10.2 Construction of the Example 211natural filtration of the couple (B, W) and is supposed to satisfy the usual
conditions. This means thatF 0 contains all null sets ofF∞and that the
filtration is right continuous. The processLdefined as
Lt=exp(
Bt−^12 t)
is known to be a strict local martingale with respect to the filtrationF.
Indeed, the processLtends almost surely to 0 at infinity, hence it cannot be
a uniformly integrable martingale. Let us define the stopping timeτas
τ=inf{
t∣
∣Lt=^1
2}
.
Clearlyτ<∞a.s.. Using the Brownian motionWwe similarly construct
Mt=exp(
Wt−^12 t)
.
The stopping timeσis defined as
σ=inf{t|Mt=2}.InthecasetheprocessMdoes not hit the level 2 the stopping timeσequals
∞and we therefore have thatMσeither equals 2 or equals 0, each with
probability^12. The stopped processMσdefined byMtσ=Mt∧σis a uniformly
integrable martingale. It follows that also the processY=Mτ∧σis uniformly
integrable and becauseτ<∞a.s. we have thatY is almost surely strictly
positive on the interval [0,∞].
The processXis now defined as the processLstopped at the stopping time
τ∧σ. Note that the processesLandMare independent since they were con-
structed using independent Brownian motions. Stopping the processes using
stopping times coming from the other Brownian motion destroys the inde-
pendence and it is precisely this phenomenon that will allow us to make the
counter-example.
Theorem 10.2.1.The processesXandY, as defined above, satisfy the prop-
erties listed in Theorem 10.1.1
(1)The processXis a strict local martingale underP, i.e.EP[X∞]< 1 and
X∞> 0 a.s..
(2)The processYis a uniformly bounded integrable martingale.
(3)The processXYis a uniformly integrable martingale.
Proof.LetusfirstshowthatX is not uniformly integrable. For this it is
sufficient to show thatE[X∞]=E[Lτ∧σ]<1. This is quite easy. Indeed
E[Lτ∧σ]=∫
{σ=∞}Lτ+∫
{σ<∞}Lσ∧τ.In the first term the variableLτequals^12 and hence this term equals^12 P[σ=
∞]=^14. The second term is calculated using the martingale property ofL
and the optional stopping time theorem.