for the stability of the discrete equations based on the spectrum of the matrix representing the spatial discretization. Somewhat surprisingly, we find that a finite difference (FD) discretization always satisfies these necessary conditions for stability, despite the fact that the matrix equations are not unconditionally diagonally dominant.The eigenvalues can be used to determine necessary conditions for stability but are known to be unreliable for determining sufficient conditions for stability. For finite dimensional problems analysis of the spectrum can lead to sufficient conditions but in the PDE context, the size of the matrix becomes unbounded as the grid is refined. In our case the matrix is non-symmetric and non-normal, and for non-normal matrices, counterexamples can be given where, even if the eigenvalues are less than one in magnitude, instability results as the dimension of the matrix becomes large (see [11,7,9,17]). For some values of the market parameters we are able to show that sufficient conditions for stability are satisfied using numerical range arguments [7,8,16,3]. In other cases, these arguments cannot be applied and we follow [2] and *

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mentioning

confidence: 96%

Analysis of the stability of the linear boundary condition for the Black–Scholes equation

Windcliff¹,

Forsyth²,

Vetzal³

2004

JCF

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“…and the integral operator discretized to form a matrix J such that We now write the discrete version of the time step Equation 4.2 so that it matches the formulation of [28,30,48]. The Crank-Nicolson time discretization is written using a rational polynomial ϕ(∆τ M) defined similarly to [28]…”

Section: General Discrete Formmentioning

confidence: 99%

“…Definition 1 Modern stability analysis (for example [28,30,48]) defines general categories of stability under an arbitrary norm · using a rational polynomial ϕ(z) (e.g. as in Equation 4.6).…”

Section: Stabilitymentioning

confidence: 99%

Numerical solution of two asset jump diffusion models for option valuation

Clift

Forsyth

2008

Applied Numerical Mathematics

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Under the assumption that two financial assets evolve by correlated finite activity jumps superimposed on correlated Brownian motion, the value of a contingent claim written on these two assets is given by a two dimensional parabolic partial integro-differential equation (PIDE). An implicit, finite difference method is derived in this paper. This approach avoids a dense linear system solution by combining a fixed point iteration scheme with an FFT. The method prices both American and European style contracts independent (under some simple restrictions) of the option payoff and distribution of jumps. Convergence under the localization from the infinite to a finite domain is investigated, as are the theoretical conditions for the stability of the discrete approximation under maximum and von Neumann analysis. The analysis shows that the fixed point iteration is rapidly convergent under typical market parameters. The rapid convergence of the fixed point iteration is verified in some numerical tests. These tests also indicate that the method used to localize the PIDE is inexpensive and easily implemented.The coefficients σ 1 > 0, σ 2 > 0 and −1 ≤ ρ v ≤ 1 are the volatility magnitudes and correlation, respectively, of the Brownian motion processes on the two assets, and r ≥ 0 represents a risk-free rate of return. For simplicity, we do not include dividends in Equation 2.4. The operator λH represents the effects of finite activity asset price jumps generated by a Poisson process with mean arrival rate λ > 0. For brevity, we write (e J − 1) = e J1 − 1, e J2 − 1 T (2.7) and then we can write the integral term aswhere jump magnitudes J are distributed according to a probability density function g(J). In this study we make the standard assumption that g(J) is independent of y, and assume that g(J) satisfies the technical conditions of [23] §II.1.2 Definition 1.6 (see also [11] Proposition 3.18) in particular, that we may write separate integrals for the second term of HUand the third termThe values κ 1 , κ 2 < ∞ correct for the mean drift due to the first term of operator HU . This first term is written separately aswhich are equivalent forms of a correlation product. Specific formulations of g(J) with Normal and exponentially distributed jumps, which are analogous to the one-dimensional jump models of Merton [39] and Kou [27] respectively, are given in Appendix B. An American option may be exercised for its terminal payoff at any time. We may write the American option price as the solution to a linear complementarity problem [42,52]

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Stability in the numerical solution of the heat equation with nonlocal boundary conditions

Borovykh

2002

Applied Numerical Mathematics

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Resolvent conditions for discretizations of diffusion-convection-reaction equations in several space dimensions

Cited by 6 publications

References 35 publications

Analysis of the stability of the linear boundary condition for the Black–Scholes equation

Analysis of the stability of the linear boundary condition for the Black–Scholes equation

Numerical solution of two asset jump diffusion models for option valuation

Stability in the numerical solution of the heat equation with nonlocal boundary conditions

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