Introduction to Stochastic Calculus with Applications, Second Edition
Containing many solved examples and exercises, this book gives a simple but rigorous treatment of stochastic calculus and its applications, including a range of advanced topics.
TABLE OF CONTENTS Introduction to Stochastic Calculus with Applications, Second Edition
8.5: Quadratic Variation and Covariation
8.5 Quadratic Variation and Covariation
If X, Y are semimartingales on the common space, then the quadratic covariation process, also known as the square bracket process and denoted [ X, Y]( t), is defined, as usual, by
where the limit is taken over shrinking partitions 
of the interval [0, t] when ? n = max i( t n i+1 ? t n i) ? 0 and is in probability. Taking Y = X we obtain the quadratic variation process of X.
Example 8.9
We have seen that quadratic variation of Brownian motion B( t) is [ B, B]( t) = t and of Poisson process N( t) is [ N, N]( t) = N( t).
Properties of Quadratic Variation
We give the fundamental properties of the quadratic variation process with some explanations, but omit the proofs.
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If X is a semimartingale, then [ X, X] exists and is an adapted process.
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It is clear from the definition that quadratic variation over non-overlapping intervals is the sum of the quadratic variation over each interval. As such, [ X, X] ( t) is non-decreasing function of t. Consequently [ X, X] ( t) is a function of finite variation.
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It follows from the definition (8.15) that [ X, Y] is bilinear and symmetric, that is, [ X,Y] =...
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