Goethe Center for Scientific Computing (G-CSC)

Goethe University Frankfurt

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Mathematical Foundations of Computational Finance

Project Description

Mathematical Foundations of Computational Finance

Peter Kloeden

New requirements for prediction quality, reliability and robustness of models demand new numerical simulation methods. To this end, existing methods have to be adapted to the new tasks or completely new methods have to be developed and analyzed. In the working group of Prof. Kloeden, numerical methods for financial models are designed and studied. Consistently, new and fundamental methods are hereby introduced. Already in the 1990s the book „Numerical Methods for Stochastic Differential Equations” by Kloeden and Platen provided a substantial contribution to computational finance.

For example, new volatility models with square root coefficients require new methods. Jentzen, Kloeden and Neuenkirch showed in a newer paper the existence of negative solutions, but they could also provide methods for their treatment and the possible unboundedness of solutions. Another example for the clever introduction of new methods is the paper of Higham and Kloeden on stochastic differential equations for jump diffusion processes.

Jentzen and Kloeden could extend the theory of Kloeden and Platen to partial stochastic differential equations. Equations of this type are used more and more in financial models. Currently, the working group of Prof. Kloeden intensely works on an extension to weak convergence which is especially important for financial models. Another important subject is the efficient computation of stochastic backward differential equations which are also used in financial models. Another research direction is the development of methods for the efficient computation of greeks in Monte Carlo-simulations which have been investigated in the dissertation of Carlos Sanz Chacón.

These works will be carried out, if possible, in close cooperation with interested industry, especially banks. It is supported by projects of DFG and DAAD and will be carried out by doctorands, post-docs and guest researchers.

References

  1. A. Jentzen and P. E. Kloeden.
    Pathwise Taylor schemes for random ordinary differential equations.
    BIT 49 (1):113–140, 2009.
  2. A. Jentzen and P.E. Kloeden.
    Overcoming the order barrier in the numerical approximation of SPDEs with additive space-time noise.
    Proc. Roy. Soc. London A 465:649–667, no. 2102, 2009.
  3. A. Jentzen and P.E. Kloeden.
    Taylor-expansions of solutions of stochastic partial differential equations with additive noise.
    Annals of Probability, to appear.
  4. A. Jentzen, P.E. Kloeden and A. Neuenkirch.
    Convergence of numerical approximations of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients.
    Numerische Mathematik 112 (1):41–64, 2009.
  5. A. Jentzen.
    Pathwise numerical approximations of SPDEs with additive noise under non-global Lipschitz coefficients.
    Potential Analysis, to appear.
  6. A. Jentzen.
    Taylor expansions of solutions of stochastic partial differential equations.
    DCDS Series B, to appear.
  7. P.E. Kloeden and E. Platen.
    The Numerical Solution of Stochastic Differential Equations.
    Springer–Verlag, 1992 (revised reprinting 1995, 3rd revised and updated printing 1999).
  8. P.E. Kloeden, A. Neuenkirch and R. Pavani.
    Multilevel Monte Carlo for SDEs with additive fractional noise.
    Annals of Operations Research, to appear.