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Cubature Methods and Applications

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Paris-Princeton Lectures on Mathematical Finance 2013

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2081))

Abstract

We present an introduction to a new class of numerical methods for approximating distributions of solutions of stochastic differential equations. The convergence results for these methods are based on certain sharp gradient bounds established by Kusuoka and Stroock under non-Hörmader constraints on diffusion semigroups. These bounds and some other subsequent refinements are covered in these lectures. In addition to the description of the new class of methods and the corresponding convergence results, we include an application of these methods to the numerical solution of backward stochastic differential equations. As it is well-known, backward stochastic differential equations play a central role in pricing financial derivatives.

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Notes

  1. 1.

    The process X is uniquely identified by (1) only up to a set of measure 0. Two processes X 1 and X 2 satisfying (1) are indistinguishable: the set \(\{\omega \in \Omega \vert \exists t \in [0,\infty )\) such that \(X_{t}^{1}(\omega ) = X_{t}^{2}(\omega )\}\) is a \(\mathbb{P}\)-null set (has probability zero). Similarly, the identity (2) holds \(\mathbb{P}\)-almost surely, i.e. there can be a subset of \(\Omega \) of probability zero where (2) does not hold.

  2. 2.

    Of course the sum of the weights λ i, t is 1, i.e., \(\sum _{i=1}^{n_{t}}\lambda _{i,t} = 1\).

  3. 3.

    To be more precise, following the phraseology of [27], we describe here the simplified weak Euler scheme for a scalar SDE driven by a multi-dimensional noise.

  4. 4.

    In (13) and subsequently, [z] denotes the integer part of \(z \in \mathbb{R}\).

  5. 5.

    See Proposition 118 in [18].

  6. 6.

    For any positive integer m, the set \(\mathcal{C}_{b}^{m}({\mathbb{R}}^{a}; {\mathbb{R}}^{b})\) is the set of all bounded continuous functions \(\varphi : {\mathbb{R}}^{a} \rightarrow {\mathbb{R}}^{b}\), m-times continuously differentiable with all derivatives bounded.

  7. 7.

    In (23) and subsequently, \(\partial V _{i}\) is the matrix valued map \(\partial V _{i} := (\partial _{n}V _{i}^{m})_{1\leq n,m\leq N}\).

  8. 8.

    Although not used in the sequel, the result holds for general \(f : [0,\infty ) \times \Omega \rightarrow E\) such that \(f(t) \in {\mathbb{D}}^{1,2}(E)\) for any t ∈ [0,T], i.e., not necessarily adapted with respect to the natural filtration of B. In this case, the F i(T) is the Skorohod integral and not the Itô integral of f. See, for example, Proposition 1.38 page 43 in [51].

  9. 9.

    In the following, we allow for the constant c to take different values from one line to another.

  10. 10.

    The TBBA has also been used to control on the computational effort for a class of numerical algorithm using the family \(\bar{X}\left (x\right ),\;x \in {\mathbb{R}}^{d}\;\)as defined in (96), see [47] for details.

  11. 11.

    These are the straight lines connecting the origin with ( − 1, − 1), (1, − 1), ( − 1, 1), (1, 1).

  12. 12.

    Numerical experiments were performed with single-threaded code on a Intel i7 processor at 2.8 GHz.

  13. 13.

    Indeed, the consideration is trivial if this condition is violated.

  14. 14.

    cf. Proposition 1.5.4 in Nualart [51]

  15. 15.

    Theorem 80 can be extended to cover the rate of convergence for test functions \(\varphi \in \mathcal{C}_{b}^{p}({\mathbb{R}}^{N}, \mathbb{R}),\) in the same manner as the corresponding results in Theorem 46 and Corollary 47.

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Acknowledgements

We should like to thank the editors of the volume, Vicky Henderson and Ronnie Sircar for their understanding and patience whilst handling our submission. One of the authors (D.C.) would like to thank Nizar Touzi for the invitation to give a series of lectures on the subject in February–April 2010 at the Institute Henri Poincaré. This work grew out of the notes of the IHP lectures. The material in the second section is based on the corresponding chapter in the PhD thesis of C. Nee [44]. The material in Sects. 3.1–3.3 is based on the paper [13].

The work of D. Crisan and K. Manolarakis was partially supported by the EPSRC Grant No: EP/H0005500/1. C. Nee’s work was supported by an EPSRC Doctoral Training Award.

Special thanks are due to Salvator Ortiz–Latorre who read large portions of the first draft and suggested many corrections and improvements. The authors would also like to thank the anonymous referee for the careful reading of the manuscript and the constructive remarks.

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  1. Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, UK

    D. Crisan, K. Manolarakis & C. Nee

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  1. D. Crisan
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Crisan, D., Manolarakis, K., Nee, C. (2013). Cubature Methods and Applications. In: Paris-Princeton Lectures on Mathematical Finance 2013. Lecture Notes in Mathematics, vol 2081. Springer, Cham. https://doi.org/10.1007/978-3-319-00413-6_4

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