A convergence result for the Emery topology and a variant of the proof of the fundamental theorem of asset pricing

Abstract

We show that no unbounded profit with bounded risk (NUPBR) implies predictable uniform tightness (P-UT), a boundedness property in the Emery topology introduced by Stricker (Séminaire de Probabilités de Strasbourg XIX, pp. 209–217, 1985). Combining this insight with well-known results of Mémin and Słominski (Séminaire de Probabilités de Strasbourg XXV, pp. 162–177, 1991) leads to a short variant of the proof of the fundamental theorem of asset pricing initially proved by Delbaen and Schachermayer (Math. Ann. 300:463–520, 1994). The results are formulated in the general setting of admissible portfolio wealth processes as laid down by Kabanov (Statistics and Control of Stochastic Processes, pp. 191–203, World Sci. Publ., River Edge, 1997).

Appendix: The P-UT property

This section is dedicated to state some results related to the P-UT property, which can be found in [25], [15], and [14, Chapter VI.6].

Proposition A.1

Let $(X^{n})_{n \geq0}$ be a sequence of semimartingales satisfying the P-UT property. Then

• (i) $(|X^{n}|^{*}_{1})_{n\geq0}$ is bounded in $L^{0}$;

• (ii) $([X^{n}, X^{n}]_{1})_{n \geq0}$ is bounded in $L^{0}$.

Proof

See [15, Lemme 1.2] or [25, Lemme 1.3]. □

The following proposition characterizes the P-UT property in terms of the $L^{0}$-boundedness of certain parts in the semimartingale decomposition (5.1).

Proposition A.2

Let $(X^{n})_{n \geq0}$ be a sequence of semimartingales and consider the semimartingale decomposition (5.1) for some $C >0$. Then $(X^{n})$ satisfies the P-UT property if and only if the following three conditions hold:

• (i) The sequence ${(\operatorname{TV}(\check{X}^{n,C})_{1})}_{n \geq 0}$ of total variations of $\check{X}^{n,C}$ is bounded in  $L^{0}$.

• (ii) The sequence $([M^{n,C}, M^{n,C}]_{1})_{n \geq0}$ is bounded in $L^{0}$.

• (iii) The sequence ${(\operatorname{TV}(B^{n,C})_{1})}_{n \geq0}$ of total variations of $B^{n,C}$ is bounded in  $L^{0}$.

Proof

See [25, Théorème 1.4] or [14, Theorem VI.6.15]. □

The following theorem builds the basis of Proposition 5.2.

Theorem A.3

Let $(H^{n})_{n \geq0}$ be a sequence of adapted càdlàg processes, and $(X^{n})_{n \geq0}$ a sequence of semimartingales satisfying the P-UT property. If the sequence $(H^{n},X^{n})$ converges uniformly in probability to $(H,X)$, then the stochastic integrals $(H^{n}_{-} \bullet X^{n})$ converge to $(H_{-} \bullet X)$ uniformly in probability as well. In particular, $[X^{n}, X^{n}]_{1} \to[X,X]_{1}$ in probability.

Proof

See [15, Théorème 2.6] or [25, Théorème 1.8, Corollaire 1.9]. □