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We consider the problem of pricing in a large market, which arises as a limit of small markets within which there are finitely many traded assets. We show that this framework allows accommodating both marginal-utility-based prices (for stochastic utilities) and arbitrage-free prices. Adopting a stochastic integration theory with respect to a sequence of semimartingales, we introduce the notion of marginal-utility-based prices for the large (post-limit) market and establish their existence, uniqueness, and relation to arbitrage-free prices. These results rely on a theorem of independent interest on utility maximisation with a random endowment in a large market that we state and prove first. Further, we provide approximation results for the marginal utility-based and arbitrage-free prices in the large market by those in small markets. In particular, our framework allows pricing asymptotically replicable claims, where we also show consistency in the pricing methodologies and provide positive examples.
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On the analyticity of the value function in optimal investment and stochastically dominant markets
We study the analyticity of the value function in optimal investment with expected utility from terminal wealth and the relation to stochastically dominant financial models. We identify both a class of utilities and a class of semimartingale models for which we establish analyticity. Specifically, these utilities have completely monotonic inverse marginals, while the market models have a maximal element in the sense of infinite-order stochastic dominance. We construct two counterexamples, themselves of independent interest, which show that analyticity fails if either the utility or the market model does not belong to the respective special class. We also provide explicit formulas for the derivatives of all orders of the value functions as well as their optimizers. Finally, we show that for the set of supermartingale deflators, stochastic dominance of infinite order is equivalent to the apparently stronger dominance of second order.
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- Award ID(s):
- 1848339
- PAR ID:
- 10512369
- Publisher / Repository:
- Yokohama Publishers
- Date Published:
- Journal Name:
- Pure and applied functional analysis
- Volume:
- 9
- Issue:
- 3
- ISSN:
- 2189-3764
- Page Range / eLocation ID:
- 825-862
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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