IFSM Representation of Brownian Motion with Applications to Simulation

Stefano Maria Iacus () and Davide La Torre ()
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Stefano Maria Iacus: University of Milan, Department of Economics, Business and Statistics
Davide La Torre: University of Milan, Department of Economics, Business and Statistics

A chapter in Math Everywhere, 2007, pp 115-124 from Springer

Abstract: Abstract Several methods are currently available to simulate paths of the Brownian motion. In particular, paths of the BM can be simulated using the properties of the increments of the process like in the Euler scheme, or as the limit of a random walk or via L 2 decomposition like the Kac-Siegert/Karnounen-Loeve series. In this paper we first propose a IFSM (Iterated Function Systems with Maps) operator whose fixed point is the trajectory of the BM. We then use this representation of the process to simulate its trajectories. The resulting simulated trajectories are self-affine, continuous and fractal by construction. This fact produces more realistic trajectories than other schemes in the sense that their geometry is closer to the one of the true BM’s trajectories. The IFSM trajectory of the BM can then be used to generate more realistic solutions of stochastic differential equations.

Keywords: Brownian Motion; Inverse Problem; Fractal Nature; Iterate Function System; Euler Scheme (search for similar items in EconPapers)
Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-540-44446-6_10

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DOI: 10.1007/978-3-540-44446-6_10

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