Volume 23, Issue 1 - 1

In this paper we consider the problem of finding optimal consumption strategies in an incomplete semimartingale market model under model uncertainty. The quality of a consumption strategy is measured by not only one probability measure but as common in risk theory by a class of scenario measures. We formulate a dual version of the optimization problem and prove the existence of a saddle point and give a characterization of an optimal consumption strategy in terms of solutions of the dual problem. This generalizes results of Karatzas and Zitkovic (2003) for the optimal consumption problem under a fixed probability measure.

A convertible (callable) bond is a security that the holder can convert into a specified number of underlying shares. In addition, the issuer can recall the bond, paying some compensation, or force the holder to convert it immediately. We give an explicit solution to the corresponding optimal stopping game in the context of a reduced form model driven by a Brownian motion and a compound Poisson process with exponential jumps. It turns out that the occurrence of jumps leads to optimal stopping strategies whose structure differs from the results for continuous models.

The well-known Harrison–Plisse theorem claims that in the classical discrete time model of the financial market with finite Ω there is no arbitrage iff there exists an equivalent martingale measure. The famous Dalang–Morton–Willinger theorem extends this result for an arbitrary Ω. Kabanov and Stricker [KS01] generalized the Harrison–Pliska theorem for the case of the market with proportional transaction costs. Nevertheless the corresponding extension of the Kabanov and Stricker result to the case of non-finite Ω fails, the corresponding counter-example with 4 assets was constructed by Schachermayer [S04]. The main result of this paper is that in the special case of 2 assets the Kabanov and Stricker theorem can be extended for an arbitrary Ω. This is quite a surprising result since the corresponding cone of hedgeable claims  T is not necessarily closed.

We study reward over penalty for risk ratios E [ u ( V )]/ E [ρ( V )], V ∈ V , where V ⊆ L 1 ( P ) describes a linear space of attainable returns in an arbitrage-free market, u is concave and ρ ≥ 0 is convex. It turns out that maximizing such reward over penalty ratios is essentially equivalent to maximizing the ratio α( V ) := E [ V ]/ E [ V − ] or the expected profit over expected loss ratio E [ V + ]/ E [ V − ]. The lowest upper bound α – := sup V ∈ V α( V ) can be determined by solving an appropriate dual problem over the set of bounded equivalent martingale measures for V . This observation leads to the definition of shortfall risk optimal equivalent martingale measures.

First we polish an argument of Rémillard and Theodorescu for the weak convergence of the empirical probability generating process. Then we prove a general inequality between probability generating processes and the corresponding empirical processes, which readily implies a rate of convergence and trivializes the problem of weak convergence: whenever the empirical process or its non-parametric bootstrap version, or the parametric estimated empirical process or its bootstrap version converges, so does the corresponding probability generating process. Derivatives of the generating process are also considered.