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I am studying the rough path theory of Lyons and am struggling to understand a heuristic claim made in several books and articles.

Let $(\Omega,\mathcal{F},\mathcal{F}_t, P, B_t)$ be a filtered probability space on which a standard Brownian motion $B_t$ is defined. Let $f:\mathbb{R}\to\mathbb{R}$ be a function (say, Lipschitz). Suppose $X_t$ is an $\mathcal{F}_t$-adapted process which solves the SDE $dX_t = f(X_t)dB_t$, that is, such that $X_t$ satisfies $$ X_t - X_0 = \int_0^tf(X_s)dB_s\quad\text{a.s.}, $$ where the the integral is in the sense of Itô. The Itô integral can then be obtained as the limit in $L^2$ of the Riemann-Stieltjes sum: $$ \sum_{0=t_0<t_1<\cdots<t_n=t}f(X_{t_i})(B_{t_{i+1}}-B_{t_i})\xrightarrow{\sup_i|t_i-t_{i+1}|\to 0}_{L^2(\Omega,\mathcal{F},P)}\int_0^tf(X_s)dB_s. $$ Many authors claim that rough path theory constitutes a "pathwise SDE solution theory": given a realization $B_t(\omega)$ of the driving Brownian motion, one can solve the corresponding "rough differential equation" driven by the corresponding realization of the Brownian rough path. In particular, the analogous statement is claimed to not hold for the Itô integral, because the above limit is only in $L^2$ (indeed, the above limit cannot be almost sure because Brownian motion has unbounded variation on any compact interval). On the other hand, a routine application of the Borel-Cantelli lemma shows the existence of a sequence of partitions along which the above limit is almost sure. My question is: in what sense does the realization of the Itô integrals $\int_0^tf(X_s)dB_s$ as an a.s. limit of Riemann-Stieljtes sums fail to constitute a "pathwise solution theory?" What defect of this "pathwise" construction does rough path theory mitigate?

Victor
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  • You are correct. There is always a sequence of partitions along with the limit *is* almost sure. Reading Motivation of Rough Path Theory in particular when they say, "This continuity property and the deterministic nature of solutions makes it possible to simplify and strengthen many results in Stochastic Analysis, such as the Freidlin-Wentzell's Large Deviation theory as well as results about stochastic flows." it strikes me as an interesting tool for some special applications. In 30 years of working with Ito calculus the orthodox – Kurt G. Sep 12 '22 at 05:10
  • ... definition of the integral was always good enough for me. Having said that, and probably to answer your question: Ito is not a pathwise solution theory because in the orthodox approach we cannot do wihout $L^2$. – Kurt G. Sep 12 '22 at 05:10
  • @KurtG. I’m afraid you haven’t answered my question at all. You say “we cannot do without $L^2$“—why not? In what sense and contexts do we need it? – Victor Sep 12 '22 at 08:03
  • If you dive into the standard definitions of the Ito integral you will notice what I said. There is no need to reproduce that here on MSE. – Kurt G. Sep 12 '22 at 08:25
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    When you want to show that an SDE has a well-defined flow, a priori, you only know that for any initial condition there exists a solution almost surely i.e. you have a collection of uncountable many null sets (one for each initial condition) and it is not clear at all that their union is also a null set, i.e. that there is a single null set such that the solution is defined on its complement. If you work with rough paths, there is no such issue and the existence of stochastic flows is a trivial consequence of the theory – G. Chiusole Sep 12 '22 at 09:21
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    (of course, there is no free lunch; existence and uniqueness for rough ODEs requires more regularity on the vector fields) – G. Chiusole Sep 12 '22 at 09:22
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    For the "impossibility" of Path by path stochastic integration you can also have a look at the argument in Protter's book "Stochastic Integration" via Banach-Steinhaus theorem in the first chapter. You can have a look here also (doesn't cover rough path) :https://math.stackexchange.com/questions/1653237/why-are-probabilities-needed-for-stochastic-differential-equations/1656572#1656572 or here https://math.stackexchange.com/questions/3190827/understand-better-stochastic-integral-through-a-s-convergence/3192318#3192318 – TheBridge Sep 12 '22 at 14:23
  • @G.Chiusole thanks! I see what you mean: the rough solution map is defined for all Brownian trajectories outside of a fixed null set, no matter which initial data is chosen for the SDE. By the way, something similar is true on the level of varying the vector fields, correct? – Victor Sep 12 '22 at 16:26
  • @TheBridge thanks for the references, they are helpful at first glance! – Victor Sep 12 '22 at 16:28
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    @Victor Yes, exactly. Also, the "pathwise" approach is not the only advantage of rough paths. It created a whole new philosophy of "higher order" information about paths (see e.g. section 3.4. in Friz-Hairer 2020 about Brownian motion in a magnetic field), ultimately leading to Hairer's theory of regularity structures and the reconstruction theorem, a vast generalization of the continuity result for the Ito-Lyons map. – G. Chiusole Sep 13 '22 at 09:52

1 Answers1

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The solution of an Itö-SDE $X_{t}=y+\int bdt+\int \sigma dB_{t}$ is defined outside a set of measure zero that depends on the whole equation i.e. $\Omega_{y,b,\sigma}$ with $P[\Omega_{y,b,\sigma}]=0$.

Considering another initial condition $y$ will create a new set of measure zero outside of which the solution is defined. Since the set of allowed initial conditions is not countable, it is a priori not clear whether

$$P[\bigcup_{y}\Omega_{y,b,\sigma}]=0,$$

i.e. that there exists a set of full measure on which an Itö-SDE can be solved for every initial condition $y$. For a pathwise solution theory such as rough paths, this is immediate. In rough paths, the solutions are built deterministically by postulating the existence of some iterated integrals.

see here too On the pathwise uniqueness of solutions of SDEs(Stochastic Differential Equations).

Thomas Kojar
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