Quadratic Variation

Definition

Consider a probability space \fs4 (\Omega, \mathscr{F}, P), and a stochastic process \fs4 X_t(\omega). The \fs4 p-th order variation process, denoted with \fs4 \left<X,X\right>_t^(p), is defined as the limit

\fs4 \left< X,X \right>_t^(p) = \mathrm{plim} \sum_{t_k \leq t} |X_{t_{k+1}}(\omega)-X_{t_k}(\omega)|^p \text{ as } \triangle t_k \rightarrow 0

for a dyadic partition \fs4 t_k of {0,t$}.

The quadratic variation process is just \fs4 \left<X,X \right>_t = \left<X,X \right>_t^(2).

For a Brownian motion \fs4 B_t, the quadratic variation will be the limit (in probability)

\fs4 \left< B,B \right>_t = \left< B,B \right>_t^(2) = \mathrm{plim} \sum |\triangle B_{t_k}|^2

Now \fs4 \left< B,B \right>_t = t, since

\fs4 \begin{eqnarray} E \left[ \sum \left( \triangle B_{t_k} \right)^2 - t \right] &=& 0\\E \left[ \sum \left( \triangle B_{t_k} \right)^2 - t \right]^2 & = & 2 \sum \left( \triangle t_k \right)^2 \rightarrow 0 \end{eqnarray}

We write the above result in shorthand as {d B_t(\omega)^2 = d t$}

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