El Yaraví

In a similar way Z_TQ Z T 0 Vs−dBis= Z T 0  Z_sQVs−  dB_si+ Z T 0 Z s− 0 Vz−dBzi  dZ_sQ.

If we take R-expectation in this formula we get that

ER " Z_TQ Z T 0 Vs−dBsi # = ER " Z T 0  Zs−QVs−  dBi_s # , (2.24)

where we have used the fact that the process Bi _{is predictable. Now from}

(2.23) and (2.24) we conclude that

EQ[Vγi] = E_R[(ZQV )_γi].

The process {(ZQV )t}0≤t≤T is an element of V since the density process ZQ

satisfies (2.14) and V ∈ L∞. Then lim

i→∞EQ[Vγi] = limi→∞ER[(Z Q

V )γi] = ER[(Z

Q

V )γ0] = E_Q[V_γ0].

2.2.5

Proof of theorem 2.11

Proof. Recall that the set of processes V(c) was defined in (2.15). We start with the following equalities

P H(c) = inf V ∈V(c)supθ∈T sup Q∈Q EQ[f (Hθ, Vθ)] = inf V ∈V(c)Q∈Qsup sup θ∈T EQ[f (Hθ, Vθ)] = inf V ∈V(c)Q∈Qsup sup γ∈A EQ[f (H, V )γ].

The first equality was proved in proposition 2.10. The second equality is trivial. In the last equality we have applied theorem 2.15.

In proposition 2.24 below we prove the existence of a pair (Q∗, γ∗) ∈ Q×A such that inf V ∈V(c)EQ ∗[f (H, V )_γ∗] = inf V ∈V(c)Q∈Qsup sup γ∈A EQ[f (H, V )γ].

This identity implies that

P H(c) = inf

V ∈V(c)EQ

∗[f (H, V )_γ∗] ≤ inf

V ∈V(c)supθ∈T

EQ∗[f (H_θ, V_θ)] ≤ P H(c),

which proves that Q∗ _{is a worst-case probability measure.}

Proposition 2.24 Assume the conditions of theorem 2.11. Then, there exist

a pair (Q∗, γ∗) ∈ Q × A such that inf V ∈V(c)EQ ∗[f (H, V )_γ∗] = inf V ∈V(c)Q∈Qsup sup γ∈A EQ[f (H, V )γ]. (2.25)

Proof. Let us note that the equality (2.25) can be written as inf

V ∈V(c)q

∗

[f (H, V )] = inf

V ∈V(c)_q∈L(Q×A)sup q[f (H, V )], (2.26)

whereL(Q×A) is the set of functionals of definition 2.19 and q∗ ∈L(Q×A). We are going to verify the hypotheses of theorem 2.25 below. To this end, let us specify the elements in that theorem. The compact Hausdorff topological space X corresponds to L(Q × A). The topological vector space

F corresponds to L∞ and the convex subset Y to V(c). We define a function

G :L(Q × A) × L∞_{→ R}

by

G(q, V ) := q[V ].

Note that if q ∈L(Q × A) is represented by a pair (Q, γ) ∈ Q × A then

G((Q, γ), V ) = EQ " Z T − 0 f (Hs, Vs)dAs+ Z T 0+ f (H−s, V−s)dBs # .

Now we check the conditions in theorem 2.25. For arbitrary q ∈L(Q×A) it will be convenient to work with a representing pair (Q, γ) ∈ Q × A.

1. The functional G is convex in the variable V ∈ V(c) since f (h, ·) is convex. Let (Q0_{, γ}0_{) ∈ Q × A be a fixed pair, we verify continuity of}

G((Q0, γ0_{), ·) with respect to the norm of L}∞.

Let {Vi}∞i=1⊂ V(c) be a sequence converging to V ∈ V(c) in L

∞_{. The}

random variable f (H, Vi)γ0 converges to the random variable f (H, V )_γ0

R-a.s. since f (h, ·) is continuous. We recall that V ∈ V(c) satisfies

0 ≤ V ≤ H, and that H ≤ K for some constant K > 0. Thus, we are aloud to apply Lebesgue dominated convergence theorem to conclude that

lim

i→∞EQ

0[f (H, V_i)_γ0] = E_Q0[f (H, V )_γ0].

2. Now we verify continuity of G in the first argument with respect to the weak topology σ((L∞)∗_{, L}∞). Let {qλ}λ∈Λ ⊂ L(Q × A) be a net

converging weakly to q0 _{∈ L(Q × A). For V ∈ V(c) the convergence}

G(qλ_{, V ) → G(q}0_{, V ) in the weak topology is immediate because H is}

uniformly bounded and hence f (V, H) does as well. 3. Now for l ∈ R and V ∈ V(c) we define

L(V ) := {(Q, γ) ∈ Q × A | G((Q, γ), V ) ≥ l}.

For V1_{, · · · , V}n _{∈ V(c) we prove that}

L :=

n

\

i=1

L(Vi)

is either connected or empty. Assume it is nonempty and let (Q1_{, γ}1_{), (Q}2_{, γ}2₎

be two elements in the intersection L. Since G is linear separately in

Q and γ, we see that for any λ ∈ (0, 1), the pair

(Q1, λγ1+ (1 − λ)γ2) is an element of L, and so does the pair

(λQ1+ (1 − λ)Q2, γ2). We define a function q : [0, 2] → Q × A by q(λ) := ( (Q1_{, (1 − λ)γ}1_{+ λγ}2_{) if λ ∈ [0, 1],} ((2 − λ)Q1_{+ (λ − 1)Q}2_{, γ}2_{) if λ ∈ [1, 2].}

Note that q(1) is well defined, and that q(0) = (Q1_{, γ}1_{) and q(2) =}

In order to conclude that L is connected it is enough to show that q is continuous. That is, we have to show that for any V ∈ L∞, r > 0 and

t0 ∈ [0, 2] then

B(t0, V, r) := {t ∈ [0, 2] | |q(t)(V ) − q(t0)(V )| < r}

is an open subset of the interval [0, 2]. We only verify the case t0 = 1,

the other cases being similar. First take t ≥ 1, then

q(t)(V ) = (2 − t)EQ1[V_γ2] − (t − 1)E_Q2[V_γ2],

so that

|q(t)(V ) − q(t0)(V )| = (t − 1) |EQ1[V_γ2] − E_Q2[V_γ2]| .

Then we see that any t ∈ [1, 2] satisfying the inequality

t < 1 + r |EQ1[V_γ2] − E_Q2[V_γ2]|−1

is in B(t0, V, r).

Now let us take t ≤ 1, then

q(t)(V ) = (1 − t)EQ1[V_γ1] + tE_Q1[V_γ2],

so that

|q(t)(V ) − q(t0)(V )| = (1 − t) |EQ1[V_γ1] − E_Q2[V_γ1]| .

Then we see that any t ∈ [0, 1] satisfying the inequality 1 − t < r |EQ1[V_γ1] − E_Q2[V_γ1]|−1

is in B(t0, V, r). This shows that B(t0, V, r) is in fact an open subset of

[0, 2].

We have verified all the hypotheses of the topological minimax theorem 2.25. This theorem implies (2.25).

The proof of the proposition is now complete.

In the proof of theorem 2.24 we have applied the following topological minimax theorem.

Theorem 2.25 Let X be a compact Hausdorff topological space, and let Y be

a nonempty convex subset of a Hausdorff topological vector space F . Suppose that G : X × Y → R is a function satisfying the following conditions

1. G(x, ·) is lower semicontinuous and convex.

2. G(·, y) is upper semicontinuous.

3. for l ∈ R, m ∈ N and yi ∈ Y , the set m

\

i=1

{x ∈ X | G(x, yi) ≥ c}

is either connected or empty.

Then we have

max

x∈X y∈Yinf G(x, y) = infy∈Ymaxx∈X G(x, y).

Proof. See theorem 3.2 in [54].