I'm trying hard to try understand the recursive nature between two equations in a recent macroeconomics paper, but my question mainly relates to how mathematically such recursive equations can depend on one another (and what that dependency between equations means). The paper is Reputation and Sovereign default by Amador and Phelan (2021).
There's a function q which is an expectation of $q$ chosen by two potential types (O and C). $\rho$ just shows perceived probability of types. This gives an overall expectation q(\tau) which is a Bayesian expectation of the two types which is (5). This I am fine with:
\begin{equation*} q(\tau)=\rho\left(\tau^{-}\right) q^{c}(\tau)+\left(1-\rho\left(\tau^{-}\right)\right) q^{o}(\tau) \tag{5} \end{equation*}
My problem comes from the actual definition of these two integrals which is as follows. I've simplified a few terms here to make it (slightly) easier to read. \begin{equation*} q^{c}(\tau)=\int_{0}^{\infty}\left(\int_{0}^{s}\beta e^{-\beta \tilde{s}} d \tilde{s}+e^{-\beta s} q^{o}(\tau+s)\right) \delta e^{-\delta s} d s \tag{6} \end{equation*}
\begin{align*} q^{o}(\tau)= & \int_{0}^{\infty}\left[\left(\int_{0}^{s}\beta e^{-\beta \tilde{s}} d \tilde{s}+e^{-\beta s} q^{c}(\tau+s)\right)\left(1-F_{\tau}(\tau+s)\right)\right. \\ & \left.+\int_{0}^{s}\left(\int_{0}^{\tilde{s}}\beta e^{-\beta \Delta} d \Delta\right) d F_{\tau}(\tau+\tilde{s})\right] \epsilon e^{-\epsilon s} d s . \tag{7} \end{align*}
The problem is that the two integrals depend on the other, and they depend on an adjusted form (i.e. not just $q^c(\tau)$ but $q^c(\tau+s)$).
Just to explain some terms, this is a Markov problem which depends on the state $\tau$. $\beta$ is just some constant less than one, and $1-F_{\tau}(\tau+s)$ is a survival function where $\tau$ represents some number of periods of survival and s is a date where switching potentially occurs. $1-F_a(b)$ shows the survival probability from a to b. $\delta$ is the Poisson probability of going from type C to O, and $\epsilon$ is the Poisson probability of going from O to C. Again, s is some potential date of type switching. $\Delta$ is a very small change (I think). I've read elswhere that the use the $\Delta$ is a way to solve recurrence relationships but I'm unsure. My question mainly relates to how one would go about:
a) Understanding a recurrence relationship here which appears not to converge to any pattern as it depends on the other function but as $q^o(\tau+s)$ and $q^c(\tau+s)$ (and not just a function of $\tau$).
b) What terminology for recurrence this would be referred to so that I am able to look it up in a textbook to understand what (5)'s functional form would converge to (or what it would take for it to converge to something meaningful).
Here's some extra context about each equation but hopefully this is not necessary to answer what definition/terminology describes this recursive pattern between (6) and (7):
(6): "Here, the outer integral is the expectation over the first type switch. The variable s in the outer integral represents the date of the first type switch from commitment to opportunistic. The two terms in the parentheses calculate the value of the bond conditional on s. The first term is the date τ value of the coupon stream between τ and τ + s. The second term is the date τ value of the remaining bond at date τ + s conditional on a type switch to an opportunistic government at that date."
(7): "The outer integral is the expectation over the first type switch. The terms in the square brackets calculate the value of the bond conditional on $s$. With probability $\left(1-F_{\tau}(\tau+s)\right.$ ), default does not occur before date $\tau+s$ starting from date $\tau$, and the terms in parentheses are similar to equation (6). The last term handles the case where default occurs before the type switch. The outer integral of this term is the expectation over the default date $\tilde{s}$, and the inner integral calculates the value of the coupon payments up to that date."
