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Dynamic programming and pricing of contingent claims in an incomplete market. (English) Zbl 0831.90010
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.

##### MSC:
 91G20 Derivative securities 90C39 Dynamic programming 93E25 Computational methods in stochastic control