314 14 The FTAP for Unbounded Stochastic Processes
Example 14.5.23.† There is a continuous process S,S 0 = 0 satisfying
(NFLVR)and so that
(i) P∈Meσ,
(ii) Sis not uniformly integrable underPandEP[S∞]>0,
(iii) the maximum functionS∗=sup 0 ≤t<∞|St|is notP-integrable,
(iv) there isQ 0 ∈Meσsuch thatSis uniformly Q 0 -integrable and, more
precisely,EQ 0 [(S∗)γ]<∞for someγ>1,
(v) the weight functionw=1+S∗is feasible and
sup
Q∈MeσEQ[S∞]> sup
Q∈Meσ
EQ[w]<∞EQ[S∞]=0.
The example is the same as in Chap. 10 but we need additional properties.
The space Ω supports two independent Brownian motions:BandW.We
first introduce the two stochastic exponentialsLt=exp(Bt−^12 t)andZt=
exp(Wt−^12 t). As in Chap. 10 we defineτ=inf{t|Lt≤^12 }andσ=inf{t|
Zt≥ 2 }. The processZσis bounded by 2 andLτis a strict local martingale.
The processSis defined asS=1−Lτ∧σandQ 0 is defined byddQP^0 =Zτ∧σ.
Sinceτ<∞a.s.Q 0 andPare equivalent. In Chap. 10 it is shown that
(i) and (ii) and hence (iii) hold true. Also it is shown thatSis a uniformly
Q 0 -integrable martingale. We will now show thatEQ 0 [(S∗)γ]<∞for some
γ>1. This implies thatw=1+S∗is a feasible weight function and since
EP[S∞]>0andP∈Meσwe get (v) as a consequence.
Hence we still have to show (iv). The estimate onEQ 0 [(S∗)γ] follows from
the statementEQ 0 [Lγτ∧σ]<∞for someγ>1. This in turn follows from the
following claim (see also [RY 91, Chap. II, Exercise (3.14)]).
Claim 1:LetWbe a Brownian motion. Letν=inf{t|Wt+^12 t≥ln 2},then
forβ≥0wehave
E[exp(−βν)] = 21 −√1+8β(^2).
This is seen as follows. Forα≥0 take the martingaleXt =exp(αWt−
1
2 α
(^2) t) stopped at timeν.SinceXνis bounded we getE[Xν]=1andthis
implies 1 =E[exp(αln 2−α 2 ν−^12 α^2 ν)] = 2αE[exp(−α(α 2 +1)ν)]. The equation
α(α+1)
2 =βhas one positive root forα,namelyα=
−1+√1+8β
2 .Thisgives
E[exp(−βν)] = 2
1 −√ 2 1+8β
.
Claim 2:Letf≥0 be a random variable and forβ∈C,Re(β)≥0letφ(β)=
E[exp(−βf)]. Suppose thatφhas an analytic continuation, still denoted by
φ, to the domain{β∈C||β|<β 0 }.Thenforβ∈C,|β|<β 0 we have
E[exp(βσ)] =φ(−β).
This is a standard exercise in probability theory.
†The original paper [DS 98, Example 5.14] contained an error in Example 14.5.23.
This was pointed out by S. Biagini and M. Frittelli which is gratefully acknowl-
edged, see [BF 04]. In their paper they adapted our example in a different way.