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  • 1.
    Djehiche, Boualem
    KTH, Superseded Departments, Mathematics.
    Hedging options in market models modulated by the fractional Brownian motion2001In: Stochastic Analysis and Applications, ISSN 0736-2994, E-ISSN 1532-9356, Vol. 19, no 5, p. 753-770Article in journal (Refereed)
    Abstract [en]

    We use the stochastic calculus of variations for the fractional Brownian motion to derive formulas for the replicating portfolios for a class of contingent claims in a Bachelier and a Black-Scholes markets modulated by fractional Brownian motion. An example of such a model is the Black-Scholes process whose volatility solves a stochastic differential equation driven by a fractional Brownian motion that may depend on the underlying Brownian motion.

  • 2. Moon, Kyoung-Sook
    et al.
    Szepessy, Anders
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Tempone Olariaga, Raul
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Zouraris, Georgios
    Convergence rates for adaptive weak approximation of stochastic differential equations2005In: Stochastic Analysis and Applications, ISSN 0736-2994, E-ISSN 1532-9356, Vol. 23, no 3, p. 511-558Article in journal (Refereed)
    Abstract [en]

    Convergence rates of adaptive algorithms for weak approximations of Ito stochastic differential equations are proved for the Monte Carlo Euler method. Two algorithms based either oil optimal stochastic time steps or optimal deterministic time steps are studied. The analysis of their computational complexity combines the error expansions with a posteriori leading order term introduced in Szepessy et al. [Szepessy, A.. R. Tempone, and G. Zouraris. 2001. Comm. Pare Appl. Math. 54:1169-1214] and ail extension of the convergence results for adaptive algorithms approximating deterministic ordinary differential equations, derived in Moon et al. [Moon, K.-S., A. Szepessy, R. Tempone, and G. Zouraris. 2003. Numer. Malh. 93:99-129]. The main step in the extension is the proof of the almost sure convergence of the error density. Both adaptive alogrithms are proven to stop with asymptotically optimal number of steps up to a problem independent factor defined in the algorithm. Numerical examples illustrate the behavior of the adaptive algorithms, motivating when stochastic and deterministic adaptive time steps are more efficient than constant time steps and when adaptive stochastic steps are more efficient than adaptive deterministic steps.

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