Interest rates and bonds

From (37) an Euler-approximation gives the recursion

$$x_{i,m} = x_{i,m-1} + (b_i - a_i\cdot x_{i,m-1})\Delta + \sigma_i\sqrt{x_{i,m-1}} \sqrt{\Delta}N_{i,m} \quad (i=1,2)$$ (49)

where the $N_{i,m}$ (fixed $i$ or alternatively fixed $m$) are i.i.d. $N(0,1)$ (cf. Fischer, May and Walther, 2003). For a general introduction into the numerics of stochastic differential equations we refer to Kloeden and Platen (1992).

Plugging the computed values into (38), resp. (39), returns the desired interest rates, resp. bond prices.