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We consider backward stochastic differential equations (BSDE) with nonlinear generators typically of quadratic growth in the control variable. A measure solution of such a BSDE will be understood as a probability measure under which the generator is seen as vanishing, so that the classical solution can be reconstructed by a combination of the operations of conditioning and using martingale representations. In case the terminal condition ist bounded and the generator fulfills the usual continuity and boundedness conditions, we show the measure solutions with equivalent measures just reinterpret classical ones. In case of terminal conditions that have only exponentially bounded moments, we discuss a series of examples which show that in cas of non-uniqueness classical solutions that fail to be measure solutions can coexists with different measure solution.
The extent to which catastrophic weather events occur strongly depends on global climate conditions such as average sea surface temperatures (SST) or sea level pressures. Some of the factors can be predicted up to a year in advance, and should therefore be taken into account in any reasonable management of weather related risk. In this paper we first set up a risk model that integrates climate factors. The we show how variance minimizing hedging strategies explicitly depend on the factors' prediction. Our analysis is based on a detailed study of the predictable representation property on the combined Poisson and Wiener spaces. Using tools of the stochastic calculus of variations we derive a representation formula of the Clark-Ocone type. Finally, we exemplify the theory developed in a case study of US hurricane risk. We derive hedging strategies taking into account that US hurricane activity strongly depends on the SST of the Pacific Ocean.
Financial markets with asymmetric information: information drift, additional utility and entropy
(2009)
We review a general mathematical link between utility and information theory appearing in a simple financial market model with two kinds of small investors: insiders, whose extra information is stored in an enlargement of the less informed agents' filtration. The insider's expected logarithmic utility increment is described in terms of the information drift, i.e. the drift one has to eliminate in order to perceive the price dynamics as a martingale from his perspective. We describe the information drift in a very general setting by natural quantities expressing the conditional laws of the better informed view of the world. This on th other hand allows to identify the additional utility by entropy related quantities known from information theory.
When managing energy or weather related risk often only imperfect hedging instruments are available. In the first part we illustrate problems arising with imperfect hedging by studying a toy model. We consider an airline’s problem with covering income risk due to fluctuating kerosene prices by investing into futures written on heating oil with closely correlated price dynamics. In the second part we outline recent results on exponential utility based cross hedging concepts. They highlight in a generalization of the Black-Scholes delta hedge formula to incomplete markets. Its derivation is based on a purely stochastic approach of utility maximization. It interprets stochastic control problems in the BSDE language, and profits from the power of the stochastic calculus of variations.
We solve Skorokhod's embedding problem for Brownian mostion with linear drift $(W_t + \kappa t)_{t\ge 0}$ by means of techniques of stochastic control theory. The search for a stopping time $T$ such that the law of $W_T + \kappa T$ coincides with a prescribed law $\mu$ processing the first moment is based on solutions of backward stochastic differential equations of quadratic type. Theis new approach generalizes an approach by Bass [BAS] of the classical version of Skorokhod's embedding problem using martingale representation techniques.
Dynamic risk measures
(2010)
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.
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 (2006). 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 $\sigma$-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.