Editing Stochastic differential equation (section)

== As rough paths ==
Usually the solution of an SDE requires a probabilistic setting, as the integral implicit in the solution is a stochastic integral. If it were possible to deal with the differential equation path by path, one would not need to define a stochastic integral and one could develop a theory  independently of probability theory. 
This points to considering the SDE

:<math> \mathrm{d} X_t(\omega) = \mu(X_t(\omega),t)\, \mathrm{d} t +  \sigma(X_t(\omega),t)\, \mathrm{d} B_t(\omega) </math>

as a single deterministic differential equation for every <math>\omega \in \Omega</math>, where <math>\Omega</math> is the sample space in the given probability space (<math>\Omega,\, \mathcal{F},\, P</math>). However, a direct path-wise interpretation of the SDE is not possible, as the Brownian motion paths have unbounded variation and are nowhere differentiable with probability one, so that there is no naive way to give meaning to terms like <math>\mathrm{d} B_t(\omega)</math>, precluding also a naive path-wise definition of the stochastic integral as an integral against every single <math>\mathrm{d} B_t(\omega)</math>.  However, motivated by the Wong-Zakai result<ref name="frizhairer"/> for limits of solutions of SDEs with regular noise and using [[rough paths]] theory, while adding a chosen definition of iterated integrals of Brownian motion, it is possible to define a deterministic rough integral for every single <math>\omega \in \Omega</math> that coincides for example with the Ito integral with probability one for a particular choice of the iterated Brownian integral.<ref name="frizhairer">Friz, P. and Hairer, M. (2020). A Course on Rough Paths with an Introduction to Regularity Structures, 2nd ed., Springer-Verlag, Heidelberg, DOI 
 https://doi.org/10.1007/978-3-030-41556-3</ref> Other definitions of the iterated integral lead to deterministic pathwise equivalents of different stochastic integrals, like the Stratonovich integral. This has been used for example in financial mathematics to price options without probability.<ref name="optionswithoutprobability"> Armstrong, J., Bellani, C., Brigo, D. and Cass, T. (2021). Option pricing models without probability: a rough paths approach. Mathematical Finance, vol. 31, pages 1494–1521.</ref>