2. An adaptive random bit multilevel algorithm for SDEs
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Abstract
We study the approximation of expectations E(f (X)) for solutions X of stochastic differential equations and functionals f on the path space by means of Monte Carlo algorithms that only use random bits instead of random numbers. We construct an adaptive random bit multilevel algorithm, which is based on the Euler scheme, the Lévy-Ciesielski representation of the Brownian motion, and asymptotically optimal random bit approximations of the standard normal distribution. We numerically compare this algorithm with the adaptive classical multilevel Euler algorithm for a geometric Brownian motion, an Ornstein-Uhlenbeck process, and a Cox-Ingersoll-Ross process.
Abstract
We study the approximation of expectations E(f (X)) for solutions X of stochastic differential equations and functionals f on the path space by means of Monte Carlo algorithms that only use random bits instead of random numbers. We construct an adaptive random bit multilevel algorithm, which is based on the Euler scheme, the Lévy-Ciesielski representation of the Brownian motion, and asymptotically optimal random bit approximations of the standard normal distribution. We numerically compare this algorithm with the adaptive classical multilevel Euler algorithm for a geometric Brownian motion, an Ornstein-Uhlenbeck process, and a Cox-Ingersoll-Ross process.
Chapters in this book
- Frontmatter I
- Preface: Multivariate algorithms and information-based complexity V
- Contents IX
- 1. The control variate integration algorithm for multivariate functions defined at scattered data points 1
- 2. An adaptive random bit multilevel algorithm for SDEs 15
- 3. RBF-based penalized least-squares approximation of noisy scattered data on the sphere 33
- 4. On the power of random information 43
- 5. Optimality criteria for probabilistic numerical methods 65
- 6. ε-Superposition and truncation dimensions, and multivariate method for∞-variate linear problems 89
- 7. Adaptive approximation for multivariate linear problems with inputs lying in a cone 109
Chapters in this book
- Frontmatter I
- Preface: Multivariate algorithms and information-based complexity V
- Contents IX
- 1. The control variate integration algorithm for multivariate functions defined at scattered data points 1
- 2. An adaptive random bit multilevel algorithm for SDEs 15
- 3. RBF-based penalized least-squares approximation of noisy scattered data on the sphere 33
- 4. On the power of random information 43
- 5. Optimality criteria for probabilistic numerical methods 65
- 6. ε-Superposition and truncation dimensions, and multivariate method for∞-variate linear problems 89
- 7. Adaptive approximation for multivariate linear problems with inputs lying in a cone 109
