[Paper Review] Cross-variation of Young integral with respect to long-memory fractional Brownian motions

Ivan Nourdin, Rola Zintout|arXiv (Cornell University)|Nov 12, 2013

Stochastic processes and financial applications11 references3 citations

TL;DR

This paper investigates the asymptotic behavior of the cross-variation of two-dimensional processes defined as Young integrals with respect to long-memory fractional Brownian motion (fBm) of Hurst index $ H > \frac{1}{2} $. It establishes that for $ H \leq \frac{3}{4} $, the normalized cross-variation converges to a mixed Gaussian limit, while for $ H > \frac{3}{4} $, the limit is expressed in terms of the difference of two independent Rosenblatt processes, providing a non-central limit theorem for cross-variation in the long-memory regime.

ABSTRACT

We study the asymptotic behaviour of the cross-variation of two-dimensional processes having the form of a Young integral with respect to a fractional Brownian motion of index $H extgreater{} 1/ 2$. When $H$ is smaller than or equal to $3 / 4$, we show asymptotic mixed normality. When $H$ is strictly bigger than $3/4$, we obtain a limit that is expressed in terms of the difference of two independent Rosenblatt processes.

Motivation & Objective

• To analyze the asymptotic behavior of the cross-variation process $ J_n(t) $ defined as the sum of products of increments of a two-dimensional process driven by fractional Brownian motion.
• To establish limit theorems for $ J_n(t) $ under different regimes of the Hurst parameter $ H \in (\frac{1}{2}, 1) $, particularly distinguishing between $ H \leq \frac{3}{4} $ and $ H > \frac{3}{4} $.
• To provide a theoretical foundation for statistical inference, such as testing the null hypothesis $ \sigma^{1,2} = \sigma^{2,1} = 0 $, in stochastic differential equations driven by fBm.
• To extend existing results on power variations to the case of cross-variation in two-dimensional fBm settings, especially in the non-Gaussian regime for $ H > \frac{3}{4} $.

Proposed method

• The process $ X_t $ is modeled as a Young integral with respect to a two-dimensional fractional Brownian motion $ B = (B^{(1)}, B^{(2)}) $ of index $ H > \frac{1}{2} $, ensuring pathwise integrability via Hölder continuity.
• The cross-variation $ J_n(t) $ is decomposed into three components: $ A_n(t) $, $ S_n(t) $, and remainder terms $ R_{1,n}(t), R_{2,n}(t) $, based on incremental expansions of the integrands and increments.
• The asymptotic behavior is analyzed using weighted random sum techniques from [3], combined with moment estimates and convergence in $ L^1 $ and in law.
• For $ H \leq \frac{3}{4} $, the limit is derived via central limit theorem arguments involving independent Brownian motion; for $ H > \frac{3}{4} $, the limit involves the Rosenblatt process constructed from linear combinations of the fBm components.
• The proof relies on a decomposition of the cross-variation into quadratic and cross-terms, with the key step being the application of Proposition 3.8 to show convergence of $ a_n S_n(t) $, where $ a_n $ is a normalization depending on $ H $.
• The analysis uses the fact that $ n^{2H-1} \sum (\Delta B^i_{k/n})^2 \to t $ almost surely, enabling application of Lemma 3.7 to obtain almost sure convergence of the quadratic variation terms.

Experimental results

Research questions

• RQ1 What is the asymptotic distribution of the cross-variation process $ J_n(t) $ for a two-dimensional Young integral driven by long-memory fractional Brownian motion?
• RQ2 How does the limiting behavior of $ J_n(t) $ change when the Hurst index $ H $ crosses the threshold $ \frac{3}{4} $?
• RQ3 Can a non-central limit theorem be established for the cross-variation when $ H > \frac{3}{4} $, and if so, what is the limiting process?
• RQ4 What normalization $ a_n $ ensures convergence in law of $ a_n J_n(t) $, and how does it depend on $ H $?
• RQ5 How can the cross-variation be used to test the statistical hypothesis $ \sigma^{1,2} = \sigma^{2,1} = 0 $ in SDEs driven by fBm?

Key findings

• For $ \frac{1}{2} < H \leq \frac{3}{4} $, the normalized cross-variation $ n^{2H-1} J_n(t) $ converges in probability to $ \int_0^t (\sigma^{1,1}_s \sigma^{1,2}_s + \sigma^{2,1}_s \sigma^{2,2}_s) ds $, establishing a central limit theorem with mixed Gaussian limit.
• For $ H > \frac{3}{4} $, the normalized cross-variation $ a_n J_n(t) $ converges in law to $ \int_0^\cdot \sigma^{1,1}_s \sigma^{2,2}_s dZ_s $, where $ Z $ is the difference of two independent Rosenblatt processes constructed from $ \beta^{(1)} = \frac{1}{\sqrt{2}}(B^{(1)} + B^{(2)}) $ and $ \beta^{(2)} = \frac{1}{\sqrt{2}}(B^{(1)} - B^{(2)}) $.
• The normalization $ a_n $ is $ n^{2H - 1/2} $ for $ H \in (\frac{1}{2}, \frac{3}{4}) $, $ \frac{n}{\sqrt{\log n}} $ for $ H = \frac{3}{4} $, and $ n $ for $ H \in (\frac{3}{4}, 1) $, reflecting the transition from Gaussian to non-Gaussian limits.
• The remainder terms $ R_{1,n}(t) $ and $ R_{2,n}(t) $ vanish in $ L^1 $ under the Hölder regularity assumption on $ \sigma $, ensuring they do not affect the limiting behavior.
• When $ \sigma^{1,2} = \sigma^{2,1} = 0 $, the cross-variation reduces to $ a_n S_n(t) $, and the limit is driven solely by the cross-term $ \Delta B^1_{k/n} \Delta B^2_{k/n} $, which leads to the Rosenblatt process in the high-H regime.
• The result extends prior work on power variations by handling the two-dimensional, cross-variation case and resolving the challenging $ H > \frac{3}{4} $ regime via non-central limit theory.