The Mathematics of Arbitrage

15.3 An Example 333

Proof.Fix a collection ((εn,k)^2

n− 1

k=1)n≥^1 of independent random variables,

εn,k=

{

− 2 −n with probability (1− 4 −n)

2 n(1− 4 −n) with probability 4−n

so thatE[εn,k] = 0. We construct a martingaleMsuch that at times

tn,k=

2 k− 1

2 n

,n∈N,k=1,..., 2 n−^1 ,

Mjumps by a suitable multiple ofεn,k,e.g.

Mt=

∑

(n,k):tn,k≤t

8 −nεn,k,t∈[0,1],

so thatMis a well-defined uniformly bounded martingale (with respect to its
natural filtration).
Defining the integrandsHnby

Hn=

(^2) ∑n−^1
k=1
8 nχ{tn,k},n∈N,
we obtain, for fixedn∈N,
(Hn·M)t=

∑

k:tn,k≤t

εn,k,

so thatHn·Mis constant on the intervals

[ 2 k− 1

2 n ,

2 k+1

2 n

[

and, on a set of prob-

ability bigger that 1− 2 −n,H·Mequals− 2 knon the intervals

[ 2 k− 1

2 n ,

2 k+1

2 n

[

.

Also on a set of probability bigger than 1− 2 −nwe have that [Hn·M,

Hn·M] 1 =

∑ 2 n−^1

k=1^2

− 2 n=2−n− (^1).
From the Borel-Cantelli lemma we infer that, for eacht∈[0,1], the random
variables (Hn·M)tconverge almost surely to the constant function− 2 tand
that [Hn·M, Hn·M] 1 tend to 0 a.s., which proves the final assertions of the
above claim.
We still have to estimate theH^1 -norm ofHn·M:
‖Hn·M‖H^1 ≤
(^2) ∑n−^1
k≥ 1
‖εn,k‖L^1
=2n−^1 [2−n(1− 4 −n)+2n(1− 4 −n)· 4 −n]≤ 1.

Remark 15.3.2.What is the message of the above example? First note that
passing to convex combinations (Kn)n≥ 1 of (Hn)n≥ 1 does not change the
picture: we always end up with a sequence of martingales (Kn·M)n≥ 1 bounded