Numerical Ross Recovery for Diffusion Processes Using a PDE Approach

Figures & data

Figure 1. Plots of $z^N(x)$ for $N=2$, 2.5, and 3 together with $z(x)=e^{\gamma x}$ for $\gamma =2$. The corresponding values of ${\bar{\lambda }}^N$ are given in the legends.

Table 1. Discretization parameters.

MP 1	$\mathrm{\Delta }x={10}^{-5}$
MP 2	$\mathrm{\Delta }x={10}^{-4}$
MP 3	$\mathrm{\Delta }x=2\cdot {10}^{-2}$, $\mathrm{\Delta }y={10}^{-2}$

Figure 2. Convergence plots for MP 1, 2 (with $\alpha =2$ and $\alpha =5$), and 3.

Table 2. Convergence rate.

	Theoretical (upper limit of) damping of relative error between $N=2$ and $N=3$			Actual damping of relative error in computed solution between $N=2$ and $N=3$			Actual damping/theoretical damping
	${\bar{\lambda }}^N$	$z^N$	$G^N$	${\bar{\lambda }}^N$	$z^N$	$G^N$	${\bar{\lambda }}^N$	$z^N$	$G^N$
MP1	3.18 × 10^−3	3.03 × 10^−3	3.03 × 10^−3	2.73 × 10^−3	3.79 × 10^−3	3.00 × 10^−3	0.86	1.25	0.99
MP2, $\alpha =2$	5.23 × 10^−1	4.02 × 10^−1	4.02 × 10^−1	4.30 × 10^−1	4.39 × 10^−1	3.71 × 10^−1	0.82	1.09	0.92
MP2, $\alpha =5$	1.97 × 10^−1	1.75 × 10^−1	1.75 × 10^−1	1.85 × 10^−1	1.87 × 10^−1	1.71 × 10^−1	0.93	1.07	0.98

Table 3. Number of iterations.

$N$	MP1	MP2, $\alpha =2$	MP2, $\alpha =5$	MP3
2.0	10	11	9	15
2.2	11	11	9	16
2.4	11	11	9	17
2.6	11	11	9	18
2.8	11	11	9	20
3.0	11	12	9	21