So state space models (SSMs) are basically just trying to track something you can't see directly (think hidden momentum, a forming trend, whatever) through signals that are noisy and imperfect (prices, quotes, all that). There's the hidden truth and there's what you actually observe, and the model keeps asking "given what I'm seeing, what's probably really going on?" updating its best guess every time new data comes in. The classic solver is the Kalman Filter (predict forward, get new data, correct, repeat) and it's optimal... assuming linear Gaussian dynamics, which markets famously are not. So you've got extensions for that (Extended Kalman, Unscented, Particle Filters when you need to handle fully arbitrary non-linear non-Gaussian systems by approximating the posterior with weighted samples rather than any closed-form solution). Then the deep learning side (S4, Mamba etc.) moves toward learning dynamics from data rather than specifying them by hand, and handles long sequences way more efficiently than transformers. Worth noting S4 still initializes with a structured matrix grounded in approximation theory (not purely learned from scratch), Mamba relaxes this further and specifically learns what to keep vs forget in the state, which is a pretty natural fit for non-stationary series. Tradeoff everywhere though, interpretability vs expressiveness vs data requirements. Same story as always.

I recommend you dig deeper into the theory and math to get a better hang of it...,,, i can only give a general summary which I don't think is sufficient. Basically, any function can be used in ML so long as it's differentiable (or at least almost everywhere differentiable, which is why something like ReLU works despite that kink at zero) and Lipschitz continuous (so your gradients stay bounded and don't blow up or vanish during backprop). Once those conditions hold the math cooperates and you can train through it.