Peano's theorem for SDEs

Question

It is well-known that Stochastic Differential Equations admit a unique solution under Lipschitz and linear growth conditions on the coefficients. However is there an analogue of the Peano-Existence theorem, only requiring continuity, for SDEs using an abstract fixed point argument (Schauder's fixed point theorem)? There are results in the direction of a Peano-Existence theorem (by P.Kloeden or in the book by Liu/Roeckner), however these results are always rather lengthy and don't involve a direct fixed point argument, as the Peano existence theorem typically does and instead establish convergence of an Euler Scheme. Or is it simply not possible to use an abstract fixed point theorem as this map typically requires continuity which clashes with the discontinuity of the Ito-solution map for SDEs?

Answer 1

This is perhaps just a partial answer, which is too long for a comment.

The problem is not with continuity (which can be achieved in the framework of rough paths), but with measurability.

Consider a stochastic differential equation $$ X_t = x + \int_0^t a(s,X_s) d Z_s, \tag{1} $$ where $Z$ is some rough stochastic process (like Brownian motion). One can fix $\omega$ and consider it pathwise, as an ordinary differential equation driven by a rough function $g(s) = Z_s(\omega)$. Then, writing a deterministic analogue of (1) as $$y = F_g(y),\tag{2}$$ one can employ some abstract fixed-point theorem to get a solution $y = S(g)$. The issue here is dependence of $X$ on $g$. While by some measurable selection theorem, it is usually possible to find a solution which is a measurable function of $g$ so that $X = S(Z)$ solves (1), this is still not enough to call $X$ a strong solution to (1). For the latter, $X$ has to be adapted to the filtration generated by $Z$. In terms of deterministic equation, we should ensure that $y_t$ depends measurably on $\{g_s,s\le t\})$ for all $t$, but there are no corresponding "adapted selection" theorems.

To solve this issue, instead of (2), one may instead to look for a solution map, i.e. for a function $\mathbf{X}: \mathfrak P_1\to \mathfrak P_2$, where $\mathfrak P_1$ and $\mathfrak P_2$ are suitable path spaces, which would satisfy $$ \mathbf X(\cdot) = F_{\cdot}(\mathbf X(\cdot)). $$ But this approach usually causes problems with compactness, certain kind of which must be assumed in fixed-point theorems.