The Mathematics of Arbitrage

296 14 The FTAP for Unbounded Stochastic Processes

Then there is, forε> 0 , a probability measureQ∼P,‖Q−Q 1 ‖<ε,
such thatSis a sigma-martingale with respect toQ.
In addition, for every predictable stopping timeT, the probabilitiesQand
Q 1 onFT, coincide, conditionally onFT−, i.e.,

dQ|FT

dQ 1 |FT

=

dQ|FT−

dQ 1 |FT−

a.s..

Proof. Step 1:Define the stopping timeTby

T=inf{t|‖∆St‖Rd≥ 1 }

and first suppose thatSremains constant after timeT, henceShas at most
one jump bigger than 1.
Similarly as in [JS 87, II.2.4] we decomposeSinto

S=X+Xˇ

whereXequals “Sstopped at timeT−”, i.e.,

Xt=

{

St fort<T

ST−fort≥T

andXˇthe jump ofSat timeT, i.e.,

Xˇt=∆ST· (^1) [[T,∞]].
AsXis bounded, it is a special semi-martingale, and we can find its Doob-
Meyer decomposition with respect toQ 1
X=M+B
whereMis a localQ 1 -martingale andBa predictable process of locally finite
variation.
We shall now find a probability measureQ 2 onF,Q 2 ∼P,s.t.
(i) ‖Q 2 −Q 1 ‖<ε 2 ,
(ii) Q 2 |FT−=Q 1 |FT−andddQQ^21 isFT-measurable,
(iii)Sis a sigma-martingale underQ 2.
We introduce the jump measureμassociated toXˇ,
μ(ω, dt, dx)=δ(T(ω),∆ST(ω)),
whereδt,xdenotes Dirac-measure at (t, x)∈R+×Rdand we denote by
νtheQ 1 -compensator ofμ(see [JS 87, Proposition II.1.6]). Similarly as in
[JS 87, Proposition II.2.9] we may find a locallyQ 1 -integrable, predictable
and increasing processAsuch that