On worst-case investment with applications in finance and insurance mathematics

We review recent results on the new concept of worst-case portfolio optimization, i.e. we consider the determination of portfolio processes which yield the highest worst-case expected utility bound if the stock price may have uncertain (down) jumps. The optimal portfolios are derived as solutions of non-linear differential equations which itself are consequences of a Bellman principle for worst-case bounds. They are by construction non-constant ones and thus differ from the usual constant optimal portfolios in the classical examples of the Merton problem. A particular application of such strategies is to model crash possibilities where both the number and the height of the crash is uncertain but bounded. We further solve optimal investment problems in the presence of an additional risk process which is the typical situation of an insurer

European Options in a Nonlinear Incomplete Market Model with Default

We study the superhedging prices and the associated superhedging strategies for European options in a nonlinear incomplete market model with default. The underlying market model consists of one risk-free asset and one risky asset, whose price may admit a jump at the default time. The portfolio processes follow nonlinear dynamics with a nonlinear driver $f$. By using a dynamic programming approach, we first provide a dual formulation of the seller's (superhedging) price for the European option as the supremum, over a suitable set of equivalent probability measures $Q \in {\cal Q}$, of the $f$-evaluation/expectation under $Q$ of the payoff. We also establish a characterization of the seller's (superhedging) price as the initial value of the minimal supersolution of a constrained backward stochastic differential equation with default. Moreover, we provide some properties of the terminal profit made by the seller, and some results related to replication and no-arbitrage issues. Our results rely on first establishing a nonlinear optional and a nonlinear predictable decomposition for processes which are ${\cal E}^f$-strong supermartingales under $Q$ for all $Q \in {\cal Q}$

Statistical equilibrium in simple exchange games I

Simple stochastic exchange games are based on random allocation of finite resources. These games are Markov chains that can be studied either analytically or by Monte Carlo simulations. In particular, the equilibrium distribution can be derived either by direct diagonalization of the transition matrix, or using the detailed balance equation, or by Monte Carlo estimates. In this paper, these methods are introduced and applied to the Bennati-Dragulescu-Yakovenko (BDY) game. The exact analysis shows that the statistical-mechanical analogies used in the previous literature have to be revised.Comment: 11 pages, 3 figures, submitted to EPJ

Inf-convolution of G-expectations

In this paper we will discuss the optimal risk transfer problems when risk measures are generated by G-expectations, and we present the relationship between inf-convolution of G-expectations and the inf-convolution of drivers G.Comment: 23 page

One-sided versus two-sided stochastic descriptions

It is well-known that discrete-time finite-state Markov Chains, which are described by one-sided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Fields, that is, nearest-neighbour Gibbs measures for finite-spin models, which are described by two-sided conditional probabilities. In such Markov Fields the time interpretation of past and future is being replaced by the space interpretation of an interior volume, surrounded by an exterior to the left and to the right. If we relax the Markov requirement to weak dependence, that is, continuous dependence, either on the past (generalising the Markov-Chain description) or on the external configuration (generalising the Markov-Field description), it turns out this equivalence breaks down, and neither class contains the other. In one direction this result has been known for a few years, in the opposite direction a counterexample was found recently. Our counterexample is based on the phenomenon of entropic repulsion in long-range Ising (or "Dyson") models.Comment: 13 pages, Contribution for "Statistical Mechanics of Classical and Disordered Systems

Recent progress in random metric theory and its applications to conditional risk measures

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measures. Section 2 gives the central framework in random metric theory, topological structures, important examples, the notions of a random conjugate space and the Hahn-Banach theorems for random linear functionals. Section 3 gives several important representation theorems for random conjugate spaces. Section 4 gives characterizations for a complete random normed module to be random reflexive. Section 5 gives hyperplane separation theorems currently available in random locally convex modules. Section 6 gives the theory of random duality with respect to the locally $L^{0}-$convex topology and in particular a characterization for a locally $L^{0}-$convex module to be $L^{0}-$pre$-$barreled. Section 7 gives some basic results on $L^{0}-$convex analysis together with some applications to conditional risk measures. Finally, Section 8 is devoted to extensions of conditional convex risk measures, which shows that every representable $L^{\infty}-$type of conditional convex risk measure and every continuous $L^{p}-$type of convex conditional risk measure ($1\leq p<+\infty$) can be extended to an $L^{\infty}_{\cal F}({\cal E})-$type of $\sigma_{\epsilon,\lambda}(L^{\infty}_{\cal F}({\cal E}), L^{1}_{\cal F}({\cal E}))-$lower semicontinuous conditional convex risk measure and an $L^{p}_{\cal F}({\cal E})-$type of ${\cal T}_{\epsilon,\lambda}-$continuous conditional convex risk measure ($1\leq p<+\infty$), respectively.Comment: 37 page

Large deviations for many Brownian bridges with symmetrised initial-terminal condition

Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ with some non-degenerate initial measure on some fixed time interval $[0,\beta]$ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $\sigma(i)$-th motion, where $\sigma$ is a uniformly distributed random permutation of $1,...,N$. Such systems play an important role in quantum physics in the description of Boson systems at positive temperature $1/\beta$. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the $N$ paths) and of the mean of the normalised occupation measures of the $N$ motions in terms of large deviations principles. The rate functions are given as variational formulas involving certain entropies and Fenchel-Legendre transforms. Consequences are drawn for asymptotic independence statements and laws of large numbers. In the special case related to quantum physics, our rate function for the occupation measures turns out to be equal to the well-known Donsker-Varadhan rate function for the occupation measures of one motion in the limit of diverging time. This enables us to prove a simple formula for the large-N asymptotic of the symmetrised trace of ${\rm e}^{-\beta \mathcal{H}_N}$, where $\mathcal{H}_N$ is an $N$-particle Hamilton operator in a trap

Multivariate risks and depth-trimmed regions

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this abstract axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from the multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship.Comment: 26 pages. Substantially revised version with a number of new results adde

Representation of the penalty term of dynamic concave utilities

In this paper we will provide a representation of the penalty term of general dynamic concave utilities (hence of dynamic convex risk measures) by applying the theory of g-expectations.Comment: An updated version is published in Finance & Stochastics. The final publication is available at http://www.springerlink.co