  The goal of this paper is to define stochastic integrals and to solve
stochastic differential equations for typical paths taking values in a possibly
infinite dimensional separable Hilbert space without imposing any probabilistic
structure. In the spirit of [33, 37] and motivated by the pricing duality
result obtained in [4] we introduce an outer measure as a variant of the
pathwise minimal superhedging price where agents are allowed to trade not only
in $\omega$ but also in $\int\omega\,d\omega:=\omega^2 -\langle \omega \rangle$
and where they are allowed to include beliefs in future paths of the price
process expressed by a prediction set. We then call a property to hold true on
typical paths if the set of paths where the property fails is null with respect
to our outer measure. It turns out that adding the second term $\omega^2
-\langle \omega \rangle$ in the definition of the outer measure enables to
directly construct stochastic integrals which are continuous, even for typical
paths taking values in an infinite dimensional separable Hilbert space.
Moreover, when restricting to continuous paths whose quadratic variation is
absolutely continuous with uniformly bounded derivative, a second construction
of model-free stochastic integrals for typical paths is presented, which then
allows to solve in a model-free way stochastic differential equations for
typical paths.

Probability