Whether the method of expansion and mean-square approximation of multiple Ito and Stratonovich stochastic integrals (based on multiple generalized Fourier series) is useful for construction of high-order strong numerical methods for Ito SDEs? The method formulated and developed in the following papers (see also the bibliography in these papers):
[1] Dmitriy F. Kuznetsov. Expansion of Multiple Ito Stochastic Integrals of Arbitrary Multiplicity, Based on Generalized Multiple Fourier Series, Converging in the Mean. arXiv:1712.09746 [math.PR]. 2017, 22 pp.
Article Expansion of Multiple Ito Stochastic Integrals of Arbitrary ...
[2] Dmitriy F. Kuznetsov. Mean-Square Approximation of Multiple Ito and Stratonovich Stochastic Integrals from the Taylor-Ito and Taylor-Stratonovich Expansions, Using Legendre Polynomials. arXiv:1801.00231 [math.PR]. 2017, 26 pp. Article Mean-Square Approximation of Multiple Ito and Stratonovich S...
[3] Dmitriy F. Kuznetsov. Exact Calculation of Mean-Square Error of Approximation of Multiple Ito Stochastic integrals for the Method, Based on the Multiple Fourier Series. arXiv:1801.01079 [math.PR]. 2018, 19 pp.
Article Exact Calculation of Mean-Square Error of Approximation of M...
[4] Dmitriy F. Kuznetsov. Expansions of Multiple Stratonovich Stochastic Integrals, Based on Generalized Multiple Fourier Series. arXiv:1712.09516 [math.PR]. 2017, 25 pp.
Article Expansions of multiple Stratonovich stochastic integrals, ba...
[5] Expansion of Multiple Stratonovich Stochastic Integrals of Arbitrary Multiplicity, Based on Generalized Repeated Fourier Series, Converging Pointwise. arXiv:1801.00784 [math.PR]. 2018, 23 pp.
Article Expansion of Multiple Stratonovich Stochastic Integrals of A...
[6] Dmitriy F. Kuznetsov. Application of the Fourier Method to the Mean-Square Approximation of Multiple Ito and Stratonovich Stochastic Integrals. arXiv:1712.08991 [math.PR]. 2017, 15 pp.
Article Application of the Fourier Method to the Mean-Square Approxi...
[7] Dmitriy F. Kuznetsov. Direct combined approach for expansions of multiple Stratonovich stochastic integrals of multiplicities 2 - 4, based on generalized multiple Fourier series. arXiv.org:1801.05654 [math.PR]. 2018, 22 pp.
Article Direct combined approach for expansion of multiple Stratonov...
[8] Dmitriy F. Kuznetsov. The hypothesis about expansion of multiple Stratonovich stochastic integrals of arbitrary multiplicity. arXiv.org:1801.03195 [math.PR]. 2018, 14 pp.
Article The Hypothesis About Expansion of Multiple Stratonovich Stoc...
[9] Dmitriy F. Kuznetsov. Expansion of Triple Stratonovich Stochastic Integrals, Based on Generalized Multiple Fourier Series, Converging in the Mean: General Case of Series Summation. arXiv.org:1801.01564 [math.PR]. 2018, 26 pp. Article Expansion of Triple Stratonovich Stochastic Integrals, Based...
[10] Dmitriy F. Kuznetsov. Application of the Direct Combined Approach to Expansion of Double Stratonovich Stochastic Integrals. arXiv:1801.07248 [math.PR]. 2018, 9 pp.
Preprint Application of the Direct Combined Approach to Expansion of ...
[11] Dmitriy F. Kuznetsov. Expansion of Multiple Stratonovich Stochastic Integrals of Multiplicity 2, Based on Double Fourier-Legendre Series, Summarized by Prinsheim Method. arXiv:1801.01962 [math.PR]. 2018, 21 pp.
Preprint Expansion of Multiple Stratonovich Stochastic Integrals of M...
[12] Dmitriy F. Kuznetsov. Expansions of Multiple Stratonovich Stochastic Integrals From the Taylor-Stratonovich Expansion, Based on Multiple Trigonometric Fourier Series. Comparison With the Milstein Expansion. arXiv:1801.08862 [math.PR]. 2018, 18 pp. Preprint Expansions of Multiple Stratonovich Stochastic Integrals Fro...
Yes. The construction of higher order method for solving SDEs requires the inclusion of multiple stochastic integrals. The problems is the approximation of these integrals in case of the presence of more than one Brownian motion. The use of the Fourier series requires the passage or the transformation of BM into Brownian Bridge. We can cooperate in numerical and computational stochastics, see some of my works : http://www.zahri.net/NumericalSchemesforSDEs.pdf and the attached paper, regards Mostafa Zahri
Dear Mostafa, first of all thank you very much for your answere and I am very sorry for my English. I think, that even in the case of 1-dimensional Wiener process (I am talking about Ito SDEs and Taylor-Ito technique of strong order 2.0) we can not solve the problem via formulas based on Hermite polynomials. The problem appears starting the strong 1.0 scheme (Milstein method) for m-dimentional Wiener process (m>1). The technique based on trigonometric expansion of Brownian Bridge process (proposed by G.N. Milstein in 1988 in his monograph; I will call this approach as Milstein expansion) gives good results for the case of Milstein method. However if we step forward and consider 1.5, 2.0, 2.5 strong numerical schemes, then we can see, that Milstein expansion leads to iterative operation of limit transitions (for example for triple stochastic integrals with respect to Wiener process). Yes? So, we get the problem of correct truncation of expansions of multiple stochastic integrals. Moreover Milstein expansion leads to trigonometry. Seems I solve the problem of iterative limit transitions in the papers [1], [2], [4], [6], [7] (see above). In these papers we have only 1 operation of limit transition. As a result in [3] I calculate exaclty the mean-square errors of approximation of multiple Ito stochastic integrals in principle of arbitrary multiplicity. Another interesting thing. When we deal with strong numerical methods of order 1.5, 2.0., 2.5 e.t.c., then we have such collection of multiple stochastic integrals, that some of them is a mixture of integrations with respect to WIener process and with respect to t (time). Expansions using Legendre polynomials for these stochastic mixed integrals are essentially simpler than Milstein expansions ([2], see above). Anyway the problem is complex and interesting. May be for practice we no need so deep investigation, but this is interesting.
Dear Dimitriy,
thanks for the detailed answer . In my paper for solving SPDEs, I tried to find a physical justification for separating noises, resp. the construction of SDEs systems driven by at most one noise. This leads to the use of the multiple stochastic integrals of the same BM. which is a gain for the construction of higher order stochastic schemes. here is my email in Sharjah(UAE) [email protected].
Regards
Mostafa
Dear Mostafa,
my range of interests is more narrow. I started in 1993 from Ito SDEs and still work in this field . I want to mention the well known books:
[1]. G.N.Milstein "Numerical integration of stochastic differential equations" 1988 [in Russian], 1995 [English edition, Kluwer].
[2]. P.E.Kloeden and E.Platen "Numerical solution of stochastic differential equations" 1995. Springer.
If we consider strong numerical method of order 1.0 for Ito SDEs (Milstein scheme), then the Brownian Bridge process expansion technique is well for approximation of stochastic integrals of 1st and 2nd multiplicity. However if we want to get more in the sense of accuracy, then we go to strong numerical method of order 1.5. And here we get some problems with approximation of triple stochastic integrals. If you look attentively section 5.8 "Strong approximation of multiple stochastic integrals" [2] (pp. 198 - 205), then you can conclude, that approximation of triple stochastic integrals with respect to multi-dimentional Wiener process is unsatisfactory. Why? I can to mention two reasons. 1). Brownian Bridge process expansion technique leads to iterative limit transitions, so the authors could not consider triple stochastic integral as one mean-square limit if p tends to infinity [2] (pp. 203 - 204). This problem was solved in my papers (see the formulation of this topic. Especially in the works [1], [4]). Thus, we may go forward. However there is reason 2). Why to use so complex approximations for the double integrals with respect to dtdW^(i) and dW^(i)dt? Obviously, that these 2 integrals consist of sums of 2 independent Gaussian random values if we use orthogonal polynomials instead of trigonometry. Yes? If we consider strong scheme of order 2.5, then we get integrals dtdtdW^(i), dtdW^(i)dt and dW^{i)dtdt as part of the set of needed integrals. Obviously these 3 integrals can be represented as sums of 3 independent Gaussian random values if we again to use Legendre polynomials instead of trigonometry (usually we always do this in the case of 1-dimentional Wiener process), and so on. Moreover approximations of another and more complex integrals (dW^{i)dW^(j)dt, dW^(i)dtdW^(i), dtdW^(i)dW^(j) ... ) will be more simpler for Legendre polynomials. I collected some practical material in this direction in the work [2] (see the formulation of this topic). Recommend for modelling. My e-mail: [email protected]
Cite
Samra Harkat
École Nationale Supérieure d'Hydraulique
https://arxiv.org/pdf/1801.00784.pdf
Cite
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