The existence of dominating local martingale measures

Abstract

We prove that for locally bounded processes the absence of arbitrage opportunities of the first kind is equivalent to the existence of a dominating local martingale measure. This is related to and motivated by results from the theory of filtration enlargements.

Appendix: Incomplete filtrations

Here we collect some classical observations which allow transferring to our setting results of other authors that were obtained under complete filtrations. There are at least three important monographs which avoid the use of complete filtrations as far as possible, namely Jacod [25], Jacod and Shiryaev [27] and von Weizsäcker and Winkler [51]. Here we follow [27].

Let $(\varOmega, \mathcal {F}, (\mathcal {F}_{t})_{t \ge0}, P)$ be a filtered probability space equipped with a right-continuous filtration $(\mathcal {F}_{t})$. Write $\mathcal {F}^{P}$ for the $P$-completion of ℱ and $\mathcal{N}^{P}$ for the $P$-nullsets of $\mathcal {F}^{P}$. Then $\mathcal {F}_{t}^{P} = \mathcal {F}_{t} \vee\mathcal{N}^{P}$, $t\ge0$, satisfies the usual conditions. It is well known and easy to show that for every random variable $X$ on $(\varOmega, \mathcal {F}^{P})$, there exists a random variable $Y$ on $(\varOmega, \mathcal {F})$ with $P[X=Y]=1$.

Recall that the optional $\sigma$-algebra over $(\mathcal {F}_{t})$ is the $\sigma$-algebra on $\varOmega\times[0,\infty)$ that is generated by all processes of the form $X_{t}(\omega) = 1_{A}(\omega) 1_{[r,s)}(t)$ for some $0\le r < s < \infty$ and $A \in \mathcal {F}_{r}$, and the predictable $\sigma$-algebra is generated by all processes of the form $X_{t}(\omega) = 1_{A}(\omega) 1_{\{0\}}(t) + 1_{B}(\omega) 1_{(r,s]}(t)$ for some $0\le r < s < \infty$, $A \in \mathcal {F}_{0}$ and $B \in \mathcal {F}_{r}$. Similarly, we define the predictable and optional $\sigma$-algebras over $(\mathcal {F}^{P}_{t})$.

The first result relates stopping times under $(\mathcal {F}_{t})$ and under $(\mathcal {F}^{P}_{t})$.

Lemma A.1

(Lemma I.1.19 of [27])

Any stopping time on the completion $(\varOmega, (\mathcal {F}^{P}_{t}))$ is almost surely equal to a stopping time on $(\varOmega, (\mathcal {F}_{t}))$.

We also have a comparable result at the level of processes.

Lemma A.2

Any predictable (optional) process on the completion $(\varOmega, (\mathcal {F}^{P}_{t}))$ is indistinguishable from a predictable (optional) process on $(\varOmega, (\mathcal {F}_{t}))$.

Proof

The predictable case is Lemma I.2.17 of [27]. The proof of the optional case works exactly in the same way: the claim is trivial for the generating processes described above, and we can use the monotone class theorem to pass to indicator functions of general optional sets. Then we use monotone convergence to pass to general optional processes. □

This allows us to deduce a similar result for càdlàg processes.

Lemma A.3

Let $S$ be an $(\mathcal {F}^{P}_{t})$-adapted process that is almost surely càdlàg. Then $S$ is indistinguishable from an $(\mathcal {F}_{t})$-adapted process (which is then, of course, almost surely càdlàg as well).

Proof

Since $(\mathcal {F}^{P}_{t})$ is complete, $S$ admits an indistinguishable version $\widetilde{S}$ which is $(\mathcal {F}_{t}^{P})$-adapted and càdlàg for every $\omega\in\varOmega$. This $\widetilde {S}$ is optional; so now the result follows from Lemma A.2. □