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

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.

### 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

Zbl 1202.60038
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### References:

 [1] Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201 [2] Burdzy, K. and Swanson, J. (2010). A change of variable formula with Itô correction term. Ann. Probab. 38 1817-1869. · Zbl 1204.60044 [3] Houdré, C. and Villa, J. (2003). An example of infinite dimensional quasi-helix. In Stochastic Models ( Mexico City , 2002). Contemporary Mathematics 336 195-201. Amer. Math. Soc., Providence, RI. · Zbl 1046.60033 [4] Lei, P. and Nualart, D. (2009). A decomposition of the bifractional Brownian motion and some applications. Statist. Probab. Lett. 79 619-624. · Zbl 1157.60313 [5] Nourdin, I. and Nualart, D. (2010). Central limit theorems for multiple Skorokhod integrals. J. Theoret. Probab. 23 39-64. · Zbl 1202.60038 [6] Nourdin, I. and Réveillac, A. (2009). Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case $$H=1/4$$. Ann. Probab. 37 2200-2230. · Zbl 1200.60023 [7] Nourdin, I., Réveillac, A. and Swanson, J. (2010). The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter $$1/6$$. Electron. J. Probab. 15 2117-2162. · Zbl 1225.60089 [8] Nualart, D. (2006). The Malliavin Calculus and Related Topics , 2nd ed. Springer, Berlin. · Zbl 1099.60003 [9] Swanson, J. (2007). Weak convergence of the scaled median of independent Brownian motions. Probab. Theory Related Fields 138 269-304. · Zbl 1116.60010 [10] Swanson, J. (2007). Variations of the solution to a stochastic heat equation. Ann. Probab. 35 2122-2159. · Zbl 1135.60041 [11] Swanson, J. (2011). Fluctuations of the empirical quantiles of independent Brownian motions. Stochastic Process. Appl. 121 479-514. · Zbl 1230.60022
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