Minimal Supersolutions of BSDEs with Lower Semicontinuous Generators

We study the existence and uniqueness of minimal supersolutions of backward stochastic differential equations with generators that are jointly lower semicontinuous, bounded below by an affine function of the control variable and satisfy a specific normalization property

Minimal supersolutions of convex BSDEs

We study the nonlinear operator of mapping the terminal value $\xi$ to the corresponding minimal supersolution of a backward stochastic differential equation with the generator being monotone in $y$, convex in $z$, jointly lower semicontinuous and bounded below by an affine function of the control variable $z$. We show existence, uniqueness, monotone convergence, Fatou's lemma and lower semicontinuity of this operator. We provide a comparison principle for minimal supersolutions of BSDEs.Comment: Published in at http://dx.doi.org/10.1214/13-AOP834 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Minimal Supersolutions of Convex BSDEs under Constraints

We study supersolutions of a backward stochastic differential equation, the control processes of which are constrained to be continuous semimartingales of the form $dZ = {\Delta}dt + {\Gamma}dW$. The generator may depend on the decomposition $({\Delta},{\Gamma})$ and is assumed to be positive, jointly convex and lower semicontinuous, and to satisfy a superquadratic growth condition in ${\Delta}$ and ${\Gamma}$. We prove the existence of a supersolution that is minimal at time zero and derive stability properties of the non-linear operator that maps terminal conditions to the time zero value of this minimal supersolution such as monotone convergence, Fatou's lemma and $L^1$-lower semicontinuity. Furthermore, we provide duality results within the present framework and thereby give conditions for the existence of solutions under constraints.Comment: 23 page

Futures Cross-Hedging with a Stationary Basis

This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.When managing risk, frequently only imperfect hedging instruments are at hand. We show how to optimally cross-hedge risk when the spread between the hedging instrument and the risk is stationary. For linear risk positions we derive explicit formulas for the hedge error, and for nonlinear positions we show how to obtain numerically efficient estimates. Finally, we demonstrate that even in cases with no clear-cut decision concerning the stationarity of the spread, it is better to allow for mean reversion of the spread rather than to neglect it.Peer Reviewe

Introduction: Nietzsche's Life and Works

An introduction to Nietzsche's life and works

Nietzsche's Ethics of Affirmation

This chapter looks at Nietzsche's notion of the affirmation of life. It begins with the origins of the concept in Schopenhauer and in the Schopenhauerian philosophy known to Nietzsche. It then examines affirmation in three phases of Nietzsche's writing: early, middle and late. It relates affirmation to other key Nietzschean concepts like the Apollonian and the Dionysian, eternal recurrence, amor fati and will to power