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IFSM Representation of Brownian Motion with Applications to Simulation

In: Math Everywhere

Author

Listed:
  • Stefano Maria Iacus

    (University of Milan, Department of Economics, Business and Statistics)

  • Davide La Torre

    (University of Milan, Department of Economics, Business and Statistics)

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.

Suggested Citation

  • Stefano Maria Iacus & Davide La Torre, 2007. "IFSM Representation of Brownian Motion with Applications to Simulation," Springer Books, in: Giacomo Aletti & Alessandra Micheletti & Daniela Morale & Martin Burger (ed.), Math Everywhere, pages 115-124, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-44446-6_10
    DOI: 10.1007/978-3-540-44446-6_10
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