\relax \providecommand\zref@newlabel[2]{} \providecommand\hyper@newdestlabel[2]{} \citation{tsaban_harnessing_2022} \citation{davies_signature_2021} \citation{zhao_space-based_2023} \citation{wu2022sustainable} \citation{strubell2019energy} \citation{e_multilevel_2019} \citation{e_multilevel_2021} \citation{hutzenthaler_strong_2021} \@writefile{toc}{\contentsline {chapter}{\numberline {1}Introduction.}{1}{chapter.1}\protected@file@percent } \@writefile{lof}{\addvspace {10\p@ }} \@writefile{lot}{\addvspace {10\p@ }} \@writefile{toc}{\contentsline {section}{\numberline {1.1}Motivation}{1}{section.1.1}\protected@file@percent } \citation{bhj20} \citation{karatzas1991brownian} \citation{da_prato_zabczyk_2002} \citation{grohs2019spacetime} \citation{grohs2019spacetime} \citation{golub2013matrix} \citation{durrett2019probability} \citation{graham_concrete_1994} \@writefile{toc}{\contentsline {section}{\numberline {1.2}Notation, Definitions \& Basic notions.}{3}{section.1.2}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {1.2.1}Norms and Inner Products}{3}{subsection.1.2.1}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {1.2.2}Probability Space and Brownian Motion}{4}{subsection.1.2.2}\protected@file@percent } \newlabel{def:brown_motion}{{1.2.11}{5}{Brownian Motion Over a Stochastic Basis}{theorem.1.2.11}{}} \newlabel{def:brown_motion@cref}{{[definition][11][1,2]1.2.11}{[1][5][]5}} \newlabel{1.9}{{1.2.14}{6}{Strong Solution of Stochastic Differential Equation}{theorem.1.2.14}{}} \newlabel{1.9@cref}{{[definition][14][1,2]1.2.14}{[1][6][]6}} \newlabel{1.5}{{1.9}{6}{Strong Solution of Stochastic Differential Equation}{equation.1.2.9}{}} \newlabel{1.5@cref}{{[equation][9][1,2]1.9}{[1][6][]6}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.2.3}Lipschitz and Related Notions}{7}{subsection.1.2.3}\protected@file@percent } \newlabel{def:1.13}{{1.2.16}{7}{Globally Lipschitz Function}{theorem.1.2.16}{}} \newlabel{def:1.13@cref}{{[definition][16][1,2]1.2.16}{[1][7][]7}} \newlabel{def:1.14}{{1.2.17}{8}{Locally Lipschitz Function}{theorem.1.2.17}{}} \newlabel{def:1.14@cref}{{[definition][17][1,2]1.2.17}{[1][8][]8}} \citation{da_prato_zabczyk_2002} \@writefile{toc}{\contentsline {subsection}{\numberline {1.2.4}Kolmogorov Equations}{9}{subsection.1.2.4}\protected@file@percent } \newlabel{(1.7)}{{1.17}{9}{Kolmogorov Equation}{equation.1.2.17}{}} \newlabel{(1.7)@cref}{{[equation][17][1,2]1.17}{[1][9][]9}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.2.5}Linear Algebra Notation and Definitions}{10}{subsection.1.2.5}\protected@file@percent } \newlabel{def:1.1.23}{{1.2.25}{11}{Column and Row Notation}{theorem.1.2.25}{}} \newlabel{def:1.1.23@cref}{{[definition][25][1,2]1.2.25}{[1][11][]11}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.2.6}$O$-type Notation and Function Growth}{13}{subsection.1.2.6}\protected@file@percent } \newlabel{1.1.20.1}{{1.2.31.1}{13}{Bounded functions and $O$-type notation}{corollary.1.2.31.1}{}} \newlabel{1.1.20.1@cref}{{[corollary][1][]1.2.31.1}{[1][13][]13}} \newlabel{1.1.20.2}{{1.2.31.2}{14}{}{corollary.1.2.31.2}{}} \newlabel{1.1.20.2@cref}{{[corollary][2][]1.2.31.2}{[1][14][]14}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.2.7}The Concatenation of Vectors \& Functions}{15}{subsection.1.2.7}\protected@file@percent } \newlabel{sum_of_frown_frown_of_sum}{{1.2.34.1}{15}{}{corollary.1.2.34.1}{}} \newlabel{sum_of_frown_frown_of_sum@cref}{{[corollary][1][]1.2.34.1}{[1][15][]15}} \newlabel{concat_fun_fun_concat}{{1.2.35.1}{16}{}{corollary.1.2.35.1}{}} \newlabel{concat_fun_fun_concat@cref}{{[corollary][1][]1.2.35.1}{[1][16][]16}} \newlabel{par_cont}{{1.2.36}{17}{}{theorem.1.2.36}{}} \newlabel{par_cont@cref}{{[lemma][36][1,2]1.2.36}{[1][17][]17}} \@setckpt{Introduction}{ \setcounter{page}{18} \setcounter{equation}{38} \setcounter{enumi}{3} \setcounter{enumii}{0} \setcounter{enumiii}{0} \setcounter{enumiv}{0} \setcounter{footnote}{0} \setcounter{mpfootnote}{0} \setcounter{part}{1} \setcounter{chapter}{1} \setcounter{section}{2} \setcounter{subsection}{7} \setcounter{subsubsection}{0} \setcounter{paragraph}{0} \setcounter{subparagraph}{0} \setcounter{figure}{0} \setcounter{table}{0} \setcounter{@pps}{0} \setcounter{@ppsavesec}{0} \setcounter{@ppsaveapp}{0} \setcounter{AM@survey}{0} \setcounter{parentequation}{0} \setcounter{footdir@label}{0} \setcounter{tmpA}{0} \setcounter{tmpB}{0} \setcounter{tmpC}{0} \setcounter{tmpD}{0} \setcounter{tmpE}{0} \setcounter{tmpF}{0} \setcounter{Hijriday}{0} \setcounter{Hijrimonth}{0} \setcounter{Hijriyear}{0} \setcounter{subfigure}{0} \setcounter{lofdepth}{1} \setcounter{subtable}{0} \setcounter{lotdepth}{1} \setcounter{section@level}{2} \setcounter{Item}{26} \setcounter{Hfootnote}{0} \setcounter{Hy@AnnotLevel}{0} \setcounter{bookmark@seq@number}{11} \setcounter{NAT@ctr}{0} \setcounter{ALG@line}{0} \setcounter{ALG@rem}{0} \setcounter{ALG@nested}{0} \setcounter{ALG@Lnr}{1} \setcounter{ALG@blocknr}{1} \setcounter{ALG@storecount}{0} \setcounter{ALG@tmpcounter}{0} \setcounter{@stackindex}{0} \setcounter{ROWcellindex@}{0} \setcounter{lstnumber}{1} \setcounter{theorem}{36} \setcounter{corollary}{0} \setcounter{lstlisting}{0} }