Table Of ContentLOCAL MARTINGALES IN DISCRETE TIME
VILMOS PROKAJANDJOHANNESRUF
7 Abstract. Foranydiscrete-timeP–localmartingaleS thereexistsaprobabilitymeasureQ∼
1
PsuchthatS isaQ–martingale. Anewproofforthisresultisprovided. Thisproofalsoyields
0
2 that, for any ε>0, themeasure Q can be chosen so that dQ/dP≤1+ε.
n
a
J
5 1. Introduction and related literature
1
Let (Ω,F,P) denote a probability space equipped with a discrete-time filtration (F ) ,
R] where Ft ⊂ F. Moreover, let S = (St)t∈N0 denote a d-dimensional P–local martingale,twth∈eNr0e
P d ∈N. ThenthereexistsaprobabilitymeasureQ,equivalenttoP,suchthatS isaQ–martingale.
. This follows from more general results that relate appropriate no-arbitrage conditions to the
h
t existence of an equivalent martingale measure; see Dalang et al. (1990) and Schachermayer
a
(1992) for the finite-horizon case and Schachermayer (1994) for the infinite-horizon case. These
m
results are sometimes baptized fundamental theorems of asset pricing.
[
More recently, Kabanov (2008) and Prokaj and Ra´sonyi (2010) have provided a direct proof
1 for the existence of such a measure Q; see also Section 2 in Kabanov and Safarian (2009).
v
The proof in Kabanov (2008) relies on deep functional analytic results, e.g., the Krein-Sˇmulian
5
2 theorem. The proof in Prokaj and Ra´sonyi (2010) avoids functional analysis but requires non-
0 trivial measurable selection techniques.
4
As this note demonstrates, in one dimension, an important but special case, the Radon-
0
. Nikodym derivative Z∞ = dQ/dP can be explicitly constructed. Moreover, in higher dimensions,
1
themeasurable selection results can besimplified. This is donehereby appropriately modifying
0
7 an ingenious idea of Rogers (1994).
1 More precisely, the following theorem will be proved in Section 3.
:
v
Xi Theorem 1. For all ε > 0, there exists a uniformly integrable P–martingale Z = (Zt)t∈N0,
bounded from above by 1+ε, with Z = lim Z > 0, such that ZS is a P–martingale and
∞ t↑∞ t
r such that E [Z |S |p]< ∞ for all t,p ∈ N .
a P t t 0
The fact that the bound on Z can be chosen arbitrarily close to 1 seems to be a novel
observation. Considering a standard random walk S directly yields that there is no hope for a
stronger version of Theorem 1 which would assert that ZS is not only a P–martingale but also
a P–uniformly integrable martingale.
A similar version of the following corollary is formulated in Prokaj and Ra´sonyi (2010); it
would also bea direct consequence of Kabanov and Stricker (2001). To state it, let us introduce
the total variation norm k·k for two equivalent probability measures Q ,Q as
1 2
kQ1−Q2k= EQ1[|dQ2/dQ1−1|].
Corollary 2. For all ε > 0, there exists a probability measure Q, equivalent to P, such that S
is a Q–martingale, kP−Qk< ε, and E [|S |p]< ∞ for all t,p ∈ N .
Q t 0
Date: January 17, 2017.
2010 Mathematics Subject Classification. Primary: 60G42; 60G48.
We thank Yuri Kabanov for many helpful comments. J.R. is grateful for the generous support provided by the
Oxford-Man Instituteof QuantitativeFinance at theUniversity of Oxford.
1
2 VILMOSPROKAJANDJOHANNESRUF
To reformulate Corollary 2 in more abstract terms, let us introduce the spaces
M = {Q ∼ P : S is a Q–local martingale};
loc
Mp = {Q ∼ P : S is a Q–martingale with E [|S |p]< ∞ for all t ∈ N }, p > 0.
Q t 0
Then Corollary 2 states that the space Mp is dense in M with respect to the total
p>0 loc
variation norm k·k.
T
Proof of Corollary 2. Consider the P–uniformly integrable martingale Z of Theorem 1, with ε
replaced by ε/2. Then the probability measure Q, given by dQ/dP= Z∞, satisfies the conditions
of the assertion. Indeed, we only need to observe that
E [|Z −1|] = 2E [(Z −1)1 ]≤ ε,
P ∞ P ∞ {Z>1}
where we used that E [Z −1] = 0 and the assertion follows. (cid:3)
P ∞
2. Generalized conditional expectation and local martingales
For sake of completeness, we review the relevant facts related to local martingales in discrete
time. To start, note that for a sigma algebra G ⊂ F and a nonnegative random variable Y,
not necessarily integrable, we can define the so called generalized conditional expectation
E [Y |G] = limE [Y ∧k|G].
P P
k↑∞
Next, for a general random variable W with E [|W||G] < ∞, but not necessarily integrable,
P
we can define the generalized conditional expectation
E [W |G]= E [W+|G]−E [W−|G].
P P P
For a stopping time τ and a stochastic process X we write Xτ to denote the process obtained
from stopping X at time τ.
Definition 3. A stochastic process S = (S ) is
t t∈N0
• a P–martingale if E [|S |]< ∞ and E [S |F ] = S for all t ∈ N ;
P t P t+1 t t 0
• a P–local martingale if there exists a sequence (τ ) of stopping times such that
n n∈N
lim τ = ∞ and Sτn1 is a P–martingale;
n↑∞ n {τn>0}
• a P–generalized martingale if E [|S ||F ]< ∞ and E [S |F ] = S for all t ∈ N .
P t+1 t P t+1 t t 0
Proposition 4. Any P–local martingale is a P–generalized martingale.
This proposition dates back to Theorem II.42 in Meyer (1972); see also Theorem VII.1 in
Shiryaev (1996). Its reverse direction would also be true but will not be used below. A direct
corollary of the proposition is that a P–local martingale S with E [|S |] < ∞ for all t ∈ N is
P t 0
indeed a P–martingale.
For sake of completeness, we will provide a proof of the proposition here.
Proof of Proposition 4. Let S denote a P–local martingale. Fix t ∈ N and a localization
0
sequence (τ ) . For each n∈ N, we have, on the event {τ > t},
n n∈N n
E [|S ||F ] = limE [|S |∧k|F ]= lim E [|Sτn |∧k|F ] = E [|Sτn ||F ] < ∞.
P t+1 t P t+1 t P t+1 t P t+1 t
k↑∞ k↑∞
Since lim τ = ∞, we get E [|S ||F ]< ∞.
n↑∞ n P t+1 t
The next step we only argue for the case d = 1, for sake of notation, but the general case
follows in the same manner. As above, again for fixed n ∈ N, on the event {τ > t}, we get
n
E [S |F ] = lim E [S+ ∧k|F ]−E [S− ∧k|F ]
P t+1 t P t+1 t P t+1 t
k↑∞
Ä = limä EP[(Stτ+n1∧k)∨(−k)|Ft]= St.
k↑∞
Thanks again to lim τ = ∞, the assertion follows. (cid:3)
n↑∞ n
LOCAL MARTINGALES IN DISCRETE TIME 3
Example 5. Assume that (Ω,F,P) supports two independent random variables U and θ such
that U is uniformly distributed on [0,1], and P[θ = −1] = 1/2 = P[θ = 1]. Moreover, let us
assume that F = {∅,Ω}, F = σ(U), and F = σ(U,θ) for all t ∈ N\{1}. Then the stochastic
0 1 t
process S = (St)t∈N0, given by St = θ/U1t≥2 is easily seen to be a P–generalized martingale and
a P–local martingale with localization sequence (τ ) given by
n n∈N
τ = 1×1 +∞×1 .
n {1/U>n} {1/U≤n}
However, we have EP[|S2|] = EP[1/U] = ∞; hence S is not a P–martingale.
Now, consider the process Z = (Z ) , given by Z = 1 +2U1 . A simple computation
t t∈N0 t t=0 t≥1
shows that Z is a strictly positive P–uniformly integrable martingale. Moreover, since Z S =
t t
2θ1 , we have E [Z |S |] ≤ 2 for all t ∈ N and ZS is a P–martingale. If we require the
t≥2 P t t 0
Radon-Nikodym to be bounded by a constant 1+ε ∈ (1,2], we could consider Z = (Z )
t t∈N0
with Zt = 1t=0 +(U∧ε)/(ε−ε2/2)1t≥1. This illustrates the validity of Theorem 1 in the context
of this example.
“ “
To see a difficulty in provingTheorem 1, let us consider a local martingale S′ = (S′) with
“ t t∈N0
two jumps instead of one; for example, let us define
θ
S′ = 1 −1 1 + 1 .
t {U>1/2} {U<1/2} t≥1 U t≥2
Again, it is simple to see that tÄhis specification maäkes S′ indeed a P–local and P–generalized
martingale. However, now we have EP[Z1S1′] = 1/2 6= 0; hence ZS′ is not a P–martingale.
Similarly, neither is ZS′. Nevertheless, as Theorem 1 states, there exists a uniformly integrable
P–martingale Z′ such that Z′S′ is a P–martingale.
More details on th“e previous example are provided in Ruf (2017).
3. Proof of Theorem 1
We start with three preparatory lemmata.
Lemma 6. Let Q denote some probability measure on (Ω,F), let G,H be sigma algebras with
G ⊂ H ⊂ F, let W denote a H –measurable d-dimensional random vector with
E [|W||G] < ∞ and E [W |G]= 0. (3.1)
Q Q
Suppose that (α ) is a bounded family of H –measurable random variables with lim α =
k k∈N k↑∞ k
1. Then for any ε> 0 there exists a family (V ) of random variables such that
k k∈N
(i) V is H –measurable and takes values in (1−ε,1) for each k ∈N;
k
(ii) limk↑∞1{EQ[VkαkW|G]=0} = 1.
We shall provide two proofs of this lemma, the first one applies only to the case d = 1, but
avoids the technicalities necessary for the general case.
Proof of Lemma 6 in the one-dimensional case. With the convention 0/0 := 1, define, for each
k ∈ N, the random variable
E [α W+|G]
Q k
C =
k E [α W−|G]
Q k
and note that
E [W+|G] 1
lim|C −1| = Q −1 = E [W+|G]−E [W−|G] = 0.
k↑∞ k (cid:12)EQ[W−|G] (cid:12) EQ[W−|G] Q Q
(cid:12) (cid:12) (cid:12) (cid:12)
Next, set (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)
V = (1−ε)∨ 1 (1∧C−1)+1 (1∧C ) ,
k {W≥0} k {W<0} k
and note that on the event {1−ε Ä≤ Ck ≤ 1/(1−ε)} ∈ G we indeed haväe EQ[VkαkW |G] = 0,
which concludes the proof. (cid:3)
4 VILMOSPROKAJANDJOHANNESRUF
Proof of Lemma 6 in the general case. Theproof is similar to theproof of theDalang–Morton–
Willinger theorem based on utility maximisation, see Rogers (1994) and Delbaen and Schacher-
mayer (2006, Section 6.6) for detailed exposition. But instead of using the exponential utility,
we choose a strictly convex function (the negative of the utility) which is smooth and whose
derivative takes values in (1−ε,1). Indeed, in what follows we fix the convex function
ε π
f(a)= a 1+ arctan(a)− , a ∈ R.
π 2
Then f is smooth and direct comÅputatiÅon shows, thatãfã is convex with derivative f′ taking
values in the interval (1−ε,1).
We formulated the statement with generalized conditional expectations. However, changing
the probability appropriately with a G–measurable density we can assume, without loss of
generality, that W ∈ L1(Q). Indeed, the probability measure Q′, given by
dQ′ e−EQ[|W||G]
= ,
dQ EQ[e−EQ[|W||G]]
satisfies that W ∈ L1(Q′). Moreover, the (generalized) conditional expectations with respect to
G arethe same underQ and Q′. So in whatfollows we assumethat |W|is an integrable random
variable.
For W there is a maximal G–measurable orthogonal projection R of Rd such that RW = 0
almost surely. For a proof, see Proposition 2.4 in Rogers (1994) or Section 6.2 in Delbaen and
Schachermayer (2006). The orthocomplement of the range of R is called the predictable range
of W.
Let B denote the d–dimensional Euclidean unit ball and set α = 1. For each k ∈ N∪{∞},
∞
consider the random function (or field) h over B, defined by the formula
k
h (u,·) = h (u) = E [f(α W ·u)|G]+|Ru|2 for all u ∈B.
k k Q k
Since f is continuous, for each k ∈ N∪{∞}, h has a version that is continuous in u for each
k
ω ∈ Ω; see Lemma 9 below. Then for each compact subset C of B and each k ∈ N∪{∞} there
is a G–measurable random vector UC taking values in C such that h (UC) = min h (u).
k k k u∈C k
This is a kind of measurable selection; for sake of completeness we give an elementary proof
below in Lemma 11.
Next, for each k ∈ N, let U be a G–measurable minimiser of h in the unit ball B and define
k k
V = f′(α W ·U ).
k k k
With this definition, (i) follows directly. For (ii) we prove below that
E [V α W |G]+2RU = 0, on {|U | < 1}, k ∈ N; (3.2)
Q k k k k
lim U = 0, almost surely. (3.3)
k
k↑∞
Then, on the event {|U |< 1}, (3.2) and the G–measurability of R yield
k
|E [V α W |G]|2 = −E [V α W |G]·Ru= −E [V α RW |G]·u= 0,
Q k k Q k k Q k k
giving us (ii).
Thus, in order to complete the proof it suffices to argue (3.2)–(3.3). For (3.2), note that h
k
is continuously differentiable almost surely for each k ∈ N, see Lemma 10 below; morever, its
derivative attheminimumpointU , whichequals theleft-hand sideof (3.2), mustbezero when
k
U is inside the ball B.
k
For (3.3) observe that h has a unique minimiser over B which is the zero vector. To see
∞
this, observe that
h (u) = E [f(W ·(I −R)u)|G]+|Ru|2,
∞ Q
LOCAL MARTINGALES IN DISCRETE TIME 5
where I denotes the d-dimensional identity matrix. So to see that the zero vector is the unique
minimiser it is enough to show that inf h (u) > 0 = h (0) almost surely for any δ ∈(0,1].
|u|≥δ ∞ ∞
Let U be a G–measurable minimiser of h over {u : |u| ∈ [δ,1]}. Then
∞
E [f(W ·(I −R)U)|G]> 0, on {(I −R)U 6= 0};
Q
|RU|2 ≥ δ2 > 0, on {(I −R)U = 0}.
The first part follows from the strict convexity of f in conjunction with Jensen’s inequality,
taking into account that E [W |G] = 0 and that W ·(I −R)U has non-trivial conditional law
Q
on {(I −R)U 6= 0} by the maximality of R. Whence inf h (u) > 0 = h (0), as required.
|u|≥δ ∞ ∞
Finally, as lim α = 1 and f is Lipschitz continuous we have
k↑∞ k
lim sup|h (u)−h (u)| = lim sup |h (u)−h (u)| = 0 almost surely.
k ∞ k ∞
k↑∞u∈B k↑∞u∈B∩Qd
Hence, any G–measurable sequence (U ) of minimisers of h converges to zero, the unique
k k∈N k
minimiser of h , almost surely. This shows (3.3) and completes the proof. (cid:3)
∞
Lemma 7. Let Q denote some probability measure on (Ω,F), let G,H be sigma algebras with
G ⊂ H ⊂ F, let Y denote a one-dimensional random variable with Y ≥ 0 and E [Y |H ] < ∞,
Q
and let W denote a H –measurable d-dimensional random vector such that (3.1) holds. Then,
for any ε > 0, there exists a random variable z such that
(i) z is H –measurable and takes values in (0,1+ε);
(ii) Q[z < 1−ε] < ε;
(iii) E [z|G]= 1;
Q
(iv) E [zW |G] = 0;
Q
(v) E [zY |G]< ∞.
Q
Proof. For each k ∈ N, define the (0,1]–valued, H –measurable random variable
1
αk = 1{EQ[Y|H]≤k}+ E [Y |H ]1{EQ[Y|H]>k}
Q
and note that lim α = 1. Lemma 6 now yields the existence of a family (V ) of
k↑∞ k k k∈N
H –measurable random variables such that Vk ∈ (1/(1+ε/2),1) and limk↑∞1{EQ[VkαkW|G]=0} =
1. Note that this yields a G–measurable random variable K, taking values in N, such that
EQ[VKαKW |G] =0, EQ[VKαK |G] >1/(1+ε), and Q[EQ[Y |H ] > K]< ε. Setting now
V α
K K
z =
E [V α |G]
Q K K
yields a random variable with the claimed properties. (cid:3)
Lemma 8. Fix n ∈ N , let Q denote some probability measure on (Ω,F) such that S is a Q–
0
local martingale, and let Y denote a nonnegative random variable with E [Y |F ]< ∞. Then,
Q n
for each ε > 0, there exists a probability measure Q′, equivalent to Q, with density Z(n) = dQ′/dQ
such that
(i) Z(n) ∈ (0,1+ε);
(ii) Q[Z(n) < 1−ε] < ε;
(iii) S is a Q′–local martingale;
(iv) EQ′[Y]< ∞.
Proof. Inthisproof,we usetheconvention F = {∅,Ω} and∆S = 0. Setε > 0besufficiently
−1 0
small such that
(n+1)ε ≤ ε, (1+ε)n+1 ≤ 1+ε, (1−ε)n+1 ≥ 1−eε.
With εreplaced by εandwithG = F , H = F , andW = ∆S , henceE [|W||G] < ∞and
n−1 n n Q
E [W |G] = 0 by Proposeition 4, let z deenote the correspondingerandom variable of Lemma 7.
Q n
If n = 0, we define Qe′ by dQ′/dQ = z, and the lemma is proven.
6 VILMOSPROKAJANDJOHANNESRUF
Ifn> 0, weproceediteratively. Considert ∈ {0,··· ,n−1} andassumethatwehaverandom
variables z ,··· ,z such that, in particular, E [Y n z | F ] < ∞. We now obtain a
t+1 n Q i=t+1 i t
random variable z by again applying Lemma 7, with ε replaced by ε and with G = F ,
t t−1
H = F , W = ∆S , and Y replaced by Y n z . Q
t t i=t+1 i
With the family (z0,··· ,zn) now given, lQet us define Z(n) = ni=0zi aend Q′ by dQ′/dQ = Z(n).
To argue that S is a Q′–local martingale, we may consider any sequence of stopping times that
Q
localizes S. Since all other assertions follow directly from the construction of Z(n) and the
choice of ε, the lemma is proven. (cid:3)
Proof of Theorem 1. We inductively construct a sequence (Q(n)) of probability measures,
e n∈N0
equivalenttoP,andasequence(ε(n)) ofpositiverealsusingLemma8. Tostart, setQ(−1) =
n∈N0
P. Now, fixn ∈ N forthemomentandsupposethatwehave Q(n−1) and(ε(m)) suchthat
0 0≤m<n
n−1(1+ε(m))< 1+ε. Choose ε(n) to be sufficiently small such that n (1+ε(m)) < 1+ε,
m=0 m=0
and for any A ∈ F with Q(n−1)[A] ≤ ε(n) we have P[A] < 2−n. Then apply Lemma 8 with ε
Q Q
replaced by ε(n), and with Q = Q(n−1) and Y = e|Sn| to obtain a probability measure Q(n) with
density Z(n), that is dQ(n) =Z(n)dQ(n−1) = ( n Z(m))dP.
m=0
Due to the fact
Q
P |1−Z(n)| > ε(n) ≤ 2−n as Q(n−1) |1−Z(n)| > ε(n) ≤ ε(n),
the Borel-Cantelli lemma yields |1 − Z(n)| < ∞; hence the infinite product Z =
î ó n∈N0 î ó ∞
∞ Z(n) converges and is positive P–almost surely. It is clear that Z ≤ 1+ε.
n=0 P ∞
We define the probability measure Q by dQ/dP = Z∞ and denote the corresponding density
Q
process by Z = E [Z |F ], for each t ∈ N . As Z(m) < 1+ε we have Q ≤ (1+ε)Q(t)
t P ∞ t 0 m>t
and as a result
Q
E Z e|St| = E e|St| ≤(1+ε)E e|St| < ∞
P t Q Q(t)
by the choice of Q(t); hence E [Z |S |p] < ∞ for all t,p ∈ N .
î P tó t î ó 0î ó
It remains to argue that ZS is a P–martingale or, equivalently, that S is a Q–martingale.
Sincewealreadyhave establishedE [|S |]< ∞forallt ∈ N , itsufficestofixt ∈ Nandtoprove
Q t 0
that E [S |F ] = S . To this end, recall that S is a Q(n)–local martingale for each n ∈ N
Q t t−1 t−1 0
by Lemma 8(iii) and note that dominated convergence, Bayes formula, and Proposition 4 yield
n
E [S |F ]Z = E [S Z |F ]= lim E S Z(m) F
Q t t−1 t−1 P t ∞ t−1 P t t−1
n↑∞ " (cid:12) #
mY=0 (cid:12)
dQ(n) (cid:12)(cid:12) n
= lim E [S |F ] = S l(cid:12)im E Z(m) F
n↑∞ Q(n) t t−1 dP (cid:12)(cid:12)Ft−1 t−1n↑∞ P"mY=0 (cid:12)(cid:12) t−1#
(cid:12) (cid:12)
= St−1Zt−1. (cid:12) (cid:12)
(cid:12) (cid:12)
This completes the proof. (cid:3)
Appendix A
In this appendix, we provide some measurability results necessary for the proof of Lemma 6.
Lemma 9. Let G be a sigma algebra with G ⊂ F and let ξ be a random element in C(K), where
(K,m) isacompact metric space. Suppose thatE [sup |ξ(u)|] < ∞andletη(u) = E [ξ(u)|G]
P u∈K P
for all u ∈K. Then (η(u)) has a continuous modification.
u∈K
Proof. Let D be a countable dense subset of K. We show that there is Ω′ ∈ G with full
probability such that (η(u)) is uniformly continuous over D on Ω′. Then we can define
u∈D
lim η(u ) on Ω′,
n
η˜(u) = uunn→∈Du
0 otherwise.
LOCAL MARTINGALES IN DISCRETE TIME 7
It is a routine exercise to check that η˜ is well defined and a continuous modification of η.
One way to get Ω′ is the following. Let µ be the modulus of continuity of ξ, that is,
µ(δ) = sup |ξ(u)−ξ(u′)|, δ > 0.
u,u′∈K,m(u,u′)≤δ
Obviously µ(δ) → 0 everywhere as δ ↓ 0. Dominated convergence, in conjunction with the
bound µ ≤ 2sup |ξ(u)|, yields µ˜(δ) = E[µ(δ)|G]→ 0 as δ ↓0 almost surely. Now define
u∈K
1 1
Ω′ = lim µ˜ = 0 ∩ |η(u)−η(u′)| ≤ µ˜ .
n↑∞ n n
ß Å ã ™ Ñn\∈N u,u′∈D,m\(u,u′)≤1/nß Å ã™é
Clearly Ω′ has full probability and the claim is proved. (cid:3)
In the setting of Lemma 9 when K ⊂ Rd and ξ is a random element in C1(K) then under
mild conditions η(u) = E[ξ(u)|G] has a version taking values in C1(K). This is the content
of the next lemma. Recall that a function f defined on K belongs to C1(K) if f is continuous
and there is continuous Rd–valued function on K which agrees with the gradient f′ of f in the
interior of K.
Lemma 10. Let G be a sigma algebra with G ⊂ F and let ξ be a random element in C1(K),
where K ⊂ Rd is a compact subset set. Suppose that
E sup|ξ(u)| +E sup|ξ′(u)| < ∞
P P
u∈K u∈K
ñ ô ñ ô
and let η(u) = E [ξ(u)|G] for all u∈ K. Then (η(u)) has a version taking values in C1(K)
P u∈K
and the continuous version of (E[ξ′(u)|G]) gives the gradient of η almost surely.
u∈K
Proof. By Lemma 9 both η(u) = E[ξ(u)|G] and η′(u) = E[ξ′(u)|G] have continuous versions.
We prove that, apart from a null set, η′ is indeed the gradient of η. To this end, let D be a
countable dense subset of the interior of K and denote by I(a,b) a directed segment going from
a to b, for each a,b ∈ K. Then, by assumption, for a,b ∈ D, with I(a,b) ⊂ intK we get
η(b)−η(a) = E[ξ(a)−ξ(b)|G] =E ξ′(u)du G = η′(u)du, almost surely.
ñZI(a,b) (cid:12)(cid:12) ô ZI(a,b)
Hence, there exists an event Ω′ ∈ G with P[Ω′] = 1 su(cid:12)(cid:12)ch that
(cid:12)
η(b,ω)−η(a,ω) = η′(u,ω)du, for all a,b ∈D, with I(a,b) ⊂ intK and ω ∈ Ω′.
ZI(a,b)
By continuity this idenity extends to all a,b ∈ intK with I(a,b) ⊂ intK on Ω′. Using again
the continuity of η′(.,ω) yields that η′ is indeed the gradient of η on Ω′. (cid:3)
Lemma 11. Let (K,m) be a compact metric space and η a random element in C(K). Then
there is a measurable minimiser of η, that is a random element U in K, such that η(U) =
min η(u).
u∈K
Proof. To shorten the notation, for each x ∈ K and δ ≥ 0, let
B(x,δ) = {u ∈K : m(u,x) ≤δ}, η(x,n) = min{η(u) : u∈ B(x,2−n)}.
For each n ∈N let D be a finite 2−n-net in K; that is, K ⊂ B(x,2−n). For each n ∈ N
n x∈Dn
fix an order of the finite set D . We shall use the fact that for any closed set F the minimum
n
S
over F, that is, min η(u), is a random variable.
u∈F
We define a sequence (U ) of random elements in K by recursion, such that
n n∈N
• η(U ,n)= min η(u), and
n u∈K
• m(U ,U )≤ 2−n +2−(n+1).
n n+1
8 VILMOSPROKAJANDJOHANNESRUF
Then (U ) has a limit U which is a measurable minimiser of η over K.
n n∈N
For n = 1 let U be the first element in
1
{v ∈ D : η(v,1) = minη(u)}.
1
u∈K
Since this set is not empty, U is well defined. If U ,...,U are defined for some n ∈ N set U
1 1 n n+1
to be the first element in
v ∈ D : η(v,n+1) = minη(u), m(v,U )≤ 2−n +2−(n+1)
n+1 n
u∈K
This set is not eßmpty as ™
B(U ,2−n)⊂ B(v,2−(n+1)),
n
v∈D[n+1
m(v,Un)≤2−n+2−(n+1)
so U is well defined. We conclude that the sequence with the above properties exists and its
n+1
limit is a measurable minimiser. (cid:3)
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Vilmos Prokaj, Department of Probability Theory and Statistics, Eo¨tvo¨s Lora´nd University,
Budapest
E-mail address: [email protected]
JohannesRuf,DepartmentofMathematics,London SchoolofEconomicsandPoliticalScience
E-mail address: [email protected]
