I've got a french notebook too.

Wiki tricks

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My mathematical notes

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Diffusion with a jump volatility

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differentiating a portfolio   (when the volatility is  ), we obtain:

 

The   term capture the possible jump of volatility (which has no direct instantaneous impact on  , but has on  , because it could turn it into  ). this term can only be captured in expectation, and because  , we obtain the desired Black Scholes equations ?