We study various properties of a dynamic convex risk measure for bounded random variables which describe the discounted terminal values of financial positions. In particular we characterize time-consistency by a joint supermartingale property of the risk measure and its penalty function. Moreover we discuss the limit behavior of the risk measure in terms of asymptotic safety and of asymptotic precision, a property which may be viewed as a non-linear analogue of martingale convergence. These results are illustrated by the entropic dynamic risk measure.

This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty functions, and by supermartingale properties of risk processes and penalty functions.A natural extension of a static risk measure is given by a conditional risk measure, which takes into account the information available at the time of risk assessment. As its static counterpart, a conditional convex risk measure can be represented as the worst conditional expected loss over a class of suitably penalized probability measures; see [31,30,17,6,26,12]. In the dynamical setting described by some filtered probability space, risk assessment is updated over the time in accordance with the new information. This leads to the notion of dynamic risk measure, which is a sequence of conditional risk measures adapted to the underlying filtration.A crucial question in the dynamical framework is how risk evaluations at different times are interrelated. Several notions of time consistency were introduced and studied in the literature. One of todays most used notions is strong time consistency, which corresponds to the dynamic programming principle; see [4,15,17,26,12,7,20,13,16] and references therein. As shown in [15,7,20], strong time consistency can be characterized by additivity of the acceptance sets and penalty functions, and also by a supermartingale property of the risk process and the penalty function process. Similar characterizations of the weaker notions of time consistency, so called rejection and acceptance consistency, were given in [18,29].Rejection consistency, also called prudence in [29], seems to be a particularly suitable property from the point of view of a regulator, since it ensures that one always stays on the safe side when updating risk assessment. The weakest notions of time consistency considered in the literature are weak acceptance and weak rejection consistency, which require that if some position is accepted (or rejected) for any scenario tomorrow, it should be already accepted (or rejected) today; see [4,35,33,9,32].As pointed out in [25,19], risk assessment in the multi-period setting should also account for uncertainty about the time value of money. This requires to consider entire cash flow processes rather than total amounts at terminal dates as risky objects, and it leads to a further extention of the notion of risk measure. Risk measures for processes were studied in [4,30,10,11,12,13,24,25,1]. The new feature in this framework is that not only the amounts but also the timing of payments matters; cf. [12,13,25,1].However, as shown in [4] in the static and in [1] in the dynamical setting, risk measures for processes can be identified with risk measures for random variables on an appropriate product space. This allows a natural translation of results obtained in the framework of ...

We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito, Delbaen, and Kupper [10]. These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably penalized probability measures on the optional σ-field. This yields an explicit analysis both of model and discounting ambiguity. We focus on supermartingale criteria for time consistency. In particular we show how "bubbles" may appear in the dynamic penalization, and how they cause a breakdown of asymptotic safety of the risk assessment procedure.

Abstract. We study superhedging of contingent claims with physical delivery in a discrete-time market model with convex transaction costs. Our model extends Kabanov's currency market model by allowing for nonlinear illiquidity effects. We show that an appropriate generalization of Schachermayer's robust no-arbitrage condition implies that the set of claims hedgeable with zero cost is closed in probability.Combined with classical techniques of convex analysis, the closedness yields a dual characterization of premium processes that are sufficient to superhedge a given claim process. We also extend the fundamental theorem of asset pricing for general conical models.Key words. superhedging, physical delivery, illiquidity, transaction costs, convex duality AMS subject classifications. 91B25, 52A07, 46A20DOI. 10.1137/0907541821. Introduction. This paper studies superhedging of contingent claims with physical delivery in markets with temporary illiquidity effects. Our market model is a generalization of the currency market model of Kabanov [12]. In Kabanov's model, price dynamics and transaction costs are modeled implicitly by solvency cones, i.e., sets of portfolios which can be transformed into the zero portfolio by self-financing transactions at a given time and state. An essential difference between Kabanov's model and more traditional models of mathematical finance (including, e.g., the transaction cost models of Jouini and Kallal [11], Cvitanić and Karatzas [6], and Kaval and Molchanov [18]) is that Kabanov's model focuses on contingent claims with physical delivery, i.e., claims whose payouts are given in terms of portfolios of assets instead of a single reference asset like cash. Accordingly, notions of arbitrage as well as the corresponding dual variables are defined in terms of vector-valued processes; see, e.g., [15,31].Astic and Touzi [2] extended Kabanov's model by allowing general convex solvency regions in the case of finite probability spaces. Nonconical solvency regions allow for the modeling of temporary illiquidity effects where marginal trading costs may depend on the magnitude of a trade as, e.g., in Ç etin, Jarrow, and Protter [3], Ç etin and Rogers [4], Ç etin, Soner, and Touzi [5], Rogers and Singh [27], or Pennanen [19,20]. These models cover nonlinear illiquidity effects, but they assume that agents have no market power in the sense that their trades do not affect the costs of subsequent trades; see [27] for further motivation of this assumption.

We provide a characterization of the family of non-negative local martingales that have continuous running supremum and vanish at infinity. This is done by describing the class of random times that identify the times of maximum of such processes. In this way we extend to the case of general filtrations a result proved by Nikeghbali and Yor [21] for continuous filtrations. Our generalization is complementary to the one presented by Kardaras [15], and is obtained by means of similar tools.