ABSTRACT
In this chapter we will define our stochastic integral I t = ∫ 0 t H s d X s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203738283/0626314b-605c-46bd-9776-73db63c067fb/content/eq297.tif"/> . To motivate the developments, think of Xs as being the price of a stock at time s and Hs as the number of shares we hold, which may be negative (selling short). The integral It then represents the net profits at time t, relative to our wealth at time 0. To check this note that the infinitesimal rate of change of the integral dIt = Ht dXt = the rate of change of the stock times the number of shares we hold.
