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Change of measure under the hard-to-borrow model

As the Securities and Exchange Commission(SEC) has implemented a new regulation on short-sellings, short-sellers are required to repurchase stocks once the clearing risk rises to a certain level. Avellaneda and Lipkin proposed a fully coupled SDE system to describe the mechanism which is referred as Hard-To-Borrow(HTB) models. Guiyuan Ma obtained the PDE system for both American and European options. There is a technical error in Guiyuan Ma where two correlated Brownian motion should be converted before change of measure. In this paper, I will provide supplement conditions.