Simulation of stopped diffusions

Buchmann, F.M.

doi:10.1016/j.jcp.2004.07.009

Cited by 21 publications

(18 citation statements)

References 26 publications

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“…The problem of simulating stopped diffusion has also been studied in e.g. [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.…”

Section: Stopped Diffusionmentioning

confidence: 99%

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%

Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

Hoel

Schwerin

Szepessy

et al. 2013

McMa

View full text Add to dashboard Cite

show abstract

“…Using exact SSA to sample the next reaction then requires numerically integrating the stochastic differential equation until the integral value crosses the given threshold for the first time. This is related to determining the first passage time of a random walk, and techniques such as those presented by Buchmann might need to be adopted [42]. This would result in a fully stochastic, yet still hybrid continuous-discrete simulation algorithm.…”

Section: Discussionmentioning

confidence: 99%

Exact stochastic simulations of intra-cellular transport by mechanically coupled molecular motors

Helmuth

Reboux

Sbalzarini

2011

Journal of Computational Science

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Numerous processes in live cells depend on active, motor-driven transport of cargo and organelles along the filaments of the cytoskeleton. Understanding the resulting dynamics and the underlying biophysical and biochemical processes critically depends on computational models of intra-cellular transport. A number of motorcargo models have hence been developed to reproduce experimentally observed transport dynamics on various levels of detail. Computer simulations of these models have so far exclusively relied on approximate time-discretization methods. Using a consensus motorcargo model that unites several existing models from the literature we demonstrate that this simulation approach is not correct. The numerical errors do not vanish even for arbitrarily small time steps, rendering the algorithm inconsistent. We propose a novel exact simulation algorithm for intra-cellular transport models that is also computationally more efficient than the approximate one. Furthermore, we introduce a robust way of analyzing the different time scales in the model dynamics using velocity autocorrelation functions. AbstractNumerous processes in live cells depend on active, motor-driven transport of cargo and organelles along the filaments of the cytoskeleton. Understanding the resulting dynamics and the underlying biophysical and biochemical processes critically depends on computational models of intra-cellular transport. A number of motor-cargo models have hence been developed to reproduce experimentally observed transport dynamics on various levels of detail. Computer simulations of these models have so far exclusively relied on approximate time-discretization methods. Using a consensus motor-cargo model that unites several existing models from the literature we demonstrate that this simulation approach is not correct. The numerical errors do not vanish even for arbitrarily small time steps, rendering the algorithm inconsistent. We propose a novel exact simulation algorithm for intra-cellular transport models that is also computationally more efficient than the approximate one. Furthermore, we introduce a robust way of analyzing the different time scales in the model dynamics using velocity autocorrelation functions.

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“…This is due to the possibility that the process may attain the barrier and come back, within the time step. Approximating the continuous sample paths of Brownian motion using discrete random walks gives the values only at the beginning and the end of the time step and so we have no information about the behaviour of the continuous process during the time step (Buchmann, 2005).…”

Section: Brownian Bridge Techniquementioning

confidence: 99%

“…For more analysis on using this technique in barrier options, we refer the reader to (Glasserman, 2003;Gobet, 2009;Baldi, 1995;Moon, 2008) and for other applications, we refer to (Alzubaidi and Shardlow, 2014) for neuroscience models and (Mannella, 1999;Jansons and Lythe, 2003;Buchmann, 2005) for physical systems.…”

Section: Introductionmentioning

confidence: 99%

Efficient Monte Carlo Algorithm Using Antithetic Variate and Brownian Bridge Techniques for Pricing the Barrier Options with Rebate Payments

Alzubaidi¹

2016

Journal of Mathematics and Statistics

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Abstract:The down-and-out call barrier option with rebate payment on dividend-paying stock is simulated using a new version of the Monte Carlo algorithm. The standard Monte Carlo method for simulating such an option suffers from two sources of errors: Hitting time error inherent from time stepping and the Monte Carlo statistical error. We present a modified version of Monte Carlo method that can reduce these errors efficiently using the Brownian bridge technique for the hitting time error and the antithetic variate approach for the statistical error. We found that the Brownian bridge technique is responsible for improving the order of convergence in hitting time from one half to one and the antithetic variate technique can speed up the Monte Carlo simulation by reducing the variance of the computed payoff, giving almost twice as much accuracy. The standard error and the coefficient of variation are applied in order to measure the effectiveness of the volatility of the underlying option.

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Simulation of stopped diffusions

Cited by 21 publications

References 26 publications

Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

Exact stochastic simulations of intra-cellular transport by mechanically coupled molecular motors

Efficient Monte Carlo Algorithm Using Antithetic Variate and Brownian Bridge Techniques for Pricing the Barrier Options with Rebate Payments

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