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Walter Schachermayer

## Martingale Measures for discrete-time processes with infinite horizon.

W. Schachermayer
Mathematical Finance, Vol. 4 (1994), No. 1, pp. 25-55.

### Abstract:

Let $(S_t)_{t \in I}$ be an $\R^d$--valued adapted stochastic process on $(\Om,\Cal F,(\Cal F_t)_{t \in I},P)$. A basic problem, occuring notably in the analysis of securities markets, is to decide whether there is a probability measure $Q$ on $\Cal F$ equivalent to $P$ such that $(S_t)_{t \in I}$ is a martingale with respect to $Q$.

It is known since the fundamental papers of Harrison--Kreps (79), Harrison--Pliska (81) and Kreps (81) that there is an intimate relation of this problem with the notions of "no arbitrage" and "no free lunch" in financial economics.

We introduce the intermediate concept of "no free lunch with bounded risk". This is a somewhat more precise version of the notion of "no free lunch": It requires that there should be an absolute bound of the maximal loss occuring in the trading strategies considered in the definition of "no free lunch". We shall give an argument why the condition of "no free lunch with bounded risk" should be satisfied by a reasonable model of the price process $(S_t)_{t \in I}$ of a securities market.

We can establish the equivalence of the condition of "no free lunch with bounded risk" with the existence of an equivalent martingale measure in the case when the index set $I$ is discrete but (possibly) infinite. A similar theorem was recently obtained by Delbaen (92) for the case of continuous time processes with continuous paths. We can combine these two theorems to get a similar result for the continuous time case when the process $(S_t)_{t \in \R_+}$ is bounded and -- roughly speaking -- the jumps occur at predictable times.

### Preprints:

[PostScript (382 k)] [PS.gz (139 k)] [PDF (402 k)] [DOI: 10.1111/j.1467-9965.1994.tb00048.x]

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