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tje quadratic variation of which is.

Elliott et al. [24] obtained the optional quadratic variation of (M_{t}) as the matrix process [M,M]_{t}=sum_{s=0}^{t} bigtriangleup M_{s} triangle M_{s}^, where '∗' means transpose.

where the square brackets denote the quadratic variation process.

The quadratic variation and realized quadratic variation have been widely used in stochastic analysis and statistics of stochastic processes.

At the instantaneous level one may only speak of variation and quadratic variation but not directly of drift and volatility.

Suppose that the quadratic variation of S i exists and vanishes identically.

An important particular example is the quadratic variation which lies in the second Wiener chaos.

The quadratic variation process 〈W〉 t is also a continuous process with independent and stationary increments.

where W t is a G-Brownian motion and 〈W〉 t is its quadratic variation process.

Its quadratic variation process is also a continuous process with independent and stationary increments.

Then one can obtain the desired strong consistence by the definition of the quadratic variation.

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