Quadratic Variation Limits Versus the Supremum Over Partitions

Summary

The document raises a foundational stochastic calculus question: how quadratic variation is defined for a function over finer and finer partitions, and how that differs from taking the supremum of squared increments across all partitions. In the limit definition, the mesh of the partition—the length of its largest interval—shrinks toward zero, and the sums must converge to the same value as partitions are refined. The supremum definition instead selects the largest sum among partitions, without requiring their intervals to become uniformly small.

The text cites a finance textbook and asks for clarification, but supplies no worked example or resolution. It therefore identifies an important conceptual distinction without explaining which functions satisfy either definition or how the definitions apply to stochastic processes such as Brownian motion. Readers should treat it as a question prompt rather than a complete account of quadratic variation.

Key ideas

• Quadratic variation can be defined as a limit of squared increments over partitions whose mesh tends to zero.
• The mesh is the width of the largest subinterval in a partition.
• Taking a supremum over all partitions is a different operation from taking a fine-partition limit.
• The document poses the distinction but does not provide an answer or examples.

Tags

# Quadratic variation


# Quadratic variation












The following question is more math than quant, but since it arises from a mathematical finance textbook, I've figured the good people in this sub might be able to help me. So here goes.

In the 3rd edition of his textbook "Introduction to Stochastic Calculus with Applications" (Imperial College Press 2012), Fima Klebaner defines quadratic variation as follows (p. 8):

> If $g$ is a function of real variable, define its quadratic variation over the interval $[0, t]$ as the limit (when it exits) $$ [g](t) = \lim_{\delta_n \rightarrow 0}\sum_{i = 1}^n\left(g(t^n_i) - g(t^n_{i - 1})\right)^2, $$ where the limit is taken over partitions: $0 = t^n_0 < t^n_1 < \cdots < t^n_n = t$, with $\delta_n = \max_{1 \leq i \leq n} \left(t^n_i - t^n_{i - 1}\right)$.

Klebaner goes on to remark that this definition is not the same as $$ \sup \sum_{i = 1}^n \left(g(t^n_i) - g(t^n_{i - 1})\right)^2 $$ where supremum is taken over all partitions.

I don't understand the highlighted definition. I feel that it is lacking in precision. If anyone understands it, as well as the distinction between it and the other definition, I would appreciate if you could explain it to me. Thanks.

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