Closed form solutions of the Riccati equations are used to find the bond price function for specific one factor short rate interest rate models such as the Vasicek and CIS model.
Example taken from Filipović, Damir; Mayerhofer, Eberhard, Affine diffusion processes: theory and applications, Albrecher, Hansjörg (ed.) et al., Advanced financial modelling. Berlin: Walter de Gruyter (ISBN 978-3-11-021313-3/hbk; 978-3-11-021314-0/ebook). Radon Series on Computational and Applied Mathematics 8, 125-164 (2009). ZBL1205.91068.
The state space is $\mathbb{R}$, and we set $r=X$ for the Vasicek short rate model $$ dr=(b+\beta r)dt+\sigma dW. $$ The affine system reads $$ \Phi(t,\ u)=\frac{1}{2}\sigma^{2}\int_{0}^{t}\Psi^{2}(s,\ u)ds+b\int_{0}^{t}\Psi(s,\ u)ds $$ $$ \partial_{t}\Psi(t,\ u)=\beta\Psi(t,\ u)-1, $$ $$ \Psi(0,\ u)=u $$ which admits a unique global solution with $$ \Psi(t,\ u)=\mathrm{e}^{\beta t}u-\frac{\mathrm{e}^{\beta t}-1}{\beta} $$ $$ \Phi(t,\ u)=\frac{1}{2}\sigma^{2}(\frac{u^{2}}{2\beta}(\mathrm{e}^{2\beta t}-1)+\frac{1}{2\beta^{3}}(\mathrm{e}^{2\beta t}-4\mathrm{e}^{\beta t}+2\beta t+3) $$ $$ -\ \frac{u}{\beta^{2}}(\mathrm{e}^{2\beta t}-2\mathrm{e}^{\beta t}+2\beta))+b(\frac{\mathrm{e}^{\beta t}-1}{\beta}u+\frac{\mathrm{e}^{\beta t}-1-\beta t}{\beta^{2}}) $$ for all $u\in \mathbb{C}$. Hence (4.6) holds for all $u\in \mathbb{C}$ and $t\leq T$. In particular, by Corollary 4.2, the bond prices $P(t,\ T)$ can be determined by $A$ and $B,$ $$ B(t)=-\Psi(t,\ 0)=\frac{\mathrm{e}^{\beta t}-1}{\beta}, $$ $$ A(t)=-\Phi(t,\ 0)=-\frac{\sigma^{2}}{4\beta^{3}}(\mathrm{e}^{2\beta t}-4\mathrm{e}^{\beta t}+2\beta t+3)+b\frac{\mathrm{e}^{\beta t}-1-\beta t}{\beta^{2}}. $$
Problem: For certain values $b=0.0012587$, $β=0.00022$ and $σ=0.0041$, and with $r_0=-0.0031$, the closed form solution of $P(t,T)$ does not make economical sense, as in general investments will increase in value in a exponential way (except from actually losing money in the first years). The bond price at $t$ is equal to the amount invested at $t$ to receive EUR 1 at $T$.
The blue line depicted in this plot is made using the observed forward rate (the rate at which the investment increases in a time period) and shows this exponential form. If I use the bond price formula obtained with the Riccati equations, I get the orange line, a related but 'wrong' form. By wrong I mean that the starting point and endpoint are the same, but In the same way you can walk the edges of a rectangle in two ways, two paths are possible from start to end. The small difference at $t=0$ can be ignored and it can safely be assumed that the solution is correct at $t=0$ for all $T$. I ask myself if its possible that the Riccati equations may have more than one solution to choose from?
