The Mathematics of Arbitrage

14.2 Sigma-martingales 283

Proposition 14.2.5.For a semi-martingaleXthe following assertions are
equivalent.

(i) Xis a local martingale.
(ii) X=φ·Mwhere theM-integrable, predictableR+-valued processφis
increasingandMis a local martingale.
(ii’) X=φ·Mwhere theM-integrable, predictableR+-valued processφis
locally boundedandMis a local martingale.
(iii) X=φ·Mwhere theM-integrable, predictableR+-valued processφis
increasingandMis a martingale.
(iii’)X=φ·Mwhere theM-integrable, predictableR+-valued processφis
locally boundedandMis a martingale.
(iv) X=φ·Mwhere theM-integrable, predictableR+-valued processφis
increasingandMis a martingale inH^1.
(iv’)X=φ·Mwhere theM-integrable, predictableR+-valued processφis
locally boundedandMis a martingale inH^1.

We will not prove this proposition as its proof is similar to the proof of
the next proposition.

Proposition 14.2.6.For a semi-martingaleXthe following are equivalent

(i) Xis a sigma-martingale.
(ii) X=φ·Mwhere theM-integrable, predictableR+-valued processφis
strictly positive andMis a local martingale.
(iii) X=φ·Mwhere theM-integrable, predictableR+-valued processφis
strictly positive andMis a martingale.
(iv) X=φ·Mwhere theM-integrable, predictableR+-valued processφis
strictly positive andMis a martingale inH^1.

Proof.Since (iv) implies (iii) implies (ii), and since obviously (i) is equivalent to
(iii), we only have to prove that (ii) implies (iv). So suppose that there is a local
martingaleMas well as a non-negativeM-integrable predictable processφ
such thatX=φ·M.Let(Tn)n≥ 1 be a sequence that localisesMin the sense
thatTnis increasing, tends to∞and for eachn,MTnis inH^1 .PutT 0 =0

and forn≥1, defineNnas theH^1 -martingaleNn=(φ (^1) ]]Tn− 1 ,Tn]])·MTn.
Let nowN=

∑

n≥ 1 anN

n, where the strictly positive sequenceanis chosen

such that

∑

an‖Nn‖H^1 <∞. The processNis anH^1 -martingale. We now

putψ= (^1) {φ=0}+φ

∑

na

− 1
n^1 ]]Tn− 1 ,Tn]]. It is easy to check thatX=ψ·Nand
thatψis strictly positive.

Corollary 14.2.7.A local sigma-martingale is a sigma-martingale. More pre-
cisely, ifXis a semi-martingale and if(Tk)k≥ 1 is an increasing sequence of

stopping times, tending to∞such that each stopped processXTkis a sigma-
martingale, thenXitself is a sigma-martingale.