Heston Riccati equation

Let \begin{align*} dY_{t} &= \left(r - \frac{1}{2} V_{t}\right) dt + \sqrt{V_{t}}dW_{t}\\ dV_{t} &= \kappa(\theta - V_{t}) dt + \rho \sigma \sqrt{V_{t}}dW_{t} + \sigma\sqrt{1-\rho^{2}}\sqrt{V_{t}}dB_{t} \end{align*} where $$W$$ and $$B$$ are two Brownian motions correlated by $$\rho$$. Due to the affinity of the log-transformed Heston model Duffie, Pan, and Singleton (2000) show a way to find the characteristic function of the Heston model. The characteristic function is then $$f(u) = \mathrm{e}^{a(t) + b_{1}(t)X_{0} + b_{2}(t)V_{0}}.$$ They also specify the Riccati ODEs which in this case will be \begin{align*} \frac{db_{1}(t)}{dt} &= 0,\\ \frac{db_{2}(t)}{dt} &= \frac{1}{2}b_{1}(t) + \frac{1}{2}\kappa b_{2}(t) - \frac{1}{2}b_{1}(t)^{2} - \rho\sigma b_{1}(t)b_{2}(t) - \frac{1}{2}\sigma^{2}b_{2}(t)^{2},\\ \frac{da(t)}{dt} &= r - rb_{1}(t) - \sigma\theta b_{2}(t).\\ \end{align*} with initial conditions $$a(0) = 0$$, $$b_{1}(0) = u$$, and $$b_{2}(0) = u$$. So, the solution for $$b_{1}$$ would be $$b_{1}(t) = u$$, then inserting this into the Riccati for $$b_{2}$$ we get $$\frac{db_{2}(t)}{dt} = \frac{1}{2}(u - u^{2}) + \left(\frac{1}{2}\kappa - \rho\sigma u\right)b_{2}(t) - \frac{1}{2}\sigma^{2}b_{2}(t)^{2}.$$ My issue is how do I solve this? I would appreciate if someone would help me derive the solution for $$b_{2}(t)$$ and maybe tell me if I also did it the right way this far. I would also like to note that I'm new to Riccati equations so any help would be appreciated.

For the proof, it suffices to follow this procedure with \begin{align} q_0(t) &= \frac{1}{2}u(1-u)\\ q_1(t) &= \frac{1}{2}\kappa-\rho\sigma u\\ q_2(t) &= -\frac{1}{2}\sigma^2\\ \end{align}

For this specific equation, all the $$q_i(t)$$ are constant functions, then you can even use this answer to solve the problem.

We note that for Riccacti equation, we need to know a particular solution first. You can use $$f(t) = \text{const} :=f$$ as the solution, then $$f$$ satisfies $$0 = \frac{1}{2}(u - u^{2}) + \left(\frac{1}{2}\kappa - \rho\sigma u\right)f - \frac{1}{2}\sigma^{2}f^2 \tag{1}$$

and $$(1)$$ is just a quadratic equation and can be solved easily.

• Thank you so much! I following your reference made it easier to understand. May 1 at 21:55
• @MarcAllan You're welcome!
– NN2
May 1 at 23:37

In [1] on p. 290-291 you find a discussion of the Cox-Ingersoll-Ross model in which the the Riccati equation $$\textstyle n_t(t,T)-\frac{1}{2}\sigma^2\, n^2(t,T)-b\,n(t,T)+1=0\,,\quad n(T,T)=0$$ is said to have the solution $$n(t,T)=\frac{\sinh(\gamma t)}{\gamma\cosh(\gamma t)+\frac{1}{2}b\sinh(\gamma t)}\,,\quad\gamma=\sqrt{b^2+2\sigma^2}\,,\quad \tau=T-t\,.$$