Proof of theorem 4.3

In this section we prove theorem 4.3. For convenience of the reader we recall the statement.

There exists a càdlàg P-supermartingale

U↑(H, P) := {U_t↑(H, P)}0≤t≤T

such that

U_τ↑(H, P) = ess sup_{P ∈P}ess sup_{θ∈T [τ,T ]}EP[Hθ | Fτ], R − a.s.,

for any stopping time τ ∈ T . Moreover, U↑(H, P) is the smallest P-supermartingale

above H in the sense that S ≥ U↑(H, P) whenever S is a càdlàg P-supermartingale

such that S ≥ H as in definition 1.1.

Proof. We simplify notation and write U↑ = U↑(H, P). Let P1 ∈ P

be fixed but arbitrary. Let us recall that in (4.11) we defined the family of random variables Z↑ = {Z_θ↑}θ∈T.

1. In this first step we show that the process {Z_t↑}0≤t≤T has a càdlàg mod-

ification. We use the fact that {Z_t↑}0≤t≤T has the P1-supermartingale

property as stated in lemma 4.15. The stopping time defined in (4.20) and the argument involved in (4.22) are important in this step and they were first considered by Föllmer and Kramkov[23]. The existence of a càdlàg modification will follow after proving that the correspondence

t → EP1[Z

↑

t] is right-continuous (see e.g., theorem 3.1 in Lipster and

Shiryayev[43]).

Let {ti}∞i=1⊂ [t, T ] be a decreasing sequence converging to t. We have

that EP1[Z ↑ t] ≥ lim_i→∞EP1[Z ↑ ti], since Z_t↑ is a P1-supermartingale.

Now we show the opposite inequality. From lemma 4.15 we know that for any  > 0, there exists a stopping time τ with t ≤ τ ≤ T and a probability measure P2 ∈ P with P2 = P1 in Ft such that

EP1[Z

↑

t] ≤  + EP1[EP2[Hτ | Ft]] =  + EP2[Hτ]. (4.19)

Now we define

τ(i) := τ 1{τ ≥ti}+ T 1{τ <ti} ∈ T [ti, T ], (4.20)

and let Pi be the pasting of P1 and P2 in Fti. Then according to lemma

4.16 we get that

EPi[Hτ(i)] ≤ EP1[Z

↑

ti]. (4.21)

so that lim infi→∞EPi[Hτ(i)] ≤ lim infi→∞EP1[Z

↑

ti]. Now in order to ob-

tain the inequality EP1[Z

↑

t] ≥ limi→∞EP1[Z

↑

ti] it only remains to show

that EP2[Hτ] ≤ lim infi→∞EPi[Hτ(i)].

Let F denote the density process of P2 with respect to P1, notice that

lims&tFs = Ft = 1, R-a.s.. According to lemma 4.8, the density of Pi

with respect to P1 is equal to F_FT

ti, then EP2[Hτ] = EP1[FTHτ] = EP1 " lim i→∞ FT Fti H_τ(i) # (4.22) ≤ lim infi→∞EP1

"

FT

Fti

H_τ(i)

#

where in the inequality we have applied Fatou’s lemma. From (4.19) and (4.21) we conclude the opposite inequality EP1[Z

↑

t] ≤ limi→∞EP1[Z

↑

ti].

2. Let {U_t↑}0≤t≤T be a càdlàg modification of the process {Z ↑

t}0≤t≤T, and

let τ ∈ T be a fixed stopping time. We now show that

U_τ↑ = ess sup_{P ∈P}ess sup_{θ∈T [τ,T ]}EP[Hθ | Fτ].

For an arbitrary stopping time θ ∈ T , let us define the usual dyadic discretizations θi = 2i_{T −1} X j=0 j + 1 2i 1{_2ij≤θ<j+1_2i } + T 1{θ=T }. (4.23) Clearly {θi_}∞

i=1 is a decreasing sequence of stopping times converging

to θ, R-a.s. Note also that U_θ↑i = Z

↑

θi R-a.s. since the stopping time θi

takes only a finite number of values.

Let τ ∈ T be an arbitrary stopping time. In order to prove that

Z_τ↑ ≤ U↑

τ we have to show that EP[Hθ | Fτ] ≤ Uτ↑ for θ ∈ T [τ, T ] and

P ∈ P, i.e., EP[1AHθ] ≤ EP[1AUτ↑] for any A ∈ Fτ. Indeed:

EP[1AHθ] = EP[ lim i→∞1AHθ

i] ≤ E_P[lim inf_i→∞1_AZ↑

θi]

= EP[lim infi→∞1AU

↑

θi] ≤ lim infi→∞EP[1AU

↑

θi]

= lim infi→∞EP[1AEP[U

↑

θi | Fτ]] ≤ EP[1AUτ↑],

where in the first equality we used the fact that H is right continuous, in the following inequality the definition of Z_θ↑i, in the next equality

that U_θ↑i = Z

↑

θi, in the following inequality Fatou’s lemma, and in the

last inequality the P -supermartingale property of U↑.

In order to prove equality of these variables, it suffices to demonstrate that EP[Uτ↑] ≤ EP[Zτ↑] for P ∈ P fixed, since we now know that Z

↑

τ ≤

U_τ↑. Using again the usual dyadic discretisations {τi_}∞

i=1 of τ we get

the following inequalities:

EP[Zτ↑] ≥ lim infi→∞EP[Z

↑

τi] = lim infi→∞EP[U

↑

τi] ≥ EP[Uτ↑],

where we have applied the P -supermartingale property of Z↑, and where the last inequality follows by Fatou’s lemma since U↑ is right continuous.

3. We now prove the last part of the theorem. Let S be a càdlàg P- supermartingale such that S ≥ H. Then it follows that EP[Hθ | Ft] ≤

EP[Sθ | Ft] ≤ St P -a.s. for θ ∈ T [t, T ]. This implies that St≥ ZtP for

any P ∈ P, and thus St ≥ Z

↑

t. Since the processes U↑ and S are right

continuous we obtain that S ≥ U↑_.