Bibliography

References grouped by topic. Each citation below is linked from the relevant Mathematical Reference page. Items marked with [*] are the primary references for that topic.

Stochastic Calculus and SDEs

  • Itô, K. (1944). Stochastic integral. Proc. Imperial Academy, 20(8), 519--524. [@ito1944stochastic]
  • Itô, K. (1951). On a formula concerning stochastic differentials. Nagoya Math. J., 3, 55--65. [@ito1951formula] [*]
  • Itô, K. (1951). On stochastic differential equations. AMS Memoirs, 4. [@ito1951stochastic]
  • Stratonovich, R. L. (1966). A new representation for stochastic integrals and equations. SIAM J. Control, 4(2), 362--371. [@stratonovich1966new]
  • Harrison, J. M. & Pliska, S. R. (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications, 11(3), 215--260. [@harrison1981martingales]
  • Kac, M. (1949). On distributions of certain Wiener functionals. Trans. AMS, 65(1), 1--13. [@kac1949distributions]
  • Feynman, R. P. (1948). Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys., 20(2), 367. [@feynman1948space]

Textbooks

  • Shreve, S. E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer. [@shreve2004stochastic] [*]
  • Steele, J. M. (2001). Stochastic Calculus and Financial Applications. Springer. [@steele2001stochastic]
  • Øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications (6th ed.). Springer. [@oksendal2013stochastic] [*]
  • Klebaner, F. C. (2012). Introduction to Stochastic Calculus with Applications. Springer. [@klebaner2012introduction]

Stochastic Optimal Control

Foundational Theory

  • Bellman, R. (1952). On the theory of dynamic programming. PNAS, 38(8), 716--719. [@bellman1952theory] [*]
  • Bellman, R. (1966). Dynamic programming. Science, 153(3731), 34--37. [@bellman1966dynamic]
  • Pontryagin, L. S. (1962). Mathematical Theory of Optimal Processes. [@pontryagin2018mathematical]
  • Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. REStat, 247--257. [@merton1969lifetime]

Controlled Diffusion and HJB

  • Krylov, N. V. (1980). Controlled Diffusion Processes. Springer. [@krylov1980controlled]
  • Fleming, W. H. & Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions. Springer. [@fleming2006controlled] [*]
  • Fleming, W. H. & Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer. [@fleming2012deterministic]
  • Yong, J. & Zhou, X. Y. (1999). Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer. [@yong1999stochastic] [*]
  • Pham, H. (2009). Continuous-Time Stochastic Control and Optimization with Financial Applications. Springer. [@pham2009continuous] [*]
  • Dynkin, E. B. (1965). Markov Processes. Springer. [@dynkin1965markov]
  • Dynkin, E. B. & Yushkevich, A. A. (1979). Controlled Markov Processes. Springer. [@dynkin1979controlled]
  • Bertsekas, D. (2012). Dynamic Programming and Optimal Control, Vol. I. [@bertsekas2012dynamic]
  • Øksendal, B. & Sulem, A. (2009). Applied Stochastic Control of Jump Diffusions (3rd ed.). Springer. [@oksendal2009applied] [*]

Viscosity Solutions

  • Crandall, M. G. & Lions, P.-L. (1983). Viscosity solutions of Hamilton-Jacobi equations. Trans. AMS, 277(1), 1--42. [@crandall1983viscosity] [*]
  • Crandall, M. G., Ishii, H. & Lions, P.-L. (1992). User's guide to viscosity solutions of second order PDEs. Bull. AMS, 27(1), 1--67. [@user2013users] [*]

Reinforcement Learning (Context)

  • Sutton, R. S. & Barto, A. G. (2018). Reinforcement Learning: An Introduction (2nd ed.). MIT Press. [@sutton2018reinforcement]
  • Schulman, J. et al. (2017). Proximal policy optimization algorithms. [@schulman2017proximal]

Market Making

Foundational Models

  • Avellaneda, M. & Stoikov, S. (2008). High-frequency trading in a limit order book. Quantitative Finance, 8(3), 217--224. [@avellaneda2008high] [*]
  • Ho, T. & Stoll, H. R. (1981). Optimal dealer pricing under transactions and return uncertainty. J. of Financial Economics, 9(1), 47--73. [@ho1981optimal] [*]

Extensions and Analysis

  • Guéant, O., Lehalle, C.-A. & Fernandez-Tapia, J. (2013). Dealing with the inventory risk: a solution to the market making problem. Math. Fin. Econ., 7(4), 477--507. [@gueant2013dealing] [*]
  • Guilbaud, F. & Pham, H. (2013). Optimal high-frequency trading with limit and market orders. Quantitative Finance, 13(1), 79--94. [@guilbaud2013optimal]
  • Bayraktar, E. & Ludkovski, M. (2011). Liquidation in limit order books with controlled intensity. Math. Finance, 24(4), 627--650. [@bayraktar2011liquidating]
  • Cartea, Á., Jaimungal, S. & Ricci, J. (2014). Buy low, sell high: A high frequency trading perspective. SIAM J. Financial Math., 5(1), 415--444. [@cartea2014buy]
  • Cartea, Á. & Sánchez-Betancourt, L. (2021). Shadow prices for optimal market making. SIAM J. Financial Math., 12(3). [@cartea2021shadow]
  • Lehalle, C.-A. & Mounjid, O. (2017). Limit order strategic placement with adverse selection risk. Market Microstructure and Liquidity. [@lehalle2017limit]
  • Guéant, O. (2017). Optimal market making. Applied Mathematical Finance, 24(2), 112--138. [@gueant2017optimal]
  • Bergault, P., Evangelista, D., Guéant, O. & Vieira, D. (2021). Closed-form approximations in multi-asset market making. Applied Mathematical Finance, 28(2), 101--126. [@bergault2021closed] [*]
  • Bergault, P. & Guéant, O. (2021). Size matters for OTC market makers: general results and dimensionality reduction techniques. Mathematical Finance, 31(3). [@bergault2021size]

Textbooks and Surveys

  • Cartea, Á., Jaimungal, S. & Penalva, J. (2015). Algorithmic and High-Frequency Trading. Cambridge. [@cartea2015algorithmic] [*]
  • Guéant, O. (2016). The Financial Mathematics of Market Liquidity. CRC Press. [@gueant2016financial]
  • Lehalle, C.-A. & Laruelle, S. (2013). Market Microstructure in Practice. World Scientific. [@lehalle2013market]
  • O'Hara, M. (1995). Market Microstructure Theory. Blackwell. [@ohara1995market]
  • Menkveld, A. J. (2013). High frequency trading and the new market makers. J. Financial Markets, 16(4), 712--740. [@menkveld2013high]
  • Brogaard, J., Hendershott, T. & Riordan, R. (2014). High-frequency trading and price discovery. Rev. Financial Studies, 27(8), 2267--2306. [@brogaard2014high]

Reinforcement Learning Approaches

  • Spooner, T. et al. (2018). Market making via reinforcement learning. AAMAS. [@spooner2018market]
  • Briola, A. et al. (2021). Deep reinforcement learning for active high frequency trading. [@briola2021deep]
  • Qin, X. et al. (2023). EarnHFT: Efficient hierarchical RL for HFT. [@qin2023earnhft]

Hawkes Processes

  • Hawkes, A. G. (2018). Hawkes processes and their applications to finance: a review. Quantitative Finance, 18(2), 193--198. [@hawkes2018hawkes] [*]
  • Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. JASA, 83(401), 9--27. [@ogata1988statistical] [*]
  • Rizoiu, M.-A. et al. (2018). SIR-Hawkes: Linking epidemic models and Hawkes processes. WWW. [@rizoiu2018sir]

Stochastic Volatility Models

  • Heston, S. L. (1993). A closed-form solution for options with stochastic volatility. Rev. Financial Studies, 6(2), 327--343. [@heston1993closed] [*]
  • Lord, R. et al. (2010). A comparison of biased simulation schemes for stochastic volatility models. Quantitative Finance, 10(2), 177--194. [@lord2010comparison]

Credit Risk and Default

Structural and Reduced-Form Models

  • Merton, R. C. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29(2), 449--470. [@merton1974pricing] [*]
  • Jarrow, R. A., Lando, D. & Turnbull, S. M. (1997). A Markov model for the term structure of credit risk spreads. Review of Financial Studies, 10(2), 481--523. [@jarrow1997markov]
  • Bielecki, T. R. & Rutkowski, M. (2004). Credit Risk: Modeling, Valuation and Hedging. Springer. [@bielecki2004credit] [*]

Optimal Investment and Contagion

  • Kraft, H. & Steffensen, M. (2007). Bankruptcy, counterparty risk, and optimal investment. Finance and Stochastics, 11(1), 131--163. [@kraft2007bankruptcy]
  • Capponi, A. & Figueroa-López, J. E. (2014). Dynamic portfolio optimization with a defaultable security and regime-switching. Mathematical Finance, 24(2), 207--249. [@capponi2014dynamic]
  • Bo, L. & Capponi, A. (2016). Optimal investment in credit derivatives portfolio under contagion risk. Mathematical Finance, 26(4), 785--834. [@bo2016optimal]
  • Sircar, R. & Zariphopoulou, T. (2010). Utility valuation of multiname credit derivatives and application to CDOs. Quantitative Finance, 10(2), 195--208. [@sircar2010utility]

Numerical Methods

Finite Difference and HJB Discretization

  • Forsyth, P. A. & Labahn, G. (2007). Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance. J. Computational Finance, 11(2). [@forsyth2007numerical] [*]
  • Achdou, Y. et al. (2013). Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications. Springer. [@achdou2013hamilton]
  • Barles, G. & Souganidis, P. E. (1991). Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Analysis, 4(3), 271--283. [@barles1991convergence]
  • Kushner, H. J. & Dupuis, P. G. (2001). Numerical Methods for Stochastic Control Problems in Continuous Time. Springer. [@kushner2001numerical] [*]
  • Howard, R. A. (1960). Dynamic Programming and Markov Processes. MIT Press. [@howard1960dynamic]
  • Courant, R., Friedrichs, K. & Lewy, H. (1928). Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann., 100, 32--74. [@courant1928partiellen]
  • Rannacher, R. (1984). Finite element solution of diffusion problems with irregular data. Numerische Math., 43(2), 309--327. [@rannacher1984finite]

BSDE Theory and Numerics

  • Pardoux, E. & Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Systems & Control Letters, 14(1), 55--61. [@pardoux1990adapted] [*]
  • Pardoux, E. & Peng, S. (2005). BSDEs and quasilinear parabolic PDEs. Lecture Notes in Control and Inf. Sci., 176, 200--217. [@pardoux2005backward]
  • El Karoui, N., Peng, S. & Quenez, M. C. (1997). Backward stochastic differential equations in finance. Mathematical Finance, 7(1), 1--71. [@el1997backward]
  • Ma, J., Protter, P. & Yong, J. (1994). Solving forward-backward SDEs explicitly---a four step scheme. Probab. Theory Rel. Fields, 98(3), 339--359. [@ma1994solving]
  • Ma, J. & Yong, J. (1999). Forward-Backward Stochastic Differential Equations and Their Applications. Springer. [@ma1999forward]
  • Gobet, E., Lemor, J.-P. & Warin, X. (2005). A regression-based Monte Carlo method to solve BSDEs. Annals of Applied Probability, 15(3), 2172--2202. [@gobet2005empirical] [*]

Deep Learning and High-Dimensional PDEs

  • Han, J., Jentzen, A. & E, W. (2018). Solving high-dimensional PDEs using deep learning. PNAS, 115(34), 8505--8510. arXiv:1707.02568 [@han2018solving]
  • Han, J. & Jentzen, A. (2017). Deep learning-based numerical methods for high-dimensional parabolic PDEs and BSDEs. arXiv:1706.04702 [@han2017deep]
  • Al-Aradi, A. et al. (2022). Extensions of the deep Galerkin method. [@al2022extensions]
  • Raissi, M., Perdikaris, P. & Karniadakis, G. E. (2019). Physics-informed neural networks. J. Computational Physics, 378, 686--707. arXiv:1711.10561 [@raissi2019physics]
  • Karniadakis, G. E. et al. (2021). Physics-informed machine learning. Nature Reviews Physics, 3(6), 422--440. [@karniadakis2021physics]
  • Song, Y. et al. (2021). Score-based generative modeling through SDEs. ICLR. arXiv:2011.13456 [@song2020score]
  • Sirignano, J. & Spiliopoulos, K. (2018). DGM: A deep learning algorithm for solving partial differential equations. Journal of Computational Physics, 375, 1339--1364. arXiv:1708.07469 [@sirignano2018dgm] [*]
  • Cheridito, P., Dupret, J.-L. & Hainaut, D. (2025). Deep learning for continuous-time stochastic control with jumps. NeurIPS. arXiv:2505.15602 [@cheridito2025jumps]

Neural Operators and Architectures

  • Vaswani, A. et al. (2017). Attention is all you need. NeurIPS. arXiv:1706.03762 [@vaswani2017attention]
  • Amos, B., Xu, L. & Kolter, J. Z. (2017). Input convex neural networks. ICML. arXiv:1609.07152 [@amos2017input]
  • Zaheer, M. et al. (2017). Deep sets. NeurIPS. arXiv:1703.06114 [@zaheer2017deep]
  • Lee, J. et al. (2019). Set transformer: A framework for attention-based permutation-invariant neural networks. ICML. arXiv:1810.00825 [@lee2019set]
  • Li, Z. et al. (2021). Fourier neural operator for parametric partial differential equations. ICLR. arXiv:2010.08895 [@li2021fourier]
  • Lu, L., Jin, P., Pang, G., Zhang, Z. & Karniadakis, G. E. (2021). Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature Machine Intelligence, 3, 218--229. arXiv:1910.03193 [@lu2021deeponet]
  • Kovachki, N. et al. (2023). Neural operator: Learning maps between function spaces with applications to PDEs. Journal of Machine Learning Research, 24(89), 1--97. arXiv:2108.08481 [@kovachki2023neural]

Group Equivariance and Hamiltonian Architectures

  • Noether, E. (1918). Invariante Variationsprobleme. Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl., 235--257. [@noether1918invariante]
  • Olver, P. J. (1993). Applications of Lie Groups to Differential Equations (2nd ed.). Springer. [@olver1993applications]
  • Cohen, T. & Welling, M. (2016). Group equivariant convolutional networks. ICML. arXiv:1602.07576 [@cohen2016group]
  • Weiler, M. & Cesa, G. (2019). General E(2)-equivariant steerable CNNs. NeurIPS. arXiv:1911.08251 [@weiler2019general]
  • Thomas, N., Smidt, T., Kearnes, S., Yang, L., Li, L., Kohlhoff, K. & Riley, P. (2018). Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds. arXiv:1802.08219 [@thomas2018tensor]
  • Fuchs, F., Worrall, D., Fischer, V. & Welling, M. (2020). SE(3)-transformers: 3D roto-translation equivariant attention networks. NeurIPS. arXiv:2006.10503 [@fuchs2020se3]
  • Satorras, V. G., Hoogeboom, E. & Welling, M. (2021). E(n)-equivariant graph neural networks. ICML. arXiv:2102.09844 [@satorras2021en]
  • Finzi, M., Stanton, S., Izmailov, P. & Wilson, A. G. (2020). Generalizing convolutional neural networks for equivariance to Lie groups on arbitrary continuous data. ICML. arXiv:2002.12880 [@finzi2020lie]
  • Worrall, D. & Welling, M. (2019). Deep scale-spaces: Equivariance over scale. NeurIPS. arXiv:1905.11697 [@worrall2019scale]
  • Greydanus, S., Dzamba, M. & Yosinski, J. (2019). Hamiltonian neural networks. NeurIPS. arXiv:1906.01563 [@greydanus2019hamiltonian]
  • Cranmer, M., Greydanus, S., Hoyer, S., Battaglia, P., Sperberg-McQueen, D. & Ho, S. (2020). Lagrangian neural networks. ICLR Deep Differential Equations Workshop. arXiv:2003.04630 [@cranmer2020lagrangian]
  • Bronstein, M. M., Bruna, J., Cohen, T. & Velickovic, P. (2021). Geometric deep learning: Grids, groups, graphs, geodesics, and gauges. arXiv:2104.13478 [@bronstein2021geometric] ^bronstein2021geometric
  • Wang, R., Walters, R. & Yu, R. (2022). Approximately equivariant networks for imperfectly symmetric dynamics. ICML. arXiv:2201.11969 [@wang2022approximately]

Neural Operators for Optimal Control

  • Hwang, R., Lee, J. Y., Shin, J. Y. & Hwang, H. J. (2021). Solving PDE-constrained control problems using operator learning. AAAI, 36(4). arXiv:2111.04941 [@hwang2021solving]
  • Wang, S., Bhouri, M. A. & Perdikaris, P. (2021). Fast PDE-constrained optimization via self-supervised operator learning. arXiv:2110.13297 [@wang2021fast]
  • Lanthaler, S. & Stuart, A. M. (2023). The parametric complexity of operator learning. arXiv:2306.15924 [@lanthaler2023parametric] [*]
  • Lee, J. Y. & Kim, Y. (2024). Hamilton-Jacobi based policy-iteration via deep operator learning. arXiv:2406.10920 [@lee2024hamilton] [*]
  • Hoischen, N., Bevanda, P., Sosnowski, S., Hirche, S. & Houska, B. (2024). Data-driven stochastic optimal control in reproducing kernel Hilbert spaces. arXiv:2407.16407 [@hoischen2024datadriven]
  • Kratsios, A., Neufeld, A. & Schmocker, P. (2025). Generative neural operators of log-complexity can simultaneously solve infinitely many convex programs. arXiv:2508.14995 [@kratsios2025generative]
  • Xu, W., Han, J. & Lai, R. (2025). Self-supervised amortized neural operators for optimal control: Scaling laws and applications. arXiv:2512.24897 [@xu2025amortized]
  • Cohen, S. N., de Feo, F., Hebner, J. & Sirignano, J. (2026). Deep Hilbert-Galerkin methods for infinite-dimensional PDEs and optimal control. arXiv:2603.19463 [@cohen2026hilbert]
  • Kratsios, A., Livieri, G. & Schmocker, P. (2026). NeuralChaos: Optimal adapted approximation of square integrable predictable processes. arXiv:2607.14361 [@kratsios2026neuralchaos]
  • Gao, S., Zhou, M. & Lai, R. (2026). Self-supervised in-context operator learning for stochastic mean-field control. arXiv:2608.18282 [@gao2026selfsupervised]
  • Mohanty, S. K. (2026). Explainable artificial intelligence for financial integral equations: A fixed-point neural operator approach. arXiv:2604.27127 [@mohanty2026explainable]

Monte Carlo Methods

  • Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer. [@glasserman2004monte]

General Numerical Analysis

  • Quarteroni, A., Sacco, R. & Saleri, F. (2007). Numerical Mathematics. Springer. [@quarteroni2007numerical]

Financial Mathematics (Broader Context)

  • Black, F. & Scholes, M. (1973). The pricing of options and corporate liabilities. JPE, 81(3), 637--654. [@black1973pricing]
  • Schwartz, E. S. (1997). The stochastic behavior of commodity prices. J. Finance, 52(3), 923--973. [@schwartz1997stochastic]
  • Vasicek, O. (1977). An equilibrium characterization of the term structure. J. Financial Economics, 5(2), 177--188. [@vasicek1977equilibrium]
  • Sharpe, W. F. (1966). Mutual fund performance. J. Business, 39(1), 119--138. [@sharpe1966mutual]
  • Sortino, F. A. & Price, L. N. (1994). Performance measurement in a downside risk framework. J. Investing, 3(3), 59--64. [@sortino1994performance]

Auxiliary

  • Gould, J. P. (1968). Adjustment costs in the theory of investment of the firm. REStud, 35(1), 47--55. [@gould1968adjustment]
  • Abel, A. B. & Eberly, J. C. (1993). A unified model of investment under uncertainty. NBER WP 4296. [@abel1993unified]
  • Gode, D. K. & Sunder, S. (1993). Allocative efficiency of markets with ZI traders. JPE, 101(1), 119--137. [@gode1993allocative]
  • Klabnik, S. & Nichols, C. (2018). The Rust Programming Language. No Starch Press. [@klabnik2023rust]
  • Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. J. Basic Eng., 82(1), 35--45. [@kalman1960new]