Paper: arXiv 2311.08826

Authors: Akihiro Kaneko

Abstract

Numerical methods for computing the solutions of Markov backward stochastic differential equations (BSDEs) driven by continuous-time Markov chains (CTMCs) are explored. The main contributions of this paper are as follows: (1) we observe that Euler-Maruyama temporal discretization methods for solving Markov BSDEs driven by CTMCs are equivalent to exponential integrators for solving the associated systems of ordinary differential equations (ODEs); (2) we introduce multi-stage Euler-Maruyama methods for effectively solving “stiff” Markov BSDEs driven by CTMCs; these BSDEs typically arise from the spatial discretization of Markov BSDEs driven by Brownian motion; (3) we propose a multilevel spatial discretization method on sparse grids that efficiently approximates high-dimensional Markov BSDEs driven by Brownian motion with a combination of multiple Markov BSDEs driven by CTMCs on grids with different resolutions. We also illustrate the effectiveness of the presented methods with a number of numerical experiments in which we treat nonlinear BSDEs arising from option pricing problems in finance.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents high theoretical and mathematical complexity by linking BSDEs driven by CTMCs to systems of ODEs via multi-stage Euler-Maruyama integrators and sparse grids. However, empirical rigor is limited primarily to stylized numerical experiments on option pricing models rather than real market data backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Solve High-Dim & Stiff Markov BSDEs Driven by CTMCs"] --> B["Data/Inputs: Nonlinear Financial BSDEs (Option Pricing Models)"]
    B --> C["Observed Equivalence: Euler-Maruyama Discretization = Exponential Integrators for ODE Systems"]
    C --> D["Method 1: Multi-Stage Euler-Maruyama Methods (Handles Stiffness from Spatial Discretization)"]
    C --> E["Method 2: Multilevel Spatial Discretization on Sparse Grids (Approximates High-Dim Brownian BSDEs)"]
    D --> F["Computational Validation: Numerical Experiments"]
    E --> F
    F --> G["Key Outcome: Efficient & Accurate Numerical Framework for Stiff & High-Dim BSDEs"]

Cited References (50)

  1. Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations (1996)
  2. Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory (2017)
  3. Linear Quadratic Optimal Stochastic Control with Random Coefficients (1976) — link
  4. An Introductory Approach to Duality in Optimal Stochastic Control (1978) — link
  5. Sparse grids (2004) — link
  6. Undiscounted Markov chain BSDEs to stopping times (2014) — link
  7. Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions (2010) — link
  8. Solutions of backward stochastic differential equations on Markov chains (2008) — link
and 42 more
  1. Ergodic BSDEs Driven by Markov Chains (2013) — link
  2. On Markovian solutions to Markov Chain BSDEs (2012) — link
  3. Computational Science and Engineering (2007)
  4. Continuous-time random walks for the numerical solution of stochastic differential equations (2018)
  5. Continuous-Time Markov Chain and Regime Switching Approximations with Applications to Options Pricing (2019) — link
  6. Differential and Integral Inequalities (2012)
  7. Nonlinear reserving in life insurance: Aggregation and mean-field approximation (2016) — link
  8. Backward Stochastic Differential Equations in Finance (1997) — link
  9. Error analysis of finite difference and Markov chain approximations for option pricing (2018) — link
  10. Exponential Time Differencing for Stiff Systems (2002) — link
  11. EXPINT—A MATLAB Package for Exponential Integrators (2007) — link
  12. A regression-based Monte Carlo method to solve backward stochastic differential equations (2005) — link
  13. A combination technique for the solution of sparse grid problems (1992)
  14. Managing smile risk (2002)
  15. Solving ordinary differential equations II (1996)
  16. The Heston stochastic-local volatility model: Efficient Monte Carlo simulation (2014) — link
  17. Hidden Markov Models: Estimation and Control (1995)
  18. Stiffness of ODEs (1993) — link
  19. Exponential integrators (2010) — link
  20. Explicit Exponential Runge–Kutta Methods for Semilinear Parabolic Problems (2005) — link
  21. Hyp Hyp Hooray (2007) — link
  22. Differential E quations.jl–a performant and feature-rich ecosystem for solving differential equations in J ulia (2017)
  23. Julia: A Fresh Approach to Numerical Computing (2017) — link
  24. A general continuous time Markov chain approximation for multi-asset option pricing with systems of correlated diffusions (2020) — link
  25. Generalized Runge-Kutta processes for stable systems with large Lipschitz constants (1967) — link
  26. Least-Squares Monte Carlo for Backward SDEs (2012) — link
  27. Valuing American options by simulation: a simple least-squares approach (2001) — link
  28. Simulation of multidimensional diffusions with sticky boundaries via Markov chain approximation (2023) — link
  29. Continuously monitored barrier options under Markov processes (2013)
  30. Non-uniform grids for PDE in finance (2000)
  31. Numerical Methods for Ordinary Differential Equations (2003)
  32. Adapted solution of a backward stochastic differential equation (1990) — link
  33. Plots.jl – A User Extendable Plotting API for the Julia Programming Language (2023) — link
  34. Pricing multi-asset options with sparse grids (2008)
  35. Sparse Grid Combination Technique for Hagan SABR/LIBOR Market Model (2017) — link
  36. Data mining with sparse grids (2001) — link
  37. A General Valuation Framework for SABR and Stochastic Local Volatility Models (2018) — link
  38. Quadrature and interpolation formulas for tensor products of certain classes of functions (1963)
  39. Stochastic Calculus and Applications (2015)
  40. Pricing Financial Instruments: The Finite Difference Method (2000)
  41. Topology (2000)
  42. Spectra and Pseudospectra (2005) — link

0 of 50 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.