What is disturbing me is how is it possible that if the solution of a differential solution is unique then, it's continuous ? Do you have any idea how, this, is possible ? Apparently, it has some connection with Volterra integral and functional equations.
I have that stochastic differential equation (or perhaps it isn't one ... I'll explain further why I don't really know):
$$ \psi(t) = \int^t_0 K(t-s) [ C \psi^2(s) - \lambda \psi(s) - \theta^2 ] ds $$
- with $K$ a kernel ( I don't really know what it exactly means, apart from wikipedia's definition. If you need any additional information on $K$ let me know please),
- $C$ a constant
since I'm dabbling with SDE, and that question is about one step in solving a SDE, perhaps this isn't a SDE. Apparently, the solution $ \psi $ of that problem is unique and continuous in $ L_{loc}^2 ( \mathbb R_+, \mathbb R ) $.
Thank you very much for anyone considering to answer me. I am very sorry if my question is not of the best accuracy. If you're lacking just a few bits, let me know and I'll try my best to give those infos.
The solution of a differential equation is, by definition, continuously differentiable. It may not exist, and even if it does exist, there may be branching points where one initial condition leads to different solutions.
The equivalent differential equation is $ψ''=K(Cψ^2 −λψ −θ^2)$ with initial conditions $ψ(0)=ψ'(0)=0$. The right side is polynomial, thus any solution is analytical and unique. Due to the quadratic term, the solution may exist only for a finite time. One can multiply by $2ψ'$ and integrate to find that $$ ψ'^2=K\left(\frac23ψ^3−λψ^2 −2θ^2ψ\right) +C\implies C=0 $$ and conclude that the solution is bounded inside some interval $[-a,0]$ and thus exists for all times and oscillates.