Deriving Brownian Covariation from Correlated Increments

Summary

The document outlines a proof that two Brownian motions with correlated increments have quadratic covariation equal to their correlation coefficient multiplied by time. It starts by examining the sum of the two processes: the variance of its increments depends on both individual Brownian variances and their correlation. Scaling that sum by the square root of its variance rate produces a standard Brownian motion.

The quadratic variation of this scaled process is therefore time. Expanding the same quadratic variation using bilinearity yields an expression involving the individual quadratic variations and the cross covariation, which can then be rearranged to obtain the claimed relation. This is a proof sketch rather than a detailed derivation: it assumes standard Brownian motion properties and bilinearity of quadratic covariation. The stated correlation range excludes negative correlations, and the sketch's scaling expression requires care at the degenerate endpoint where the sum has zero variance.

Key ideas

• The variance of the sum of correlated Brownian increments incorporates their cross-correlation.
• Scaling the sum by its standard deviation rate gives a standard Brownian motion.
• The scaled process has quadratic variation equal to elapsed time.
• Bilinearity lets that quadratic variation be expressed using the cross covariation of the original processes.

Tags

# Mutual variation of Brownian motions


# Mutual variation of Brownian motions












Let $\{W^1\}_{t\geq0}$ and $\{W^2\}_{t\geq0}$ be two Brownian motions with correlation coefficient $\rho \in [0, 1]$, i.e., $\mathbb{E}[(W^1(t)-W^1(s))(W^2(t)-W^2(s))]=\rho(t-s)$ for all $t,s \geq 0$. Show that $<W^1, W^2>_t = \rho t$ for all $t\geq0$.

## Answer by ir7 (score 1)

https://quant.stackexchange.com/a/63568

Hints:

Show first that $$E[((W^1_t + W^2_t)-(W^1_s + W^2_s))^2] = (2+2\rho)(t-s) $$

Then conclude that

$$ [(2+2\rho)^{-1/2} (W^1 + W^2)]_t =t $$

On the other hand, show (using bilinearity of quadratic covariation) that

$$ [(2+2\rho)^{-1/2} (W^1 + W^2)]_t = [(2+2\rho)^{-1/2} (W^1 + W^2), (2+2\rho)^{-1/2} (W^1 + W^2) ]_t $$ $$ = (1+\rho)^{-1} (t+ [W^1, W^2]_t)$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.