TY - GEN

T1 - Construction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carlo

AU - Hoel, Hakon

AU - Häppölä, Juho

AU - Tempone, Raul

N1 - KAUST Repository Item: Exported on 2020-10-01

PY - 2016/6/14

Y1 - 2016/6/14

N2 - A formal mean square error expansion (MSE) is derived for Euler-Maruyama numerical solutions of stochastic differential equations (SDE). The error expansion is used to construct a pathwise, a posteriori, adaptive time-stepping Euler-Maruyama algorithm for numerical solutions of SDE, and the resulting algorithm is incorporated into a multilevel Monte Carlo (MLMC) algorithm for weak approximations of SDE. This gives an efficient MSE adaptive MLMC algorithm for handling a number of low-regularity approximation problems. In low-regularity numerical example problems, the developed adaptive MLMC algorithm is shown to outperform the uniform time-stepping MLMC algorithm by orders of magnitude, producing output whose error with high probability is bounded by TOL > 0 at the near-optimal MLMC cost rate б(TOL log(TOL)) that is achieved when the cost of sample generation is б(1).

AB - A formal mean square error expansion (MSE) is derived for Euler-Maruyama numerical solutions of stochastic differential equations (SDE). The error expansion is used to construct a pathwise, a posteriori, adaptive time-stepping Euler-Maruyama algorithm for numerical solutions of SDE, and the resulting algorithm is incorporated into a multilevel Monte Carlo (MLMC) algorithm for weak approximations of SDE. This gives an efficient MSE adaptive MLMC algorithm for handling a number of low-regularity approximation problems. In low-regularity numerical example problems, the developed adaptive MLMC algorithm is shown to outperform the uniform time-stepping MLMC algorithm by orders of magnitude, producing output whose error with high probability is bounded by TOL > 0 at the near-optimal MLMC cost rate б(TOL log(TOL)) that is achieved when the cost of sample generation is б(1).

UR - http://hdl.handle.net/10754/622138

UR - http://link.springer.com/10.1007/978-3-319-33507-0_2

UR - http://www.scopus.com/inward/record.url?scp=84977600217&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-33507-0_2

DO - 10.1007/978-3-319-33507-0_2

M3 - Conference contribution

SN - 9783319335056

SP - 29

EP - 86

BT - Springer Proceedings in Mathematics & Statistics

PB - Springer Nature

ER -