67.7. REMEMBERING THE FORMULA 2299

67.7 Remembering The FormulaI find it almost impossible to remember this formula. Here is a way to do it. Recall that|∆W |2 is like ∆t. Therefore, in what follows, neglect all terms which are like dWdt, dt2,but keep terms which are dW,dt,dW 2. Then you start with dX = φdt +ΦdW. Thus forF (t,X) ,

dF = Ftdt +FX dX +12(FXX dX ,dX)

other terms from Taylor’s formula are neglected because they involve dtdW or dt2. Nowthe above equals

dF = Ftdt +FX (φdt +ΦdW )+12(FXX ΦdW,ΦdW )

Since the dW occurs twice, in that inner product, you get a dt out of it. Hence you get

dF = (Ft +FX φ)dt +12(FXX Φ,Φ)dt +FX ΦdW

Now place an∫ t

0 in front of everything and you have the Ito formula.

67.8 An Interesting FormulaSuppose everything is real valued and φ is progressively measurable and in

L2 ([0,T ]×Ω) .

Let

X (t) =∫ t

0φdW − 1

2

∫ t

0φ

2ds

and consider F (X) = eX . Then from the Ito formula,

dF =−(

eXφ

2 12

)dt +

12

eXφ

2dt + eXφdW

dF = eXφdW

and then do an integral

eX(t)−1 =∫ t

0eX

φdW

Thus

eX(t) = 1+∫ t

0eX(s)

φdW

That expression on the right is obviously a local martingale and so the expression on theleft is also. To see this, you can use a localizing sequence of stopping times which dependon the size of X (t). This will work fine because X (t) is continuous.