Fichera Theory and Its Application in Finance

Implementation of alternating direction explicit methods for higher dimensional Black-Scholes equations

“…Preprint -Preprint -Preprint -Preprint -Preprint -Preprint stability, e.g., in [6,8,5]. For higher dimensional models, especially including the mixed derivatives, it leads to a conditional stable scheme.…”

“…Numerical analysis results focusing on stability and consistency considerations are described in [6] and [8]. In [8] a numerical analysis of convection-diffusion-reaction equation with constant coefficients and smooth initial data is provided.…”

“…Numerical analysis results focusing on stability and consistency considerations are described in [6] and [8]. In [8] a numerical analysis of convection-diffusion-reaction equation with constant coefficients and smooth initial data is provided. The authors proved that the ADE method applied to the one-dimensional reaction-diffusion equation on a uniform mesh with the discretization of the diffusion according to Saul'ev [1] and the discretization of the convection term following Towler and Yang [9] is unconditionally stable.…”

“…If a convection term is added to the equation and upwind discretization for this term is used, the ADE scheme is also unconditionally stable cf. [8].…”

“…Hereby the number of sweeps does not depend on the dimension. It has been shown [8,4,6] that for the upward and downward sweep the consistency is of order O((dτ) 2 + h 2 + dτ h ), where dτ is the time step and h denotes the space step. An exceptionality of the ADE method is that the average of upward and downward solutions has consistency of order O((dτ) 2 + h 2 ).…”