If either of the processes X and Y(V) in (1.1) can have drift, the conclusion of Theorem 1.2 fails as the following example demonstrates.
LetR(V)be the difference ofX andY(V)and assume that it takes the form Rt(V) =r+µt+Bt−σVt,¯
whereB is the fixed(Ft)-Brownian motion,V ∈ Van arbitrary(Ft)-Brownian motion, ¯
σa volatility parameter different from 1, r a strictly negative starting point andµ a constant positive drift. Then the candidate extremal Brownian motions in (1.5) are given byVI =BandVII =−Band the following lemma holds.
Lemma 5.1.For any starting point r < 0, time horizon T > 0, volatility σ >¯ 0 and positive driftµ >0, the inequalityPr
τ0+(R(VI))> T <Pr τ0+(R(VII))> T holds.
Lemma 5.1 implies that Theorem 1.2 cannot hold for processes with drift. An intuitive explanation for this phenomenon is as follows: in the presence of a large drift upwards, it is better to reduce the volatility as much as possible (in this case to the level|1−σ¯|), instead of increasing it to its maximal value (equal to(1 + ¯σ)), since the drift makes the processesX andY(V)couple before timeT.
Proof. Fixr <0,T >0,¯σ >0,µ >0and define the functionF : (0,∞)→[0,1]by F(v) :=N −r+√µT T v −e−2µrv2N −r−√µT T v , v >0, and recall thatPr
τ0+(R(VI))> T=F(1/|1−σ¯|),Pr
τ0+(R(VII))> T=F(1/(1 + ¯σ)) (see e.g. [2, II.2.1, Eq. 1.1.4]), where N(·) denotes the normal cdf. To establish the lemma it is sufficient to show that F is strictly decreasing on the bounded interval [1/(1 + ¯σ),1/|1−σ¯|]. Since the derivative takes the form
F0(v) =−2µ√T n −r+√µT T v + 4µrve−2µrv2N −r−√µT T v
and clearly satisfiesF0(v)<0for allv >0, the lemma follows.5
5.2 (Ft)-adapted non-(Ft)-Markov processes on a discrete state space
In this section we construct two continuous-time (Ft)-adapted processes with a countable discrete state space, neither of which are (Ft)-Markov, and show that in both cases the strategies in Theorems 1.1 and 1.2 are suboptimal. In the first (resp. second) example, Section 5.2.1 (resp. Section 5.2.2), the constructed process is semi- Markov (resp. Markov) with respect to its natural filtration. This demonstrates that the assumption that the chainZ is an(Ft)-Markov process, not just a Markov process with respect to its “natural” filtration, is indeed necessary in Theorems 1.1 and 1.2.
5.2.1 (Ft)-semi-Markov process
Recall thatB is(Ft)-Brownian motion, fix∈(0,1)and then let the random timesTn, n∈N∪ {0}, be given byT0:= 0and
Tn:= inf{t≥Tn−1 : |Bt−BTn−1|=} forn≥1. Define the processesN = (Nt)t≥0andW = (Wt)t≥0by
Nt:= max{n∈N∪ {0} : Tn≤t} and Wt:=BTNt.
For every t > 0 we have {Tn ≤ t} ∈ Ft for all n ∈ N and hence the processW is (Ft)-adapted. FurthermoreW is a continuous-time semi-Markov process (i.e. the pair
(W, B)is(Ft)-Markov) with state spaceZand càdlàg trajectories. In particular,W has only finitely many jumps on any compact interval. Let
Z :=z0E(W), for a fixedz0>0,
whereEdenotes the Doléans-Dade stochastic exponential [9, Sec II.7, Thm. 37]. There- fore, by definition, we have
Zt=z0+ Z t 0 Zs−dWs=z0+ Z TNt 0 Zs−dWs=ZTNt,
where the second equality follows from the facts thatTNt ≤ t, and that there are no jumps of W during the time interval (TNt, t]. The process Z has a countable state space,6which can be expressed as
E:={z0(1−)n(1 +)m:m, n∈N} ⊂(0,∞)and is a
continuous-time semi-Markov process (as before,(Z, B)is(Ft)-Markov).
Consider the stochastic integralR0·ZsdBsand note that the equalityWTn−WTn−= BTn−BTn−1 holds for all n∈ N. Hence the stochastic integral can be expressed as follows: Z t 0 ZsdBs = Z TNt 0 ZsdBs+ Z t TNt ZsdBs= Nt X n=1 ZTn−1(BTn−BTn−1) +ZTNt(Bt−BTNt) = (ZTNt −z0) +ZTNt(Bt−BTNt) =Zt(1 + (Bt−BTNt))−z0.
Therefore, since by definition we have |Bt−BTNt| < and Zt > 0, the following inequalities hold:
−z0≤(1−)Zt−z0≤
Z t
0
ZsdBs for allt≥0. (5.1) As in Section 1.2.1, define σi : E → Rbyσi(z) := −iz for any z ∈ E andi = 1,2, and note that by (1.5) we haveVI =BandVII =−B. Hence, for any starting points x, y∈R, definition (1.1) and inequality (5.1) yield the following almost sure inequalities:
Xt−Yt(VI) =x−y+ Z t 0 ZsdBs≥x−y−z0, Xt−Yt(VII) =x−y−3 Z t 0 ZsdBs≤x−y+3z0.
For any time horizonT >0, counterexamples to the Conjecture in Section 1.2 (for both Problems(T)and(C)) can now be constructed in the same way as in Section 1.2.1.
5.2.2 Non-(Ft)-Markov Markov chain
In order to define a processZ, which is an(Ft)-adapted, time-homogeneous Markov chain in its own filtration and has properties analogous to the ones in the previous section, we sample the path of the Brownian motionB at a sequence of jump times of a Poisson processN. The key idea is to use the increments ofB over the jump times ofNto construct a certain compound Poisson process (in its own filtration), which is coupled withBand(Ft)-adapted. The corresponding stochastic exponential will then serve as an example ofZwith the required properties.
Fix a small >0and assume thatNis an(F
t)-Poisson process7with intensity1/. Note thatNis necessarily independent ofB by Lemma 2.5. Define(Ft)-stopping times
Tn:= inf{t≥0 :Nt=n} for anyn∈NandT−1:=T0:= 0. (5.2)
6An additional bijection is needed to define a chain with a state space that is a discrete subspace of a Euclidean space.
7Nis a Lévy process started at0with state spaceN∪ {0}, such thatN
t −Nsis independent ofFsand Poisson distributed with parameter(t−s)/for any times0≤s < t.
Recall thatTn−Tn−1,n ∈N, are IID exponentially distributed with with meanand
note that
Nt= max{n∈N∪ {0} : Tn ≤t}, implying TNt≤t < TNt+1 ∀t≥0. (5.3) Defineh() := exp(−1/2)and the functiong:
R→Rby the formula g(x) :=h()b1 +x/h()cI{x>0}+h()bx/h()cI{x<0},
wherebycdenotes the largest element inZsmaller thany∈R. The functiongsatisfies
|g(x)−x| ≤h() ∀x∈R and g(x) = 0 ⇐⇒ x= 0. (5.4) We now discretize the increments ofBusingg: define the processW= (Wt)t≥0by
Wt:= N t X n=0 g BTn−BTn−1 , t≥0.
The processWis(Ft)-adapted, i.e. the random variableWt, which can be expressed as W t = P∞ m=0I{N t=m} Pm n=0I{Tn≤t}g Bt∧Tn−Bt∧Tn−1
, isFt-measurable for every t ≥0 (N is (F
t)-adapted and recall that for any stopping times τ ≤ρ, the r.v. Bτ is
Fρ-measurable). Furthermore, the state space ofWish()Z(recall (5.4) andB0= 0), its
trajectories are piecewise constant and its jumps are IID with distributiong(BT1). The jump times ofWare given by the sequence of timesTn,n∈N, for the following reason: T1is independent ofB and exponentially distributed making the r.v. BT1 continuous. Hence, by (5.4),g(BT1)6= 0almost surely implying thatW
jumps if and only if there is a jump inN. HenceWis a cádlág(F
t)-semimartingale, equal to the sum of its jumps, which is a continuous-time random walk in its own filtration.