Let $B$ be a Brownian motion. Definining a *pathwise* stochastic integral $I(f):=\int f~dB$ for certain classes of deterministic functions is straightforward: For instance if $f=\sum\_ic\_i1\{[t\_i,t\_{i+1})\}$ is a compactly supported simple function, then
$$I(f):=\sum\_ic\_i(B(t\_{i+1})-B(t\_i)).$$
More generally if $AC\_0$ denotes the set of compactly supported absolutely continuous functions, continuity of $B$ allows to define the stochastic integral via integration by parts
$$I(f):=-\int f'(x)B(x)~dx.\tag{1}$$


Two properties of this "stochastic integral operator" on $AC\_0$ that are of interest to me are:


1. The operator defines a *pathwise* stochastic integral, i.e., for almost every outcome $\omega\in\Omega$ in our probability space, the brownian path $B\_\omega\in C[0,\infty)$ yields a linear functional $I\_\omega:AC\_0\to\mathbb R$, as $(1)$ satisfies
$$I\_\omega(af+bg)=aI\_\omega(f)+bI\_\omega(g).$$
2. The random functional $f\mapsto I(f)$ is a centered Gaussian process indexed by $AC\_0$ with covariance given by the $L^2$ inner product.



> 
> **Question.** Is there a way to extend this operator to deterministic functions with much less regularity (namely, continuous with compact support) *while preserving the above two properties simultaneously*?
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Using Itô's theory, we can of course define $I(f)\in L^2(\Omega)$ for every $f\in L^2[0,\infty)$, but my understanding is that this construction fails the above question. That is, for any fixed $f,g\in L^2$, we have linearity in the sense that
$$I(af+bg)=aI(f)+bI(g)\tag{2}$$
as elements of $L^2(\Omega)$, but this is a *weaker* linearity: the probability-one event on which $(2)$ holds depends on $a,b,f,g$, and thus from this alone we can't define $I$ as an almost sure linear operator $I\_\omega:L^2[0,\infty)\to\mathbb R$.



Pathwise stochastic integral as a linear operator on continuous functions
You can do the same for less regular functions, for example for functions with finite $p$-variation for some fixed $p \in [1,2)$ or functions in $\mathcal{C}^\alpha$ for some fixed $\alpha > 1/2$. You cannot go much below that in the sense that there exists *no* Banach space $\mathcal{B}$ with the following properties:


1. Trigonometric polynomials are dense in $\mathcal{B}$.
2. For almost every $\omega$, $f \mapsto I\_\omega(f)$ extends to a continuous linear map on $\mathcal{B}$.
3. $\mathcal{B}$ is large enough so that Brownian paths belong to $\mathcal{B}$ almost surely.


See for example [this paper](https://www.jstor.org/stable/51856) by Terry Lyons. The argument is pretty straightforward: basically one constructs two Brownian paths that are correlated in such a way that the integral of one against the other diverges and derives a contradiction from there.


