A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm. (English) Zbl 1213.65016
The authors introduce a new approach to variance reduction for many-query context. That is, problems for which the expectation \(\mathbb{E} Z^\lambda\) of a functional
\[ Z^\lambda= g^\lambda(X_T^\lambda)+\int_0^T f^\lambda(s,X^\lambda_s)\,ds \]
of the solution of an Itô stochastic differential equation
\[ X^\lambda_t = x+\int_0^t b^\lambda(s,X^\lambda_s)\, ds+\int_0^t \sigma^\lambda(s,X^\lambda_s) \,dB_s \]
parametrised by \(\lambda\) has to be evaluated by Monte-Carlo estimations for a very large number of parameter values \(\lambda\). In particular they consider variance reduction by the calculations of control variates using a reduced basis approach. The two algorithms the authors propose split the problem into an offline and online phase. The offline part is an initial stage during which high accuracy control variates are calculated on a low dimensional vector basis which spans a good linear approximation for the whole parameter space. These high accurate control variates are then used in the online phase to allow for a fast calculation of approximations to the control variates for any parameter value.
The two proposed methods differ in the calculation of the highly accurate control variate in the offline phase. In the first algorithm the control variates are calculated using a Monte Carlo method, whereas in the second algorithm the problem is transformed and the basis is obtained by numerically solving the associated backward Kolmogorov equation. It is discussed in detail how these algorithms may be practically implemented and particularly focus on the selection of the parameter values used in the offline phase via a so called greedy procedure. Further, the feasibility of the methods is illustrated by two numerical examples from the applied literature. The first is a scalar valued processes with constant drift and parametrised diffusion (calibration of the Black-Scholes model) and the second a two-dimensional system with constant diffusion and parametrised drift (molecular simulation of dumbbells in polymeric fluids).
Finally, the authors draw conclusions from the performance of the algorithms in the numerical tests and general implementation issues, e.g., memory requirements; however, no theoretical analysis of the algorithms is presented.
\[ Z^\lambda= g^\lambda(X_T^\lambda)+\int_0^T f^\lambda(s,X^\lambda_s)\,ds \]
of the solution of an Itô stochastic differential equation
\[ X^\lambda_t = x+\int_0^t b^\lambda(s,X^\lambda_s)\, ds+\int_0^t \sigma^\lambda(s,X^\lambda_s) \,dB_s \]
parametrised by \(\lambda\) has to be evaluated by Monte-Carlo estimations for a very large number of parameter values \(\lambda\). In particular they consider variance reduction by the calculations of control variates using a reduced basis approach. The two algorithms the authors propose split the problem into an offline and online phase. The offline part is an initial stage during which high accuracy control variates are calculated on a low dimensional vector basis which spans a good linear approximation for the whole parameter space. These high accurate control variates are then used in the online phase to allow for a fast calculation of approximations to the control variates for any parameter value.
The two proposed methods differ in the calculation of the highly accurate control variate in the offline phase. In the first algorithm the control variates are calculated using a Monte Carlo method, whereas in the second algorithm the problem is transformed and the basis is obtained by numerically solving the associated backward Kolmogorov equation. It is discussed in detail how these algorithms may be practically implemented and particularly focus on the selection of the parameter values used in the offline phase via a so called greedy procedure. Further, the feasibility of the methods is illustrated by two numerical examples from the applied literature. The first is a scalar valued processes with constant drift and parametrised diffusion (calibration of the Black-Scholes model) and the second a two-dimensional system with constant diffusion and parametrised drift (molecular simulation of dumbbells in polymeric fluids).
Finally, the authors draw conclusions from the performance of the algorithms in the numerical tests and general implementation issues, e.g., memory requirements; however, no theoretical analysis of the algorithms is presented.
Reviewer: Martin Riedler (Edinburgh)
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 65C05 | Monte Carlo methods |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 82D60 | Statistical mechanics of polymers |
| 91G60 | Numerical methods (including Monte Carlo methods) |
