\def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\begin{thebibliography}{10}

\bibitem{BandiniPDMPs}
E.~Bandini.
\newblock Optimal control of {P}iecewise-{D}eterministic {M}arkov {P}rocesses:
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\newblock {\em ESAIM: Control, Optimization and Calculus of Variations},
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\bibitem{BandiniCalviaColaneri}
E.~Bandini, A.~Calvia, and K.~Colaneri.
\newblock Stochastic filtering of a pure jump process with predictable jumps
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\newblock {\em Stochastic Processes Appl.}, 151:396--435, 2022.

\bibitem{BandiniRusso1}
E.~Bandini and F.~Russo.
\newblock Weak {D}irichlet processes with jumps.
\newblock {\em {Stochastic Process. Appl.}}, 127(12):4139--4189, 2017.

\bibitem{BandiniRusso2}
E.~Bandini and F.~Russo.
\newblock Special weak {D}irichlet processes and {BSDE}s driven by a random
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\newblock {\em Bernoulli}, 24(4A):2569--2609, 2018.

\bibitem{BandiniRusso_DistrDrift}
E.~Bandini and F.~Russo.
\newblock Path-dependent {SDE}s with jumps and irregular drift: well-posedness
  and {D}irichlet properties.
\newblock {\em Preprint Arxiv 2211.03444}, 2022.

\bibitem{BandiniRusso_equivalence}
E.~Bandini and F.~Russo.
\newblock {Characteristics and It{\^o}'s formula for weak Dirichlet processes:
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\newblock {\em Preprint hal-04019358}, March 2023.

\bibitem{cordoni}
D.~R. Ba{\~n}os, F.~Cordoni, G.~Di~Nunno, L.~Di~Persio, and E.~E. R{\o}se.
\newblock Stochastic systems with memory and jumps.
\newblock {\em J. Differ. Equations}, 266(9):5772--5820, 2019.

\bibitem{barrasso3}
A.~Barrasso and F.~Russo.
\newblock Martingale driven {BSDE}s, {PDE}s and other related deterministic
  problems.
\newblock {\em Stochastic Processes and their Applications}, 133:193--228,
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\bibitem{BLT}
B.~Bouchard, G.~Loeper, and X.~Tan.
\newblock A {{\(\mathbb{C}^{0, 1}\)}}-functional {It{\^o}}'s formula and its
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\newblock {\em Stochastic Process. Appl.}, 148:299--323, 2022.

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G.~Cannizzaro and K.~Chouk.
\newblock {Multidimensional SDEs with singular drift and universal construction
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\newblock {\em {Ann. Probab.}}, 46(3):1710--1763, 2018.

\bibitem{cjms}
F.~Coquet, A.~Jakubowski, J.~M{\'e}min, and L.~S{\l}omi{\'n}ski.
\newblock Natural decomposition of processes and weak {D}irichlet processes.
\newblock In {\em In memoriam {P}aul-{A}ndr\'e {M}eyer: {S}\'eminaire de
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M.~H.~A. Davis.
\newblock {\em Markov models and optimization.}, volume~49 of {\em Monographs
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\newblock Chapman $\&$ Hall., 1993.

\bibitem{diel}
F.~Delarue and R.~Diel.
\newblock Rough paths and 1d {SDE} with a time dependent distributional drift:
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\newblock {\em Probab. Theory Related Fields}, 165(1-2):1--63, 2016.

\bibitem{ds}
N.~Dunford and J.~T. Schwartz.
\newblock {\em Linear operators. {P}art {I}}.
\newblock Wiley Classics Library. John Wiley \& Sons Inc., New York, 1988.
\newblock General theory, With the assistance of William G. Bade and Robert G.
  Bartle, Reprint of the 1958 original, A Wiley-Interscience Publication.

\bibitem{er}
M.~Errami and F.~Russo.
\newblock Covariation de convolutions de martingales.
\newblock {\em C. R. Acad. Sci. Paris S\'er. I Math.}, 326(5):601--606, 1998.

\bibitem{er2}
M.~Errami and F.~Russo.
\newblock {$n$}-covariation, generalized {D}irichlet processes and calculus
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\newblock {\em Stochastic Process. Appl.}, 104(2):259--299, 2003.

\bibitem{issoglio}
F.~Flandoli, E.~Issoglio, and F.~Russo.
\newblock Multidimensional stochastic differential equations with
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\newblock {\em Trans. Amer. Math. Soc.}, 369(3):1665--1688, 2017.

\bibitem{frw1}
F.~Flandoli, F.~Russo, and J.~Wolf.
\newblock Some {SDE}s with distributional drift. {I}. {G}eneral calculus.
\newblock {\em Osaka J. Math.}, 40(2):493--542, 2003.

\bibitem{frw2}
F.~Flandoli, F.~Russo, and J.~Wolf.
\newblock Some {SDE}s with distributional drift. {II}. {L}yons-{Z}heng
  structure, {I}t\^o's formula and semimartingale characterization.
\newblock {\em Random Oper. Stochastic Equations}, 12(2):145--184, 2004.

\bibitem{fo}
H.~F{\"o}llmer.
\newblock Calcul d'{I}t\^o sans probabilit\'es.
\newblock In {\em Seminar on Probability, XV (Univ. Strasbourg, Strasbourg,
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  143--150. Springer, Berlin, 1981.

\bibitem{FolDir}
H.~F{\"o}llmer.
\newblock Dirichlet processes.
\newblock In {\em Stochastic integrals ({P}roc. {S}ympos., {U}niv. {D}urham,
  {D}urham, 1980)}, volume 851 of {\em Lecture Notes in Math.}, pages 476--478.
  Springer, Berlin, 1981.

\bibitem{fwy}
W.~F\"ollmer, C.T. Wu, and M.~Yor.
\newblock {Canonical decomposition of linear transformations of two independent
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\newblock {\em {Stochastic Process. Appl.}}, 84(1):137--164, 1999.

\bibitem{Hocquet}
P.K. Friz, A.~Hocquet, and Khoa L{\^e}.
\newblock Rough stochastic differential equations.
\newblock {\em Preprint Arxiv 2186.10340}, 2021.

\bibitem{FuhrmanTessitore}
M.~Fuhrman and G.~Tessitore.
\newblock Generalized directional gradients, backward stochastic differential
  equations and mild solutions of semilinear parabolic equations.
\newblock {\em Appl Math Optim}, 108:263--298, 2003.

\bibitem{GOR}
A.O. Gomes, A.~Ohashi, F.~Russo, and A.~Teixeira.
\newblock Rough paths and regularization.
\newblock {\em Journal of Stochastic Analysis (JOSA)}, 2 (4):1--21, 12 2021.

\bibitem{gr1}
F.~Gozzi and F.~Russo.
\newblock Verification theorems for stochastic optimal control problems via a
  time dependent {F}ukushima-{D}irichlet decomposition.
\newblock {\em Stochastic Process. Appl.}, 116(11):1530--1562, 2006.

\bibitem{gr}
F.~Gozzi and F.~Russo.
\newblock Weak {D}irichlet processes with a stochastic control perspective.
\newblock {\em Stochastic Process. Appl.}, 116(11):1563--1583, 2006.

\bibitem{chineseBook}
S.~He, J.~Wang, and J.~Yan.
\newblock {\em Semimartingale theory and stochastic calculus}.
\newblock Science Press Beijing New York, 1992.

\bibitem{JacodBook}
J.~Jacod.
\newblock {\em Calcul {S}tochastique et {P}robl\`emes de martingales}, volume
  714 of {\em Lecture Notes in Mathematics}.
\newblock Springer, Berlin, 1979.

\bibitem{jacod_book}
J.~Jacod and A.~N. Shiryaev.
\newblock {\em Limit theorems for stochastic processes}, volume 288 of {\em
  Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
  Mathematical Sciences]}.
\newblock Springer-Verlag, Berlin, second edition, 2003.

\bibitem{memin}
A.~Jakubowski, J.~Memin, and G.~Pag\`es.
\newblock Convergence en loi des suites d'integrales stochastiques sur l'espace
  {{\({\mathbb{D}}^ 1\)}} de {S}korokhod. ({C}onvergence in law of sequences of
  stochastic integrals on the {S}korokhod space {{\({\mathbb{D}}^ 1)\)}}.
\newblock {\em Probab. Theory Relat. Fields}, 81(1):111--137, 1989.

\bibitem{ks}
I.~Karatzas and S.~E. Shreve.
\newblock {\em Brownian motion and stochastic calculus}, volume 113 of {\em
  Graduate Texts in Mathematics}.
\newblock Springer-Verlag, New York, second edition, 1991.

\bibitem{LaachirRusso}
I.~Laachir and F.~Russo.
\newblock B{SDE}s, c\`adl\`ag martingale problems, and orthogonalization under
  basis risk.
\newblock {\em SIAM J. Financial Math.}, 7(1):308--356, 2016.

\bibitem{ORT1_PartI}
A.~Ohashi, F.~Russo, and A.~Teixeira.
\newblock On path-dependent {SDEs} involving distributional drifts.
\newblock {\em Mod. Stoch., Theory Appl.}, 9(1):65--87, 2022.

\bibitem{ORT1_Bessel}
A.~Ohashi, F.~Russo, and A.~Teixeira.
\newblock {SDE}s for {B}essel processes in low dimension and path-dependent
  extensions.
\newblock {\em Preprint Arxiv 2211.04859}, 2022.

\bibitem{protter}
Ph. Protter.
\newblock {\em Stochastic Integration and Differential Equations}.
\newblock Stochastic Modelling and Applied Probabilities. Springer, second
  edition, 2000.

\bibitem{ry}
D.~Revuz and M.~Yor.
\newblock {\em Continuous martingales and {B}rownian motion}, volume 293 of
  {\em Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
  Mathematical Sciences]}.
\newblock Springer-Verlag, Berlin, third edition, 1999.

\bibitem{rtrut}
F.~Russo and G.~Trutnau.
\newblock Some parabolic {PDE}s whose drift is an irregular random noise in
  space.
\newblock {\em Ann. Probab.}, 35(6):2213--2262, 2007.

\bibitem{rv95}
F.~Russo and P.~Vallois.
\newblock The generalized covariation process and {I}t\^o formula.
\newblock {\em Stochastic Process. Appl.}, 59(1):81--104, 1995.

\bibitem{rv4}
F.~Russo and P.~Vallois.
\newblock Stochastic calculus with respect to continuous finite quadratic
  variation processes.
\newblock {\em Stochastics Stochastics Rep.}, 70(1-2):1--40, 2000.

\bibitem{Russo_Vallois_Book}
F.~Russo and P.~Vallois.
\newblock {\em Stochastic Calculus via Regularizations}, volume~11.
\newblock Springer International Publishing. Springer-Bocconi, 2022.

\bibitem{Stroock_Varadhan}
D.~W. Stroock and S.~S. Varadhan.
\newblock {\em Multidimensional diffusion processes}.
\newblock Springer, 2007.

\end{thebibliography}
