Conditional full stability of positivity-preserving finite difference scheme for diffusion–advection-reaction models

Abstract

The matter of the stability for multidimensional diffusion-advection-reaction problems treated with the semi-discretization method is remaining challenge because when all the stepsizes tend simultaneously to zero the involved size of the problem grows without bounds. Solution of such problems is constructed by starting with a semi-discretization approach followed by a full discretization using exponential time differencing and matrix quadrature rules. Analysis of the time variation of the numerical solution with respect to previous time level together with the use of logarithmic norm of matrices are the basis of the stability result. Sufficient stability conditions on stepsizes, that also guarantee positivity and boundedness of the solution, are found. Numerical examples in different fields prove its competitiveness with other relevant methods.

An ETD method for multi‐asset American option pricing under jump‐diffusion model

“…For better approximation the spatial computational domain is chosen as $$$ \left[-5,5\right]\times \left[-5,5\right] $$$ with $$$ 150\times 150 $$$ nodes. In Table 3, we compare the results for $$$ \left({S}_1,{S}_2\right)=\left(100,100\right) $$$ with ones computed by the FD method (with mixed derivative removing transformation) proposed in Company et al [32] and by the high‐order computational (HOC) method of [26]. The option price for other assets' prices are reported in Table 4.…”

“…Following the ideas of Company et al [32], we consider the truncated domain $$$ \Omega =\left[{y}_{1_{\mathrm{min}}},{y}_{1_{\mathrm{max}}}\right]\times \dots \times \left[{y}_{M_{\mathrm{min}}},{y}_{M_{\mathrm{max}}}\right] $$$ and introduce the uniform in each dimension mesh $$$$ {y}_i^j={y}_{i_{\mathrm{min}}}+j{h}_i,\kern0.30em {h}_i=\frac{1}{N_i}\left({y}_{i_{\mathrm{max}}}-{y}_{i_{\mathrm{min}}}\right),j=0,\dots, {N}_i,i=1,\dots, M. $$$$ …”

“…Example 5.1 (American basket call). Consider an American basket call option [32] with the parameters given in Table 2.…”

“…For better approximation the spatial computational domain is chosen as [−5, 5] × [−5, 5] with 150 × 150 nodes. In Table 3, we compare the results for (S 1 , S 2 ) = (100, 100) with ones computed by the FD method (with mixed derivative removing transformation) proposed in Company et al [32] and by the high-order computational (HOC) method of [26]. The option price for other assets' prices are reported in Table 4.…”

An ETD Method for Vulnerable American Options

“…This method is easy to implement but computationally expensive and time-consuming to obtain the solution not in one fixed point but in some domain. Therefore, in the present study, the method of exponential time differencing proposed in [14] is employed after applying the mixed derivative terms removing transformation [13] and the semi-discretization technique [16].…”