Quadratic Variation Examples

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$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle [X]^P_t &\displaystyle= \sum_{n=1}^\infty\left(X_{\tau_n\wedge t}-X_{\tau_{n-1}\wedge t}\right)^2,\smallskip\\ \displaystyle [X,Y]^P_t &\displaystyle = \sum_{n=1}^\infty\left(X_{\tau_n\wedge t}-X_{\tau_{n-1}\wedge t}\right)\left(Y_{\tau_n\wedge t}-Y_{\tau_{n-1}\wedge t}\right). \end{array}$

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$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle\int_0^t \xi^i\xi^j\,d[X^i,X^j] &\displaystyle= \sum_{k=1}^n \int_0^t 1_{\{t_{k-1}<s\le t_{k-1}\}}c^i_kc^j_k\,d[X^i,X^j]_s\smallskip\\ &\displaystyle=\sum_{k=1}^n c^{\rm t}_k([X]_{t\wedge t_k}-[X]_{t\wedge t_{k-1}})c_k \end{array}$

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$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle [Z,Y]^P_t &\displaystyle= \sum_{n=1}^\infty (Z_{\tau_n\wedge t}-Z_{\tau_{n-1}\wedge t})(Y_{\tau_n\wedge t}-Y_{\tau_{n-1}\wedge t})\smallskip\\ &\displaystyle=\sum_{n=1}^\infty \xi_{\tau_n}(X_{\tau_n\wedge t}-X_{\tau_{n-1}\wedge t})(Y_{\tau_n\wedge t}-Y_{\tau_{n-1}\wedge t})\smallskip\\ &\displaystyle=\int_0^t\xi\,d[X,Y]^P. \end{array}$

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$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle [X,V]_t &\displaystyle= \lim_{n\rightarrow\infty}\sum_{k=1}^n (X_{kt/n}-X_{(k-1)t/n})(V_{kt/n}-V_{(k-1)t/n})\smallskip\\ &\displaystyle=\lim_{n\rightarrow\infty}\sum_{k=1}^n\int_0^t 1_{\{(k-1)t/n<s\le kt/n\}}(X_{kt/n}-X_{(k-1)t/n})\,dV_s\smallskip\\ &\displaystyle=\int_0^t\lim_{n\rightarrow\infty}\sum_{k=1}^n 1_{\{(k-1)t/n<s\le kt/n\}}(X_{kt/n}-X_{(k-1)t/n})\,dV_s\smallskip\\ &\displaystyle=\int_0^t\Delta X_s\,dV_s. \end{array}$

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