dissertation_work/Dissertation/Dissertation.aux

276 lines
16 KiB
TeX
Raw Permalink Normal View History

2024-02-17 01:32:15 +00:00
\relax
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\@writefile{toc}{\contentsline {chapter}{\numberline {1}Introduction}{4}{chapter.1}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {1.1}Notation and Definitions}{4}{section.1.1}\protected@file@percent }
\newlabel{1.9}{{1.1.11}{6}{Strong Solution of Stochastic Differential Equation}{theorem.1.1.11}{}}
\newlabel{1.9@cref}{{[definition][11][1,1]1.1.11}{[1][6][]6}}
\newlabel{1.5}{{1.1.5}{6}{Strong Solution of Stochastic Differential Equation}{equation.1.1.5}{}}
\newlabel{1.5@cref}{{[equation][5][1,1]1.1.5}{[1][6][]6}}
\newlabel{def:1.13}{{1.1.13}{7}{Lipschitz Function}{theorem.1.1.13}{}}
\newlabel{def:1.13@cref}{{[definition][13][1,1]1.1.13}{[1][7][]7}}
\citation{da_prato_zabczyk_2002}
\newlabel{def:1.14}{{1.1.14}{8}{Locally Lipschitz Function}{theorem.1.1.14}{}}
\newlabel{def:1.14@cref}{{[definition][14][1,1]1.1.14}{[1][8][]8}}
\newlabel{(1.7)}{{1.1.13}{8}{Kolmogorov Equation}{equation.1.1.13}{}}
\newlabel{(1.7)@cref}{{[equation][13][1,1]1.1.13}{[1][8][]8}}
\@writefile{toc}{\contentsline {chapter}{\numberline {2}Brownian Motion Monte Carlo}{10}{chapter.2}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {2.1}Brownian Motion Preliminaries}{10}{section.2.1}\protected@file@percent }
\newlabel{def:1.17}{{2.1.5}{11}{Of $\mathfrak {k}$}{theorem.2.1.5}{}}
\newlabel{def:1.17@cref}{{[definition][5][2,1]2.1.5}{[1][11][]11}}
\newlabel{def:1.18}{{2.1.6}{11}{Primary Setting}{theorem.2.1.6}{}}
\newlabel{def:1.18@cref}{{[definition][6][2,1]2.1.6}{[1][11][]11}}
\newlabel{Setting 1.1}{{2.1.6}{11}{Primary Setting}{theorem.2.1.6}{}}
\newlabel{Setting 1.1@cref}{{[definition][6][2,1]2.1.6}{[1][11][]11}}
\newlabel{(1.12)}{{2.1.2}{11}{Primary Setting}{equation.2.1.2}{}}
\newlabel{(1.12)@cref}{{[equation][2][2,1]2.1.2}{[1][11][]11}}
\newlabel{(1.12)}{{2.1.3}{11}{Primary Setting}{equation.2.1.3}{}}
\newlabel{(1.12)@cref}{{[equation][3][2,1]2.1.3}{[1][11][]11}}
\newlabel{(1.14)}{{2.1.4}{11}{Primary Setting}{equation.2.1.4}{}}
\newlabel{(1.14)@cref}{{[equation][4][2,1]2.1.4}{[1][11][]11}}
\newlabel{lemma1.1}{{2.1.7}{11}{}{theorem.2.1.7}{}}
\newlabel{lemma1.1@cref}{{[lemma][7][2,1]2.1.7}{[1][11][]11}}
\citation{hutzenthaler_overcoming_2020}
\newlabel{lem:1.20}{{2.1.8}{12}{}{theorem.2.1.8}{}}
\newlabel{lem:1.20@cref}{{[lemma][8][2,1]2.1.8}{[1][12][]12}}
\newlabel{(1.16)}{{2.1.6}{12}{Brownian Motion Preliminaries}{equation.2.1.6}{}}
\newlabel{(1.16)@cref}{{[equation][6][2,1]2.1.6}{[1][12][]12}}
\newlabel{eq:1.4}{{2.1}{12}{Brownian Motion Preliminaries}{equation.2.1.6}{}}
\newlabel{eq:1.4@cref}{{[section][1][2]2.1}{[1][12][]12}}
\newlabel{(1.17)}{{2.1.7}{12}{Brownian Motion Preliminaries}{equation.2.1.7}{}}
\newlabel{(1.17)@cref}{{[equation][7][2,1]2.1.7}{[1][12][]12}}
\newlabel{(1.18)}{{2.1.8}{13}{Brownian Motion Preliminaries}{equation.2.1.8}{}}
\newlabel{(1.18)@cref}{{[equation][8][2,1]2.1.8}{[1][12][]13}}
\newlabel{(1.19)}{{2.1.9}{13}{Brownian Motion Preliminaries}{equation.2.1.9}{}}
\newlabel{(1.19)@cref}{{[equation][9][2,1]2.1.9}{[1][13][]13}}
\newlabel{(1.20)}{{2.1.10}{13}{Brownian Motion Preliminaries}{equation.2.1.10}{}}
\newlabel{(1.20)@cref}{{[equation][10][2,1]2.1.10}{[1][13][]13}}
\newlabel{(1.21)}{{2.1.11}{13}{Brownian Motion Preliminaries}{equation.2.1.11}{}}
\newlabel{(1.21)@cref}{{[equation][11][2,1]2.1.11}{[1][13][]13}}
\newlabel{cor:1.20.1}{{2.1.8.1}{13}{}{corollary.2.1.8.1}{}}
\newlabel{cor:1.20.1@cref}{{[corollary][1][]2.1.8.1}{[1][13][]13}}
\citation{rio_moment_2009}
\citation{rio_moment_2009}
\@writefile{toc}{\contentsline {section}{\numberline {2.2}Monte Carlo Approximations}{14}{section.2.2}\protected@file@percent }
\newlabel{lem:1.21}{{2.2.1}{14}{}{theorem.2.2.1}{}}
\newlabel{lem:1.21@cref}{{[lemma][1][2,2]2.2.1}{[1][14][]14}}
\citation{grohsetal}
\newlabel{corollary:1.11.1.}{{2.2.1.1}{15}{}{corollary.2.2.1.1}{}}
\newlabel{corollary:1.11.1.@cref}{{[corollary][1][]2.2.1.1}{[1][15][]15}}
\newlabel{(1.26)}{{2.2.5}{15}{}{equation.2.2.5}{}}
\newlabel{(1.26)@cref}{{[equation][5][2,2]2.2.5}{[1][15][]15}}
\newlabel{cor:1.22.2}{{2.2.1.2}{15}{}{corollary.2.2.1.2}{}}
\newlabel{cor:1.22.2@cref}{{[corollary][2][]2.2.1.2}{[1][15][]15}}
\@writefile{toc}{\contentsline {section}{\numberline {2.3}Bounds and Covnvergence}{15}{section.2.3}\protected@file@percent }
\newlabel{lem:1.21}{{2.3.1}{15}{}{theorem.2.3.1}{}}
\newlabel{lem:1.21@cref}{{[lemma][1][2,3]2.3.1}{[1][15][]15}}
\newlabel{lem:1.22}{{2.3.2}{16}{}{theorem.2.3.2}{}}
\newlabel{lem:1.22@cref}{{[lemma][2][2,3]2.3.2}{[1][16][]16}}
\newlabel{lem:1.25}{{2.3.3}{18}{}{theorem.2.3.3}{}}
\newlabel{lem:1.25@cref}{{[lemma][3][2,3]2.3.3}{[1][18][]18}}
\newlabel{(1.46)}{{2.3.16}{18}{Bounds and Covnvergence}{equation.2.3.16}{}}
\newlabel{(1.46)@cref}{{[equation][16][2,3]2.3.16}{[1][18][]18}}
\newlabel{(1.47)}{{2.3.17}{18}{Bounds and Covnvergence}{equation.2.3.17}{}}
\newlabel{(1.47)@cref}{{[equation][17][2,3]2.3.17}{[1][18][]18}}
\newlabel{(1.48)}{{2.3.18}{18}{Bounds and Covnvergence}{equation.2.3.18}{}}
\newlabel{(1.48)@cref}{{[equation][18][2,3]2.3.18}{[1][18][]18}}
\newlabel{cor:1.25.1}{{2.3.3.1}{19}{}{corollary.2.3.3.1}{}}
\newlabel{cor:1.25.1@cref}{{[corollary][1][]2.3.3.1}{[1][18][]19}}
\newlabel{(2.48)}{{2.3.26}{20}{}{equation.2.3.26}{}}
\newlabel{(2.48)@cref}{{[equation][26][2,3]2.3.26}{[1][20][]20}}
\newlabel{2.3.27}{{2.3.27}{20}{}{equation.2.3.27}{}}
\newlabel{2.3.27@cref}{{[equation][27][2,3]2.3.27}{[1][20][]20}}
\newlabel{2.3.29}{{2.3.28}{20}{Bounds and Covnvergence}{equation.2.3.28}{}}
\newlabel{2.3.29@cref}{{[equation][28][2,3]2.3.28}{[1][20][]20}}
\citation{hjw2020}
\citation{hjw2020}
\citation{hjw2020}
\newlabel{2.3.34}{{2.3.34}{23}{Bounds and Covnvergence}{equation.2.3.34}{}}
\newlabel{2.3.34@cref}{{[equation][34][2,3]2.3.34}{[1][22][]23}}
\citation{crandall_lions}
\citation{Beck_2021}
\citation{Ito1942a}
\citation{Ito1946}
\citation{Beck_2021}
\citation{BHJ21}
\@writefile{toc}{\contentsline {chapter}{\numberline {3}That $u$ is a viscosity solution}{24}{chapter.3}\protected@file@percent }
\@writefile{lof}{\addvspace {10\p@ }}
\@writefile{lot}{\addvspace {10\p@ }}
\@writefile{toc}{\contentsline {section}{\numberline {3.1}Some Preliminaries}{24}{section.3.1}\protected@file@percent }
\newlabel{lemma:2.7}{{3.1.1}{24}{}{theorem.3.1.1}{}}
\newlabel{lemma:2.7@cref}{{[lemma][1][3,1]3.1.1}{[1][24][]24}}
\citation{karatzas1991brownian}
\citation{karatzas1991brownian}
\newlabel{lem:3.4}{{3.1.2}{26}{}{theorem.3.1.2}{}}
\newlabel{lem:3.4@cref}{{[lemma][2][3,1]3.1.2}{[1][26][]26}}
\newlabel{2.13}{{3.1.7}{26}{}{equation.3.1.7}{}}
\newlabel{2.13@cref}{{[equation][7][3,1]3.1.7}{[1][26][]26}}
\newlabel{2.14}{{3.1.8}{26}{}{equation.3.1.8}{}}
\newlabel{2.14@cref}{{[equation][8][3,1]3.1.8}{[1][26][]26}}
\citation{da_prato_zabczyk_2002}
\citation{karatzas1991brownian}
\newlabel{(2.19)}{{3.1.10}{27}{Some Preliminaries}{equation.3.1.10}{}}
\newlabel{(2.19)@cref}{{[equation][10][3,1]3.1.10}{[1][27][]27}}
\citation{da_prato_zabczyk_2002}
\@writefile{toc}{\contentsline {section}{\numberline {3.2}Viscosity Solutions}{28}{section.3.2}\protected@file@percent }
\newlabel{sumofusc}{{3.2.3.1}{28}{}{corollary.3.2.3.1}{}}
\newlabel{sumofusc@cref}{{[corollary][1][]3.2.3.1}{[1][28][]28}}
\newlabel{neglsc}{{3.2.3.2}{29}{}{corollary.3.2.3.2}{}}
\newlabel{neglsc@cref}{{[corollary][2][]3.2.3.2}{[1][29][]29}}
\newlabel{negdegel}{{3.2.6}{30}{}{theorem.3.2.6}{}}
\newlabel{negdegel@cref}{{[lemma][6][3,2]3.2.6}{[1][30][]30}}
\newlabel{def:viscsubsolution}{{3.2.7}{30}{Viscosity subsolutions}{theorem.3.2.7}{}}
\newlabel{def:viscsubsolution@cref}{{[definition][7][3,2]3.2.7}{[1][30][]30}}
\newlabel{def:viscsupsolution}{{3.2.8}{30}{Viscosity supersolutions}{theorem.3.2.8}{}}
\newlabel{def:viscsupsolution@cref}{{[definition][8][3,2]3.2.8}{[1][30][]30}}
\newlabel{def:viscsolution}{{3.2.9}{31}{Viscosity solution}{theorem.3.2.9}{}}
\newlabel{def:viscsolution@cref}{{[definition][9][3,2]3.2.9}{[1][31][]31}}
\newlabel{maxviscosity}{{3.2.10}{31}{}{theorem.3.2.10}{}}
\newlabel{maxviscosity@cref}{{[lemma][10][3,2]3.2.10}{[1][31][]31}}
\newlabel{ungeq0}{{3.2.11}{32}{}{theorem.3.2.11}{}}
\newlabel{ungeq0@cref}{{[lemma][11][3,2]3.2.11}{[1][32][]32}}
\newlabel{limitofun}{{3.2.13}{32}{}{equation.3.2.13}{}}
\newlabel{limitofun@cref}{{[equation][13][3,2]3.2.13}{[1][32][]32}}
\newlabel{hessungeq0}{{3.2.15}{32}{}{equation.3.2.15}{}}
\newlabel{hessungeq0@cref}{{[equation][15][3,2]3.2.15}{[1][32][]32}}
\newlabel{phieps}{{3.2.16}{32}{Viscosity Solutions}{equation.3.2.16}{}}
\newlabel{phieps@cref}{{[equation][16][3,2]3.2.16}{[1][32][]32}}
\newlabel{maxphiu}{{3.2.20}{33}{Viscosity Solutions}{equation.3.2.20}{}}
\newlabel{maxphiu@cref}{{[equation][20][3,2]3.2.20}{[1][33][]33}}
\newlabel{unleq0}{{3.2.11.1}{34}{}{corollary.3.2.11.1}{}}
\newlabel{unleq0@cref}{{[corollary][1][]3.2.11.1}{[1][34][]34}}
\newlabel{limitofun}{{3.2.27}{35}{}{equation.3.2.27}{}}
\newlabel{limitofun@cref}{{[equation][27][3,2]3.2.27}{[1][35][]35}}
\newlabel{viscsolutionvn}{{3.2.28}{35}{}{equation.3.2.28}{}}
\newlabel{viscsolutionvn@cref}{{[equation][28][3,2]3.2.28}{[1][35][]35}}
\newlabel{hessungeq0}{{3.2.29}{35}{}{equation.3.2.29}{}}
\newlabel{hessungeq0@cref}{{[equation][29][3,2]3.2.29}{[1][35][]35}}
\newlabel{hgeq0}{{3.2.30}{35}{Viscosity Solutions}{equation.3.2.30}{}}
\newlabel{hgeq0@cref}{{[equation][30][3,2]3.2.30}{[1][35][]35}}
\newlabel{unneq0}{{3.2.11.2}{36}{}{corollary.3.2.11.2}{}}
\newlabel{unneq0@cref}{{[corollary][2][]3.2.11.2}{[1][36][]36}}
\newlabel{absq}{{3.2.12}{36}{}{theorem.3.2.12}{}}
\newlabel{absq@cref}{{[lemma][12][3,2]3.2.12}{[1][36][]36}}
\citation{karatzas1991brownian}
\newlabel{ugoesto0}{{3.2.13}{37}{}{theorem.3.2.13}{}}
\newlabel{ugoesto0@cref}{{[lemma][13][3,2]3.2.13}{[1][37][]37}}
\newlabel{limsupis0}{{3.2.39}{37}{}{equation.3.2.39}{}}
\newlabel{limsupis0@cref}{{[equation][39][3,2]3.2.39}{[1][37][]37}}
\newlabel{xnasintuvxn}{{3.2.40}{37}{}{equation.3.2.40}{}}
\newlabel{xnasintuvxn@cref}{{[equation][40][3,2]3.2.40}{[1][37][]37}}
\newlabel{lipformun}{{3.2.42}{37}{Viscosity Solutions}{equation.3.2.42}{}}
\newlabel{lipformun@cref}{{[equation][42][3,2]3.2.42}{[1][37][]37}}
\newlabel{expofxisbounded}{{3.2.43}{38}{Viscosity Solutions}{equation.3.2.43}{}}
\newlabel{expofxisbounded@cref}{{[equation][43][3,2]3.2.43}{[1][37][]38}}
\newlabel{mathcalxn-mathclx0}{{3.2.44}{38}{Viscosity Solutions}{equation.3.2.44}{}}
\newlabel{mathcalxn-mathclx0@cref}{{[equation][44][3,2]3.2.44}{[1][38][]38}}
\newlabel{2.19}{{3.2.14}{39}{}{theorem.3.2.14}{}}
\newlabel{2.19@cref}{{[lemma][14][3,2]3.2.14}{[1][39][]39}}
\newlabel{2.59}{{3.2.50}{40}{}{equation.3.2.50}{}}
\newlabel{2.59@cref}{{[equation][50][3,2]3.2.50}{[1][40][]40}}
\newlabel{2.60}{{3.2.51}{40}{}{equation.3.2.51}{}}
\newlabel{2.60@cref}{{[equation][51][3,2]3.2.51}{[1][40][]40}}
\newlabel{2.62}{{3.2.53}{40}{Viscosity Solutions}{equation.3.2.53}{}}
\newlabel{2.62@cref}{{[equation][53][3,2]3.2.53}{[1][40][]40}}
\newlabel{2.63}{{3.2.54}{40}{Viscosity Solutions}{equation.3.2.54}{}}
\newlabel{2.63@cref}{{[equation][54][3,2]3.2.54}{[1][40][]40}}
\newlabel{2.64}{{3.2.55}{40}{Viscosity Solutions}{equation.3.2.55}{}}
\newlabel{2.64@cref}{{[equation][55][3,2]3.2.55}{[1][40][]40}}
\citation{BHJ21}
\citation{Beck_2021}
\newlabel{2.65}{{3.2.56}{41}{Viscosity Solutions}{equation.3.2.56}{}}
\newlabel{2.65@cref}{{[equation][56][3,2]3.2.56}{[1][40][]41}}
\newlabel{2.66}{{3.2.57}{41}{Viscosity Solutions}{equation.3.2.57}{}}
\newlabel{2.66@cref}{{[equation][57][3,2]3.2.57}{[1][41][]41}}
\newlabel{ungn}{{3.2.58}{41}{Viscosity Solutions}{equation.3.2.58}{}}
\newlabel{ungn@cref}{{[equation][58][3,2]3.2.58}{[1][41][]41}}
\newlabel{u0gn}{{3.2.59}{41}{Viscosity Solutions}{equation.3.2.59}{}}
\newlabel{u0gn@cref}{{[equation][59][3,2]3.2.59}{[1][41][]41}}
\newlabel{2.70}{{3.2.61}{41}{Viscosity Solutions}{equation.3.2.61}{}}
\newlabel{2.70@cref}{{[equation][61][3,2]3.2.61}{[1][41][]41}}
\newlabel{unu0}{{3.2.62}{42}{Viscosity Solutions}{equation.3.2.62}{}}
\newlabel{unu0@cref}{{[equation][62][3,2]3.2.62}{[1][42][]42}}
\newlabel{2.73}{{3.2.64}{42}{Viscosity Solutions}{equation.3.2.64}{}}
\newlabel{2.73@cref}{{[equation][64][3,2]3.2.64}{[1][42][]42}}
\newlabel{2.74}{{3.2.65}{42}{Viscosity Solutions}{equation.3.2.65}{}}
\newlabel{2.74@cref}{{[equation][65][3,2]3.2.65}{[1][42][]42}}
\newlabel{2.76}{{3.2.67}{43}{Viscosity Solutions}{equation.3.2.67}{}}
\newlabel{2.76@cref}{{[equation][67][3,2]3.2.67}{[1][42][]43}}
\newlabel{2.79}{{3.2.71}{43}{}{equation.3.2.71}{}}
\newlabel{2.79@cref}{{[equation][71][3,2]3.2.71}{[1][43][]43}}
\newlabel{2.81}{{3.2.73}{44}{}{equation.3.2.73}{}}
\newlabel{2.81@cref}{{[equation][73][3,2]3.2.73}{[1][44][]44}}
\newlabel{2.82}{{3.2.74}{44}{Viscosity Solutions}{equation.3.2.74}{}}
\newlabel{2.82@cref}{{[equation][74][3,2]3.2.74}{[1][44][]44}}
\newlabel{2.84}{{3.2.76}{44}{Viscosity Solutions}{equation.3.2.76}{}}
\newlabel{2.84@cref}{{[equation][76][3,2]3.2.76}{[1][44][]44}}
\citation{Beck_2021}
\citation{Beck_2021}
\newlabel{2.86}{{3.2.78}{45}{Viscosity Solutions}{equation.3.2.78}{}}
\newlabel{2.86@cref}{{[equation][78][3,2]3.2.78}{[1][44][]45}}
\newlabel{2.88}{{3.2.80}{45}{Viscosity Solutions}{equation.3.2.80}{}}
\newlabel{2.88@cref}{{[equation][80][3,2]3.2.80}{[1][45][]45}}
\newlabel{2.89}{{3.2.81}{45}{Viscosity Solutions}{equation.3.2.81}{}}
\newlabel{2.89@cref}{{[equation][81][3,2]3.2.81}{[1][45][]45}}
\newlabel{2.90}{{3.2.83}{46}{Viscosity Solutions}{equation.3.2.83}{}}
\newlabel{2.90@cref}{{[equation][83][3,2]3.2.83}{[1][46][]46}}
\newlabel{2.93}{{3.2.85}{46}{Viscosity Solutions}{equation.3.2.85}{}}
\newlabel{2.93@cref}{{[equation][85][3,2]3.2.85}{[1][46][]46}}
\newlabel{2.95}{{3.2.87}{47}{Viscosity Solutions}{equation.3.2.87}{}}
\newlabel{2.95@cref}{{[equation][87][3,2]3.2.87}{[1][46][]47}}
\@writefile{toc}{\contentsline {section}{\numberline {3.3}Solutions, characterization, and computational bounds to the Kolmogorov backward equations}{47}{section.3.3}\protected@file@percent }
\newlabel{thm:3.21}{{3.3.1}{47}{Existence and characterization of $u_d$}{theorem.3.3.1}{}}
\newlabel{thm:3.21@cref}{{[theorem][1][3,3]3.3.1}{[1][47][]47}}
\newlabel{3.102}{{3.3.2}{47}{Existence and characterization of $u_d$}{equation.3.3.2}{}}
\newlabel{3.102@cref}{{[equation][2][3,3]3.3.2}{[1][47][]47}}
\newlabel{lem:3.19}{{3.3.1.1}{48}{}{corollary.3.3.1.1}{}}
\newlabel{lem:3.19@cref}{{[corollary][1][]3.3.1.1}{[1][48][]48}}
\newlabel{3.3.7}{{3.3.7}{48}{}{equation.3.3.7}{}}
\newlabel{3.3.7@cref}{{[equation][7][3,3]3.3.7}{[1][48][]48}}
\newlabel{3.3.12}{{3.3.12}{49}{Solutions, characterization, and computational bounds to the Kolmogorov backward equations}{equation.3.3.12}{}}
\newlabel{3.3.12@cref}{{[equation][12][3,3]3.3.12}{[1][49][]49}}
\newlabel{3.3.13}{{3.3.13}{49}{Solutions, characterization, and computational bounds to the Kolmogorov backward equations}{equation.3.3.13}{}}
\newlabel{3.3.13@cref}{{[equation][13][3,3]3.3.13}{[1][49][]49}}
\citation{Beck_2021}
\newlabel{3.2.21}{{3.3.16}{51}{Solutions, characterization, and computational bounds to the Kolmogorov backward equations}{equation.3.3.16}{}}
\newlabel{3.2.21@cref}{{[equation][16][3,3]3.3.16}{[1][51][]51}}
\citation{*}
\bibdata{Dissertation.bib}
\bibcite{bass_2011}{{1}{2011}{{Bass}}{{}}}
\bibcite{Beck_2021}{{2}{2021a}{{Beck et~al.}}{{}}}
\bibcite{BHJ21}{{3}{2021b}{{Beck et~al.}}{{}}}
\bibcite{crandall_lions}{{4}{1992}{{Crandall et~al.}}{{}}}
\bibcite{da_prato_zabczyk_2002}{{5}{2002}{{Da~Prato and Zabczyk}}{{}}}
\bibcite{durrett2019probability}{{6}{2019}{{Durrett}}{{}}}
\bibcite{grohsetal}{{7}{2020}{{Grohs et~al.}}{{}}}
\bibcite{Gyngy1996ExistenceOS}{{8}{1996}{{Gy{\"o}ngy and Krylov}}{{}}}
\bibcite{hutzenthaler_overcoming_2020}{{9}{2020a}{{Hutzenthaler et~al.}}{{}}}
\bibcite{hutzenthaler_strong_2021}{{10}{2021}{{Hutzenthaler et~al.}}{{}}}
\bibcite{hjw2020}{{11}{2020b}{{Hutzenthaler et~al.}}{{}}}
\bibcite{Ito1942a}{{12}{1942a}{{It\^o}}{{}}}
\bibcite{Ito1946}{{13}{1942b}{{It\^o}}{{}}}
\bibcite{karatzas1991brownian}{{14}{1991}{{Karatzas and Shreve}}{{}}}
\bibcite{rio_moment_2009}{{15}{2009}{{Rio}}{{}}}
\bibstyle{apalike}
\@writefile{toc}{\contentsline {chapter}{Appendices}{55}{section*.3}\protected@file@percent }
\gdef \@abspage@last{58}