Paper: arXiv 2505.18297

Authors: Kristoffer Andersson, Alessandro Gnoatto, Camilo Andrés García Trillos

Abstract

We present the first deep-learning solver for backward stochastic Volterra integral equations (BSVIEs) and their fully-coupled forward-backward variants. The method trains a neural network to approximate the two solution fields in a single stage, avoiding the use of nested time-stepping cycles that limit classical algorithms. For the decoupled case we prove a non-asymptotic error bound composed of an a posteriori residual plus the familiar square root dependence on the time step. Numerical experiments are consistent with this rate and reveal two key properties: \emph{scalability}, in the sense that accuracy remains stable from low dimension up to 500 spatial variables while GPU batching keeps wall-clock time nearly constant; and \emph{generality}, since the same method handles coupled systems whose forward dynamics depend on the backward solution. These results open practical access to a family of high-dimensional, time-inconsistent problems in stochastic control and quantitative finance.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents the first deep-learning solver for backward stochastic Volterra integral equations (BSVIEs) with proven non-asymptotic error bounds. It combines advanced stochastic analysis and measure theory with modern deep-learning algorithms, showing impressive scalability up to 500 spatial dimensions.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve High-Dimensional BSVIEs & Forward-Backward Variants] --> B[Input: Coupled/Decoupled BSVIEs & Time-Inconsistent Dynamics]
    B --> C[Methodology: Single-Stage Neural Network Optimization]
    C --> D[Computational Process: GPU-Batched Loss Minimization for Dual Solution Fields]
    D --> E[Error Analysis: Proved Non-Asymptotic Error Bound O_sqrt_dt]
    E --> F[Key Outcomes: Scalable to 500D with Stable Accuracy & Constant Wall-Clock Time]

Cited References (171)

  1. Neural Network for CVA : Learning Future Values (2017) — link
  2. A numerical scheme for BSVIE s (2016) — link
  3. Backward stochastic Volterra integral equations and some related problems (2006) — link
  4. Well-posedness and regularity of backward stochastic Volterra integral equations (2008) — link
  5. A C lass of Time-Inconsistent R isk M easures and B ackward Stochastic V olterra I ntegral E quations (2013) — link
  6. Continuous-time dynamic risk measures by backward stochastic Volterra integral equations (2007)
  7. Recursive utility processes, dynamic risk measures and quadratic backward stochastic Volterra integral equations (2021)
  8. Fully dynamic risk measures: Horizon risk, time-consistency, and relations with BSDE s and BSVIE s (2024)
and 163 more
  1. Time-inconsistent stochastic optimal control problems and backward stochastic Volterra integral equations (2021)
  2. Closed-loop equilibrium strategies for general time-inconsistent optimal control problems (2021)
  3. Optimal controls for forward-backward stochastic differential equations: Time-inconsistency and time-consistent solutions (2024)
  4. Well-posedness and regularity of backward stochastic Volterra integral equations (2008)
  5. Backward stochastic Volterra integral equations—a brief survey (2013)
  6. Backward stochastic Volterra integral equations and some related problems (2006)
  7. Adapted solution of a backward stochastic nonlinear Volterra integral equation (2002)
  8. Time-inconsistent optimal control problems and the equilibrium HJB equation (2012)
  9. Affine Volterra Processes (2019) — link
  10. Volterra Equations Driven by Rough Signals (2021) — link
  11. Integrated structural approach to Credit Value Adjustment (2019) — link
  12. Partial Differential Equation Representations of Derivatives with Bilateral Counterparty Risk and Funding Costs (2011)
  13. In the balance (2011)
  14. Cross Currency Valuation and Hedging in the Multiple Curve Framework (2021) — link
  15. Approximations for adapted M -solutions of Type-II backward stochastic Volterra integral equations (2023)
  16. A generalized coupling approach for the weak approximation of stochastic functional differential equations (2024) — link
  17. Discretization of backward stochastic Volterra integral equations (2012)
  18. A NOTE ON THE SELF-FINANCING CONDITION FOR FUNDING, COLLATERAL AND DISCOUNTING (2015) — link
  19. A note on the self-financing condition for funding, collateral and discounting (2015)
  20. Arbitrage-free bilateral counterparty risk valuation under collteralization and application to credit default swaps (2014) — link
  21. Numerical S olution of S tochastic D ifferential E quations (1992) — link
  22. Arbitrage-free valuation of bilateral counterparty risk for interest-rate products: Impact of volatilities and correlations (2011)
  23. Stochastic differential equations: an introduction with applications (2013)
  24. The deep multi-FBSDE method: a robust deep learning method for coupled FBSDEs — → 2503.13193
  25. Risk Neutral Pricing of Counterparty Risk
  26. Counterparty Risk in Derivatives and Collateral Policies: The Replicating Portfolio Approach
  27. Swap Rates and Credit Quality (1996) — link
  28. Nonlinear valuation with XVAs: two converging approaches (2022)
  29. Funding, collateral and hedging: uncovering the mechanics and the subtleties of funding valuation adjustments (2012) — link
  30. Funding valuation adjustment: a consistent framework including cva, dva, collateral, netting rules and re-hypothecation (2011) — link
  31. When Capital Is a Funding Source: The XVA Anticipated BSDE s (2020)
  32. Arbitrage-Free XVA (2018)
  33. Analysis of non-linear approximated value equation under multiple risk factors and stochastic intensities (2023)
  34. Backward stochastic differential equations (2017)
  35. A review of tree-based approaches to solve forward-backward stochastic differential equations (2021)
  36. Nonlinear consistent valuation of CCP cleared or CSA bilateral trades with initial margins under credit, funding and wrong-way risks (2014)
  37. Dangers of Bilateral Counterparty Risk: the fundamental impact of closeout conventions (2010) — link
  38. Modeling Default Risk (2002)
  39. Numerical method for backward stochastic differential equations (2002)
  40. Statistical Learning of Value-at-Risk and Expected Shortfall (2024)
  41. Regression quantiles (1978)
  42. Learning value-at-risk and expected shortfall (2022) — link
  43. Approximation error analysis of some deep backward schemes for nonlinear PDE s (2022)
  44. A posteriori error estimates for fully coupled McKean–Vlasov forward-backward SDEs (2024)
  45. Adaptive Multilevel Stochastic Approximation of the Value-at-Risk — → 2408.06531
  46. A \emph F ourier cosine method for an efficient computation of solutions to BSDE s (2015)
  47. Efficient numerical \emph F ourier methods for coupled forward–backward \emph SDE s (2016)
  48. Numerical \emph F ourier method and second-order Taylor scheme for backward \emph SDE s in finance (2016)
  49. Solving forward-backward stochastic differential equations explicitly - \emph A four step scheme (1994)
  50. Error expansion for the discretization of backward stochastic differential equations (2007)
  51. Solving \emph BSDE with adaptive control variate (2010)
  52. On multilevel \emph P icard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations (2019)
  53. A forward scheme for backward \emph SDE s (2007)
  54. Discrete-time approximation and \emph M onte-\emph C arlo simulation of backward stochastic differential equations (2004)
  55. Stochastic grid bundling method for backward stochastic differential equations (2019)
  56. A probabilistic numerical method for fully nonlinear parabolic \emph PDE s (2011)
  57. Least-squares \emph M onte \emph C arlo for backward \emph SDE s (2012)
  58. On the \emph M onte \emph C arlo simulation of \emph BSDE s: \emph A n improvement on the \emph M alliavin weights (2010)
  59. Stratified regression \emph M onte-\emph C arlo scheme for semilinear \emph PDE s and \emph BSDE s with large scale parallelization on \emph GPU s (2016)
  60. Counterparty risk and funding: immersion and beyond (2016) — link
  61. XVA analysis from the balance sheet (2021)
  62. An Explicit Scheme for Pathwise XVA Computations (2024) — link
  63. Pathwise CVA regressions with oversimulated defaults (2023)
  64. On the existence and uniqueness of solutions of stochastic integral equations of the Volterra type (1979)
  65. Forward–backward stochastic neural networks: deep learning of high-dimensional partial differential equations (2024)
  66. Social discounting and the long rate of interest (2018)
  67. Numerical methods for backward stochastic differential equations: \emph A survey (2021) — link
  68. Numerical methods for backward stochastic differential equations: A survey (2023)
  69. Deep backward schemes for high-dimensional nonlinear \emph PDE s (2020)
  70. Solving high-dimensional partial differential equations using deep learning (2018)
  71. Deep x\emph VA solver–\emph A neural network based counterparty credit risk management framework (2023) — link
  72. Convergence of the backward deep BSDE method with applications to optimal stopping problems (2023)
  73. Generalized convergence of the deep BSDE method: a step towards fully-coupled FBSDE s and applications in stochastic control (2024) — link
  74. A multistep scheme to solve backward stochastic differential equations for option pricing on GPUs (2021)
  75. Backward stochastic differential equations and quasilinear parabolic partial differential equations (2005)
  76. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations (1991)
  77. Bilateral counterparty risk under funding constraints—Part I: Pricing (2015)
  78. Bilateral counterparty risk under funding constraints—Part II: CVA (2015)
  79. Collateralization, Nonlinear Valuation Under and Risk, Credit (2016)
  80. Positive XVAs (2022)
  81. A unified approach to xVA with CSA discounting and initial margin (2021)
  82. XVA Principle, Nested Monte Carlo Strategies, and GPU Optimizations (2018) — link
  83. Mild to classical solutions for XVA equations under stochastic volatility (2024)
  84. Nonlinear valuation under credit, funding, and margins: Existence, uniqueness, invariance, and disentanglement (2019)
  85. A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations — → 2404.08456
  86. A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations — → 2404.08456
  87. A numerical F ourier cosine expansion method with higher order Taylor schemes for fully coupled FBSDE s (2025) — link
  88. Convergence of the deep \emph BSDE method for coupled \emph FBSDE s (2020)
  89. Convergence of the Deep \emph BSDE method for \emph FBSDE s with non-\emph L ipschitz coefficients (2021) — link
  90. A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations (2020)
  91. Analysis of the generalization error: \emph E mpirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black–Scholes partial differential equations (2020)
  92. D\emph NN expression rate analysis of high-dimensional \emph PDE s: Application to option pricing (2021)
  93. A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of B lack– S choles partial differential equations (2023) — link
  94. A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of K olmogorov partial differential equations with constant diffusion and nonlinear drift coefficients (2021)
  95. Solving the \emph K olmogorov \emph PDE by means of deep learning (2021)
  96. Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations (2019)
  97. Deep splitting method for parabolic \emph PDE s (2021)
  98. Asymptotic expansion as prior knowledge in deep learning method for high dimensional \emph BSDE s (2019)
  99. Deep primal-dual algorithm for \emph BSDE s: Applications of machine learning to \emph CVA and \emph IM (2017)
  100. Forward-backward stochastic neural networks: \emph D eep learning of high-dimensional partial differential equations (2018) — link
  101. Three algorithms for solving high-dimensional fully coupled \emph FBSDE s through deep learning (2020)
  102. Learning \emph L ocomotion \emph C ontrollers for \emph W alking \emph U sing \emph D eep \emph FBSDE (2021) — link
  103. Machine learning for semi linear \emph PDE s (2019)
  104. Some machine learning schemes for high-dimensional nonlinear \emph PDE s (2019) — link
  105. The One Step M alliavin scheme: new discretization of BSDE s implemented with deep learning regressions (2024)
  106. A novel pricing method for E uropean options based on F ourier-cosine series expansions (2009)
  107. Convergence of a robust deep FBSDE method for stochastic control (2023)
  108. An overview on deep learning-based approximation methods for partial differential equations (2023)
  109. Counterparty credit risk in a multivariate structural model with jumps (2015)
  110. Deep learning for CVA computations of large portfolios of financial derivatives (2021)
  111. A deep learning approach for computations of exposure profiles for high-dimensional Bermudan options (2021)
  112. Learning deep stochastic optimal control policies using forward-backward \emph SDE s (2019) — link
  113. Deep \emph FBSDE \emph C ontroller for \emph A ttitude \emph C ontrol of \emph H ypersonic \emph A ircraft (2021)
  114. Strong approximations of BSDE s in a domain (2009)
  115. Time discretization and M arkovian iteration for coupled FBSDE s (2008)
  116. Convergence of the M arkovian iteration for coupled FBSDE s via a differentiation approach — → 2504.02814
  117. Pricing barrier options with deep BSDE s (2020) — link
  118. Barrier options and Greeks: Modeling with neural networks (2023)
  119. A deep neural network algorithm for semilinear elliptic PDEs with applications in insurance mathematics (2020)
  120. A control method for solving high-dimensional Hamiltonian systems through deep neural networks (2021) — link
  121. Deep \emph BSDE-ML \emph L earning and \emph I ts \emph A pplication to \emph M odel-\emph F ree \emph O ptimal \emph C ontrol (2022) — link
  122. Deep learning approximation for stochastic control problems (2016)
  123. Backward forward stochastic differential equations (1993)
  124. Characterization of bistability for stochastic multistep methods (2012)
  125. Two-sided error estimates for the stochastic theta method (2010)
  126. Convergence of the Euler scheme for a class of stochastic differential equation (2002)
  127. Stochastic differential equations (1992)
  128. Rate of convergence of the Euler approximation for diffusion processes (1991)
  129. Weak rate of convergence of the \emph E uler–\emph M aruyama scheme for stochastic differential equations with non-regular drift (2017)
  130. Introduction to stochastic control theory (2012)
  131. Approximations by superpositions of a sigmoidal function (1989)
  132. Activation functions: Comparison of trends in practice and research for deep learning (2018) — link
  133. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates (1974)
  134. Temporal difference learning for high-dimensional PIDEs with jumps (2024)
  135. Deep learning algorithms for FBSDE s with jumps: Applications to option pricing and a MFG model for smart grids (2024) — link
  136. A deep solver for BSDE s with jumps (2025)
  137. Valuing A merican options by simulation: a simple least-squares approach (2001)
  138. On the pricing of corporate debt: The risk structure of interest rates (1974)
  139. Valuing corporate securities: Some effects of bond indenture provisions (1976)
  140. Valuation adjustments: A balanced perspective (2011)
  141. Counterparty Risk and Funding: A Tale of Two Puzzles (2015)
  142. Counterparty risk and funding: A tale of two puzzles (2014)
  143. XVA: Theory, Implementation, and Practice (2019)
  144. A change-of-measure approach for wrong-way risk (2018)
  145. Disentangling wrong-way risk: pricing CVA via change of measures and drift adjustment (2018) — link
  146. Augmenting Structural Credit Models with Stochastic Recovery (2022)
  147. Deep learning-based numerical methods for high-dimensional partial differential equations (2017)
  148. Deep BSDE methods for high-dimensional nonlinear PDEs: A review (2019)
  149. Solving Coupled Nonlinear Forward-backward Stochastic Differential Equations: An Optimization Perspective with Backward Measurability Loss (2023) — link
  150. Basel III: A Global Regulatory Framework for More Resilient Banks and Banking Systems (2010)
  151. Probabilistic Theory of Mean Field Games with Applications I (2018)
  152. Deep optimal stopping (2019)
  153. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations (2021)
  154. Convergence of a Deep BSDE solver with jumps — → 2501.09727
  155. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations (2019)
  156. Deep backward schemes for high-dimensional nonlinear PDEs (2020)
  157. $\Gamma$-convergence and optimal control problems (1982)
  158. Variational analysis (2009)
  159. On the time discretization of stochastic optimal control problems: the dynamic programming approach (2019)
  160. Adam: A method for stochastic optimization (2014) — link
  161. Backward deep BSDE methods and applications to nonlinear problems (2020) — link
  162. Deep Learning algorithms for solving high dimensional nonlinear Backward Stochastic Differential Equations (2020) — link
  163. The Malliavin Calculus and Related Topics (2006)

6 of 171 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.