On the quadratic variation process of a continuous martingale
This article deals with the pathwise construction of the quadratic variation process \(<M>\) associated to a continuous local martingale (M(t),\(t\geq 0)\). More precisely, for \(n\geq 1\) let the sequence \((T_ i^ n;i\geq 0)\) of stopping times \(T_ i^ n\) be defined by \(T_ 0^ n=0\), \(T^ n_{i+1}=\inf \{t\geq T_ i^ n:| M(t)-M(T_ i^ n)| \geq 2^{-n}\}\) and let \(X_ n(t)=\sum^\infty_{i=0}(M(t\wedge T^ n_{i+1})-M(t\wedge T_ i^ n))^ 2\). The author gives a simple proof of the a.s.-convergence of \(X_ n\) to \(<M>\) and obtains in this way the following explicit expression for \(<M>\): \[ <M>(t,\omega) = \limsup_{n\uparrow\infty} 2^{-2n} K_ n(t,\omega), \quad \text{where }K_ n(t,\omega)=i \text{ if }T_ i^ n(\omega) \leq t<T^ n_{i+1}(\omega). \] Related work of the author on pathwise approximation of solutions of stochastic differential equations is published in Sankhyā, Ser. A 43, 121-132 (1981; Zbl 0507.60047).
scientific article; zbMATH DE number 3844742