Adaptive Monte Carlo Algorithms for Stopped Diffusion

Dzougoutov, Anna; Moon, Kyoung-Sook; Schwerin, Erik von; Szepessy, Anders; Tempone, Raúl

doi:10.1007/3-540-26444-2_3

Cited by 16 publications

(14 citation statements)

References 13 publications

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“…[3,4,24]. In this subsection we combine the adaptive multilevel algorithm of Section 2.2 with an error estimate derived in [8] that also takes into account the construction of mesh hierarchy sampling on existing meshes Figure 3. Experimental complexity when the algorithm in Section 2.1 is applied to the drift singularity problem in Section 3.2.…”

Section: Stopped Diffusionmentioning

confidence: 99%

“…[26]. The idea of extending the MLMC method [11] to hierarchies of adaptively refined, non uniform time discretizations that are generated by the adaptive algorithm introduced in [26,25,8] was first introduced and tested computationally by the authors in [17].…”

Section: Introductionmentioning

confidence: 99%

“…The work [11] proposed and analyzed a MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, forward Euler Monte Carlo method from O TOL −3 to O (TOL −1 log(TOL −1 )) 2 for a mean square error of O TOL 2 . Later, the work [17] presented a MLMC method using a hierarchy of adaptively refined, non uniform time discretizations that are generated by the adaptive algorithm introduced in [26,25,8], and, as such, it may be considered a generalization of Giles' work [11]. This work improves the algorithms presented in [17] and furthermore, it provides mathematical analysis of these new adaptive MLMC algorithms.…”

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confidence: 99%

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Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

Hoel

Schwerin

Szepessy

et al. 2013

McMa

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Abstract. We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic differential equations (SDE). The work [11] proposed and analyzed a MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, forward Euler Monte Carlo method from O TOL −3 to O (TOL −1 log(TOL −1 )) 2 for a mean square error of O TOL 2 . Later, the work [17] presented a MLMC method using a hierarchy of adaptively refined, non uniform time discretizations that are generated by the adaptive algorithm introduced in [26,25,8], and, as such, it may be considered a generalization of Giles' work [11]. This work improves the algorithms presented in [17] and furthermore, it provides mathematical analysis of these new adaptive MLMC algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis. Numerical tests include one case with singular drift and one with stopped diffusion, where the complexity of a uniform single level method is O TOL −4 . For both these cases the results confirm the theory, exhibiting savings in the computational cost for achieving the accuracy O (TOL) from O TOL −3 for the adaptive single level algorithm to essentially O TOL −2 log TOL −1 2 for the adaptive MLMC.

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Section: Stopped Diffusionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

Hoel

Schwerin

Szepessy

et al. 2013

McMa

Self Cite

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“…Stopped diffusion is a good example that adaptive time steps improve the convergence rate, see Buchmann and Petersen [9] and Moon et al [14]. A priori error estimates of the time discretization error in (3) were first derived by Talay and Tubaro [41].…”

Section: Convergence Rates For Adaptive Approximation 513mentioning

confidence: 99%

“…which satisfy ik t n = i c j t n X t n k c p t n X t n jp t n+1 + ik c j t n X t n j t n+1 t n < T (14) ik T = ik g X T and ikm t n = i c j t n X t n k c p t n X t n m c r t n X t n jpr t n+1 + im c j t n X t n k c p t n X t n jp t n+1 + i c j t n X t n km c p t n X t n jp t n+1…”

Section: Theorem 22mentioning

confidence: 99%