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We study minimal supersolutions of backward stochastic differential equations. We show the existence and uniqueness of the minimal supersolution, if the generator is jointly lower semicontinuous, bounded from below by an affine function of the control variable, and satisfies a specific normalization property. Semimartingale convergence is used to establish the main result.
We provide results on the existence and uniqueness of equilibrium in dynamically incomplete financial markets in discrete time. Our framework allows for heterogeneous agents, unspanned random endowments and convex trading constraints. In the special case where all agents have preferences of the same type and all random endowments are replicable by trading in the financial market we show that a one-fund theorem holds and give an explicit expression for the equilibrium pricing kernel. If the underlying noise is generated by finitely many Bernoulli random walks, the equilibrium dynamics can be described by a system of coupled backward stochastic difference equations, which in the continuous-time limit becomes a multi-dimensional backward stochastic differential equation. If the market is complete in equilibrium, the system of equations decouples, but if not, one needs to keep track of the prices and continuation values of all agents to solve it. As an example we simulate option prices in the presence of stochastic volatility, demand pressure and short-selling constraints.
We consider a class of generalized capital asset pricing models in continuous time with a finite number of agents and tradable securities. The securities may not be sufficient to span all sources of uncertainty. If the agents have exponential utility functions and the individual endowments are spanned by the securities, an equilibrium exists and the agents’ optimal trading strategies are constant. Affine processes, and the theory of information-based asset pricing are used to model the endogenous asset price dynamics and the terminal payoff. The derived semi-explicit pricing formulae are applied to numerically analyze the impact of the agents’ risk aversion on the implied volatility of simultaneously-traded European-style options.
In the paradigm of VON N EUMANN AND M ORGENSTERN, a representation of affine pref-
erences in terms of an expected utility can be obtained under the assumption of weak continu-
ity. Since the weak topology is coarse, this requirement is a priori far from being negligible.
In this work, we replace the assumption of weak continuity by monotonicity. More precisely,
on the space of lotteries on an interval of the real line, it is shown that any affine preference
order which is monotone with respect to the first stochastic order admits a representation in
terms of an expected utility for some nondecreasing utility function. As a consequence, any
affine preference order on the subset of lotteries with compact support, which is monotone
with respect to the second stochastic order, can be represented in terms of an expected util-
ity for some nondecreasing concave utility function. We also provide such representations
for affine preference orders on the subset of those lotteries which fulfill some integrability
conditions. The subtleties of the weak topology are illustrated by some examples.
To address the plurality of interpretations of the subjective notion of risk, we describe it by means of a risk order and concentrate on the context invariant features of diversification and monotonicity. Our main results are uniquely characterized robust representations of lower semicontinuous risk orders on vector spaces and convex sets. This representation covers most instruments related to risk and allow for a differentiated interpretation depending on the underlying context which is illustrated in different settings: For random variables, risk perception can be interpreted as model risk, and we compute among others the robust representation of the economic index of riskiness. For lotteries, risk perception can be viewed as distributional risk and we study the "Value at Risk". For consumption patterns, which excerpt an intertemporality dimension in risk perception, we provide an interpretation in terms of discounting risk and discuss some examples.
We present and compare two different approaches to conditional
risk measures. One approach draws from vector space based convex analysis
and presents risk measures as functions on L^p spaces while the other approach
utilizes module based convex analysis where conditional risk measures are defined on L^p type modules. Both approaches utilize general duality theory for
vector valued convex functions in contrast to the current literature in which
we fi nd ad hoc dual representations. By presenting several applications such
as monotone and sub(cash) invariant hulls with corresponding examples we
illustrate that module based convex analysis is well suited to the concept of
conditional risk measures.