121 lines
5.0 KiB
Plaintext
121 lines
5.0 KiB
Plaintext
\begin{thebibliography}{}
|
||
|
||
\bibitem[Bass, 2011]{bass_2011}
|
||
Bass, R.~F. (2011).
|
||
\newblock {\em Brownian Motion}, page 6–12.
|
||
\newblock Cambridge Series in Statistical and Probabilistic Mathematics.
|
||
Cambridge University Press.
|
||
|
||
\bibitem[Beck et~al., 2021a]{Beck_2021}
|
||
Beck, C., Gonon, L., Hutzenthaler, M., and Jentzen, A. (2021a).
|
||
\newblock On existence and uniqueness properties for solutions of stochastic
|
||
fixed point equations.
|
||
\newblock {\em Discrete \& Continuous Dynamical Systems - B}, 26(9):4927.
|
||
|
||
\bibitem[Beck et~al., 2021b]{BHJ21}
|
||
Beck, C., Hutzenthaler, M., and Jentzen, A. (2021b).
|
||
\newblock On nonlinear {Feynman}{\textendash}{Kac} formulas for viscosity
|
||
solutions of semilinear parabolic partial differential equations.
|
||
\newblock {\em Stochastics and Dynamics}, 21(08).
|
||
|
||
\bibitem[Beck et~al., 2021c]{bhj20}
|
||
Beck, C., Hutzenthaler, M., and Jentzen, A. (2021c).
|
||
\newblock On nonlinear feynman–kac formulas for viscosity solutions of
|
||
semilinear parabolic partial differential equations.
|
||
\newblock {\em Stochastics and Dynamics}, 21(08):2150048.
|
||
|
||
\bibitem[Crandall et~al., 1992]{crandall_lions}
|
||
Crandall, M.~G., Ishii, H., and Lions, P.-L. (1992).
|
||
\newblock User’s guide to viscosity solutions of second order partial
|
||
differential equations.
|
||
\newblock {\em Bull. Amer. Math. Soc.}, 27(1):1--67.
|
||
|
||
\bibitem[Da~Prato and Zabczyk, 2002]{da_prato_zabczyk_2002}
|
||
Da~Prato, G. and Zabczyk, J. (2002).
|
||
\newblock {\em Second Order Partial Differential Equations in Hilbert Spaces}.
|
||
\newblock London Mathematical Society Lecture Note Series. Cambridge University
|
||
Press.
|
||
|
||
\bibitem[Durrett, 2019]{durrett2019probability}
|
||
Durrett, R. (2019).
|
||
\newblock {\em Probability: Theory and Examples}.
|
||
\newblock Cambridge Series in Statistical and Probabilistic Mathematics.
|
||
Cambridge University Press.
|
||
|
||
\bibitem[Golub and Van~Loan, 2013]{golub2013matrix}
|
||
Golub, G. and Van~Loan, C. (2013).
|
||
\newblock {\em Matrix Computations}.
|
||
\newblock Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins
|
||
University Press.
|
||
|
||
\bibitem[Grohs et~al., 2018]{grohsetal}
|
||
Grohs, P., Hornung, F., Jentzen, A., and von Wurstemberger, P. (2018).
|
||
\newblock {A proof that artificial neural networks overcome the curse of
|
||
dimensionality in the numerical approximation of Black-Scholes partial
|
||
differential equations}.
|
||
\newblock Papers 1809.02362, arXiv.org.
|
||
|
||
\bibitem[Grohs et~al., 2023]{grohs2019spacetime}
|
||
Grohs, P., Hornung, F., Jentzen, A., and Zimmermann, P. (2023).
|
||
\newblock Space-time error estimates for deep neural network approximations for
|
||
differential equations.
|
||
\newblock {\em Advances in Computational Mathematics}, 49(1):4.
|
||
|
||
\bibitem[Grohs et~al., 2022]{Grohs_2022}
|
||
Grohs, P., Jentzen, A., and Salimova, D. (2022).
|
||
\newblock Deep neural network approximations for solutions of {PDEs} based on
|
||
monte carlo algorithms.
|
||
\newblock {\em Partial Differential Equations and Applications}, 3(4).
|
||
|
||
\bibitem[Gy{\"o}ngy and Krylov, 1996]{Gyngy1996ExistenceOS}
|
||
Gy{\"o}ngy, I. and Krylov, N.~V. (1996).
|
||
\newblock Existence of strong solutions for {It\^o}'s stochastic equations via
|
||
approximations.
|
||
\newblock {\em Probability Theory and Related Fields}, 105:143--158.
|
||
|
||
\bibitem[Hutzenthaler et~al., 2020a]{hutzenthaler_overcoming_2020}
|
||
Hutzenthaler, M., Jentzen, A., Kruse, T., Anh~Nguyen, T., and von
|
||
Wurstemberger, P. (2020a).
|
||
\newblock Overcoming the curse of dimensionality in the numerical approximation
|
||
of semilinear parabolic partial differential equations.
|
||
\newblock {\em Proceedings of the Royal Society A: Mathematical, Physical and
|
||
Engineering Sciences}, 476(2244):20190630.
|
||
|
||
\bibitem[Hutzenthaler et~al., 2021]{hutzenthaler_strong_2021}
|
||
Hutzenthaler, M., Jentzen, A., Kuckuck, B., and Padgett, J.~L. (2021).
|
||
\newblock Strong {$L^p$}-error analysis of nonlinear {Monte} {Carlo}
|
||
approximations for high-dimensional semilinear partial differential
|
||
equations.
|
||
\newblock Technical Report arXiv:2110.08297, arXiv.
|
||
\newblock arXiv:2110.08297 [cs, math] type: article.
|
||
|
||
\bibitem[Hutzenthaler et~al., 2020b]{hjw2020}
|
||
Hutzenthaler, M., Jentzen, A., and von Wurstemberger~Wurstemberger (2020b).
|
||
\newblock {Overcoming the curse of dimensionality in the approximative pricing
|
||
of financial derivatives with default risks}.
|
||
\newblock {\em Electronic Journal of Probability}, 25(none):1 -- 73.
|
||
|
||
\bibitem[It\^o, 1942a]{Ito1942a}
|
||
It\^o, K. (1942a).
|
||
\newblock Differential equations determining {Markov} processes (original in
|
||
{Japanese}).
|
||
\newblock {\em Zenkoku Shijo Sugaku Danwakai}, 244(1077):1352--1400.
|
||
|
||
\bibitem[It\^o, 1942b]{Ito1946}
|
||
It\^o, K. (1942b).
|
||
\newblock On a stochastic integral equation.
|
||
\newblock {\em Proc. Imperial Acad. Tokyo}, 244(1077):1352--1400.
|
||
|
||
\bibitem[Karatzas and Shreve, 1991]{karatzas1991brownian}
|
||
Karatzas, I. and Shreve, S. (1991).
|
||
\newblock {\em Brownian Motion and Stochastic Calculus}.
|
||
\newblock Graduate Texts in Mathematics (113) (Book 113). Springer New York.
|
||
|
||
\bibitem[Rio, 2009]{rio_moment_2009}
|
||
Rio, E. (2009).
|
||
\newblock Moment {Inequalities} for {Sums} of {Dependent} {Random} {Variables}
|
||
under {Projective} {Conditions}.
|
||
\newblock {\em J Theor Probab}, 22(1):146--163.
|
||
|
||
\end{thebibliography}
|