In a complete financial market, the pricing of contingent claims is well understood. If the stochastic process

$X$ describes discounted stock prices, ‘complete’ means that every sufficiently integrable random variable can be represented as the sum of a constant and a stochastic integral of

$X$ satisfying suitable integrability conditions. An (up to technical conditions) equivalent formulation is the statement that there exists a unique equivalent martingale measure

${P}^{*}$ for

$X$. The price of the claim (random variable)

$H$ is then

${E}^{*}\left[H\right]$ (see [

*J. M. Harrison* and

*S. R. Pliska*, Stochastic Processes Appl. 11, 215-260 (1981;

Zbl 0482.60097)]). In an incomplete market, there are many equivalent martingale measures

$Q$, leading to a range of possible prices between

${inf}_{Q}{E}_{Q}\left[H\right]$ and

${sup}_{Q}{E}_{Q}\left[H\right]$. Assuming that

$X$ is a multidimensional Itô process, the authors show that

${sup}_{Q}{E}_{Q}\left[H\right]$ is the smallest initial wealth

$w$ for which there exists a suitable integrand

$\vartheta $ such that

$w+{\int}_{0}^{T}{\vartheta}_{s}d{X}_{s}\ge H$. A similar result holds for

${inf}_{Q}{E}_{Q}\left[H\right]$. This means that one needs at least the amount

${sup}_{Q}{E}_{Q}\left[H\right]$ for the construction of a self-financing hedging strategy which covers the claim

$H$, and thus implies that

${sup}_{Q}{E}_{Q}\left[H\right]$ and

${inf}_{Q}{E}_{Q}\left[H\right]$ can be viewed as fair selling and buying prices for

$H$, respectively. Actually, the authors prove a dynamical version of the above theorem. Analogous results for a general semimartingale

$X$ have subsequently been obtained by

*D. O. Kramkov* [‘Optimal decomposition of supermartingales and hedging contingent claims in incomplete security markets’ (1994), submitted to Probability Theory and Related Fields], using a different approach.