I am taking a first course on continuous time methods for Finance and I am trying to solve the following problem:
I am able to solve it until item d). Let me show what I have so far.
We apply Ito's lemma on $r_t = X_t^2$ to get $dr_t = 2X_tdX_t + (dX_t)^2$. Define $\kappa = 2\hat{\kappa}$, $\sigma = 2\hat{\sigma}$ and $\theta = \frac{\hat{\sigma}}{2\hat{\kappa}}$. We can plug the dynamics for $X_t$ back in and derive that $dr_t = \kappa(\theta - r_t)dt + \sigma \sqrt{r_t}dZ_t$.
We are looking for the following object (where I am using $dt = \Delta$):
$\phi(t, r_t; \omega) = E\left[e^{i\omega r_{t + dt}} | r_t \right]$
Let $\omega$ be fixed. I will omit it for now just for brevity. Using the Kolmogorov Backward Equation, the characteristic function must satisfy the following PDE: $$ \phi_t(t, r_t) + \phi_r(t, r_t) \cdot \kappa(\theta - r_t) + \frac{\sigma^2}{2}r_{t}\cdot \phi_{rr}(t, r_t) = 0, \quad \text{with} \quad \phi(t + dt, r_{t + dt}) = e^{i\omega r_{t+d_t}}$$
Now, we look for an exponentially affine solution. Under this guess, we have that:
$$ \phi_t = \phi \cdot (\dot{a}_0 (t) + \dot{a}_1 (t)r)\\ \phi_r = \phi \cdot a_1 (t)\\ \phi_{rr} = \phi \cdot (a_1 (t))^2$$
Plugging this back into the PDE, doing the algebra and noticing that this must be valid for any $r_t$ yields the following two ODEs:
$$\dot{a}_1(t) - \kappa a_1 (t) + \frac{\sigma^2}{2}(a_{1}(t))^2 = 0, \quad \text{with} \quad a_1(t+dt) = i\omega \\ \dot{a}_0(t) + \kappa \theta a_1(t) = 0, \quad \text{with}\quad a_0(t+dt) = 0$$
Now what I can't understand:
We are given the solution for $\phi$. Ideally, I'd like to take derivatives with respect to $r$ and with respect to $t$, plug it in back into the PDE and check that it indeed satisfies the equation. However, I don't see time $t$ in the characteristic function. I tried to do it assuming that $\phi_t = 0$ but that didn't work. I am clearly missing something. My two questions are:
- Why does time $t$ does not appear in the final solution?
- What is the correct derivative with respect to time?
Thanks a lot in advance!