Central limit theorem for a Stratonovich integral with Malliavin calculus. (English) Zbl 1285.60050

Harnett, Daniel; Nualart, David

Ann. Probab. 41, No. 4, 2820-2879 (2013).

This article derives a change of variable formula for Stratonovich type integral processes driven by a Gaussian process that is not necessarily a semimartingale. The central technical ingredient is the multidimensional extension of a central limit theorem for iterated Skorokhod integrals obtained by I. Nourdin and D. Nualart in [J. Theor. Probab. 23, No. 1, 39–64 (2010; Zbl 1202.60038)]. More precisely, they consider the limit of the “midpoint” Riemann sums \[ \Phi_n(t) := \sum_{j=1}^{\lfloor nt/2\rfloor} f'(W_{(2j-1)/n)}\left( W_{2j/n}-W_{(2j-2)/n} \right), \] where \(f\) is a suitable function and \(W=(W_t)_{t\geq 0}\) is a Gaussian process with a technical assumptions on the covariance function. The authors show that the sequence \((W_t,\Phi_n(t))_{t\geq 0}\) has a limit \((W_t,\Phi(t))_{t\geq0}\) in distribution on the Skorokhod space \(\mathbb{D}^2[0,\infty)\) and that the second component is interpreted as Stratonovich integral \(\Phi(t)=\int_0^tf(W_s)\circ dW_s\). They derive a change of variable formula for the Stratonovich integral of the form \[ f(W_t) = f(W_0) + \int_0^t f'(W_s)\circ dW_s + \frac{1}{2}\int_0^t f''(W_s)dB_s. \] In addition to the usual Leibniz rule of the Stratonovich calculus for continuous martingales, an Itō integral of the second derivative appears. Here, \(B=(B)_{t\geq 0}\) is a scaled Brownian motion independent of \(W\), with variance \(\operatorname{E}B_t^2 =2\eta(t)\), where \(\eta(t)\) is determined by the covariance structure of \(W\). In the example of a bifractional Brownian motion with parameter \(H\leq 1/2\), \(H\times K =1/4\), the scaling function \(\eta\) is calculated explicitly.
The article is structured as follows. After a brief introduction to the Malliavin calculus of isonormal Gaussian processes, the authors proof a multidimensional version of the central limit theorem of Nourdin and Nualart [loc. cit.] stating that iterated Skorokhod integrals \(F_n= \delta^q(u_n)\) converge stably and that the limit has a conditional Gaussian distribution \(\mathcal{N}(0,\Sigma)\) given \(X\).
The main step is then to identify the second-order terms in a Taylor expansion of \(f\) as Skorokhod integrals of indicator functions of the form \[ \begin{split} f''(W_{(2j-1)/n})\left( \Delta W^2_{2j/n}-\Delta W^2_{(2j-1)/n} \right)\\ = f''(W_{(2j-1)/n})\delta^2 \left(\mathbf{1}_{[(2j-1)/n,j/n]}^{\otimes 2}-\mathbf{1}_{[(2j-2)/n,(2j-1)/n]}^{\otimes 2} \right),\end{split} \] where the authors apply their multidimensional version of the central limit theorem for Skorokhod integrals, generating the additional Gaussian term.

Reviewer: Jan Gairing (Berlin)

MSC:

60H05	Stochastic integrals
60F05	Central limit and other weak theorems
60G15	Gaussian processes
60G17	Sample path properties
60G22	Fractional processes, including fractional Brownian motion
60H07	Stochastic calculus of variations and the Malliavin calculus

Keywords:

Itō formula; Skorokhod integral; Malliavin calculus; bifractional Brownian motion; iterated integral; central limit theorem

Citations:

Zbl 1202.60038

References:

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