The Mathematics of Arbitrage

12.4 The Existence of an Absolutely Continuous Local Martingale Measure 245

continuous local martingale, then the Girsanov-Maruyama transformation is,
at least formally, given by the local martingaleLt =exp(

∫t

0 −h

′

udMu−

1

2

∫t

0 h

′
ud〈M, M〉uhu),L^0 = 1. Formally one can verify thatLSis a local
martingale. However, things are not so easy. First of all, there is no guarantee
that the processhisM-integrable, soLneed not be defined. Second, even if
Lis defined, it may only be a local martingale and not a uniformly integrable
martingale. The examples in [S 93] and in Chap. 10 show that even when an
equivalent risk neutral measure exists, the local martingaleLneed not to be
uniformly integrable. In other words a risk neutral measure need not be given
byL. Third, in case the two previous points are fulfilled, the densityL∞need
not be different from zero a.s..
What can we save in our setting? In any case, Theorem 12.3.6 shows that
in the case whenSsatisfies the no-arbitrage property for general admissible
integrands, the processhsatisfies the properties:

(1)T=inf{t|

∫t

0 h

′d〈M,M〉h=∞}>0 a.s..

(2) The [0,∞]-valued process

∫t

0 h

′

ud〈M,M〉uhuis continuous; in particular,

it does not jump to∞.

In this case the stochastic integralsh·M andh·Scan be defined on the
interval [[0,T[[ and at timeTwe have thatLTcan be defined as the left limit.
The theory of continuous martingales ([RY 91]) shows that

{LT=0}=

{∫T

0

h′td〈M, M〉tht=∞

}

.

If after timeT, i.e. fort>T, we putLt= 0, the processLis well-defined, it is
a continuous local martingale, it satisfiesdLt=−Lth′tdMtandLSis a local
martingale. The processX=L^1 −1 is also defined on the interval [[0,T[[ a n d
on the set{LT=0}its left limit equals infinity. The crucial observation is
now that on the interval [[0,T[[,wehavethatdXt=L^1 th′tdSt.
This follows simply by plugging in Itˆo’s formula (compare Chap. 11).
For eachε>0letτεbe the stopping time defined byτε=inf{t|Lt≤ε}.
Because the processXis always larger than−1, the stopped processesXτ

ε
are
outcomes of admissible integrands. IfQis an absolutely continuous probability
measure such thatSbecomes a local martingale thae, by Theorem 12.1.3 we
have that the setH=

{

Xτ

ε

∞|ε>^0

}

is bounded inL^0 ({ddQP> 0 }).Butitis
clear that on the set{LT=0},thesetHis unbounded.
As a consequence we obtain the following lemma.

Lemma 12.4.1.If the continuous semi-martingaleSsatisfies the no-arbitrage
condition with respect to general admissible integrands and ifQis an abso-
lutely continuous local martingale measure forS,then{ddQP> 0 }⊂{LT> 0 }.

In order to prove the existence of an absolutely continuous local martingale
measureQwe therefore should restrict ourselves to measures supported by