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My current understanding: (a) changing the probability measure of a diffusion process does not change the variance. (b) for a general stochastic process the variance may change. Please confirm whether this is correct. Secondly, for (a) why isn't it important that there may be covariance between the asset being diffused and the value of the numeraire? (or the ratio of numeraires)?

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  • $\begingroup$ Do you mean (co)variance or quadratic (co)variation? Also, "why isn't it important" in what regard, can you give an example of what "bothers" you? $\endgroup$ – Quantuple Aug 26 '16 at 13:51
  • $\begingroup$ For the first question, I mean the variance of the distribution after some time T. $\endgroup$ – dm63 Aug 26 '16 at 21:25
  • $\begingroup$ And by diffusion process what do you mean exactly: no jumps? A specific form for $\mu$ and $\sigma$ in the SDE: $dS_t = \mu dt + \sigma dW_t$ e.g. no dependence on $S_t$ ? $\endgroup$ – Quantuple Aug 27 '16 at 14:54
  • $\begingroup$ yes, what restrictions on μ and σ in the SDE are necessary and sufficient for the varaince to stay constant under a change of measure? $\endgroup$ – dm63 Dec 26 '16 at 14:53
  • $\begingroup$ Well the variance is not necessarily constant after a change or measure. Quadratic covariation is, at least under the conditions that underly the Girsanov theorem $\endgroup$ – Quantuple Dec 26 '16 at 19:36

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