Abstract:
We present an adaptive approach for valuing the European call option on assets with stochastic volatility. The essential feature of the method is a reduction of uncertainty in latent volatility due to a Bayesian learning procedure. Starting from a discrete-time stochastic volatility model, we derive a recurrence equation for the variance of the innovation term in latent volatility equation. This equation describes a reduction of uncertainty in volatility which is crucial for option pricing. To implement the idea of adaptive control, we use the risk-minimization procedure involving random volatility with uncertainty. By using stochastic dynamic programming and a Bayesian approach, we derive a recurrence equation for the risk inherent in writing the option. This equation allows us to find the fair price of the European call option. We illustrate numerically that the adaptive procedure leads to a decrease in option price.

Abstract:
We design an optimal strategy for investment in a portfolio of assets subject to a multiplicative Brownian motion. The strategy provides the maximal typical long-term growth rate of investor's capital. We determine the optimal fraction of capital that an investor should keep in risky assets as well as weights of different assets in an optimal portfolio. In this approach both average return and volatility of an asset are relevant indicators determining its optimal weight. Our results are particularly relevant for very risky assets when traditional continuous-time Gaussian portfolio theories are no longer applicable.

Abstract:
In an equity market model with "Knightian" uncertainty regarding the relative risk and covariance structure of its assets, we characterize in several ways the highest return relative to the market that can be achieved using nonanticipative investment rules over a given time horizon, and under any admissible configuration of model parameters that might materialize. One characterization is in terms of the smallest positive supersolution to a fully nonlinear parabolic partial differential equation of the Hamilton--Jacobi--Bellman type. Under appropriate conditions, this smallest supersolution is the value function of an associated stochastic control problem, namely, the maximal probability with which an auxiliary multidimensional diffusion process, controlled in a manner which affects both its drift and covariance structures, stays in the interior of the positive orthant through the end of the time-horizon. This value function is also characterized in terms of a stochastic game, and can be used to generate an investment rule that realizes such best possible outperformance of the market.

Abstract:
We introduce a novel framework for computing optimal randomized security policies in networked domains which extends previous approaches in several ways. First, we extend previous linear programming techniques for Stackelberg security games to incorporate benefits and costs of arbitrary security configurations on individual assets. Second, we offer a principled model of failure cascades that allows us to capture both the direct and indirect value of assets, and extend this model to capture uncertainty about the structure of the interdependency network. Third, we extend the linear programming formulation to account for exogenous (random) failures in addition to targeted attacks. The goal of our work is two-fold. First, we aim to develop techniques for computing optimal security strategies in realistic settings involving interdependent security. To this end, we evaluate the value of our technical contributions in comparison with previous approaches, and show that our approach yields much better defense policies and scales to realistic graphs. Second, our computational framework enables us to attain theoretical insights about security on networks. As an example, we study how allowing security to be endogenous impacts the relative resilience of different network topologies.

Abstract:
this research was aimed at gaining a deeper insight into perception linked to environmental uncertainty and the strategic significance of perceptual diversity. factors intervening in perception were characterised. it is specifically shown that an individual's cognitive limitations and their beliefs' affective influence gave rise to cognitive bias distorting individual perception. this model was applied to both management (perceived uncertainty) and outside observers (objective uncertainty) perceiving environmental uncertainty. the idiosyncratic nature of perceiving uncertainty and the interrelationships between various individuals' perception was thus considered (stress-management and outside observers). the significance of the heterogeneity of perception of managers working at a single company was analysed and compared to that of those working in different companies. it was found that inter-company perception of diversity enabled selective access to competitive advantages. diversity of perception at intra-company level enhanced assessment of the background of strategy management and reduced organisational coordination.

Abstract:
This paper aims to contribute to the development of valuation models for long-term investments while keeping an eye on market prices. The adopted methodology is rooted on the existence of markets for futures and options on commodities related to energy investments. These markets are getting ever-increasingly liquid with ever-longer maturities while trading contracts. We discuss the advantages of this approach relative to other alternatives such as the Net Present Value (NPV) or the Internal Rate of Return (IRR), despite a limited increase in the complexity of the models involved. More specifically, using the valuation methods well-known to energy-finance academics, the paper shows how to: break down an investment into its constituent parts, apply to each of them the corresponding risk premium, value annuities on assets with a deterministic or stochastic behavior, and value the options that are available to its owner, in order to get an overall value of the investment project. It also includes an application to improvement in coal consumption, where futures markets are used to get a numerical estimate of the parameters that are required for valuation. The results are then compared with those from traditional methodologies. Conclusions for this type of investments under uncertainty are derived.

Abstract:
Using mathematical arguments and computer simulations we show that in more adverse environments individuals perceive their resources to be more unpredictable, and that this unpredictability favours cooperation. First we show analytically that in a more adverse environment the individual experiences greater perceived uncertainty. Second we show through a simulation study that more perceived uncertainty implies higher level of cooperation in communities of selfish individuals.This study captures the essential features of the natural examples: the positive impact of resource adversity or uncertainty on cooperation. These newly discovered connections between environmental adversity, uncertainty and cooperation help to explain the emergence and evolution of cooperation in animal and human societies.The drive to understand the emergence of cooperation – actions of benefit to both actor and recipient – in communities of selfish individuals has generated a large body of theoretical and empirical research in recent decades [1-14]. This research focuses on the dynamics of interactions between individuals and pays relatively little attention to the effects of the environment. However, evidence is growing, in many taxa, that as the adversity (harshness) and uncertainty of the environment increase cooperation is enhanced and we present a model here that attempts to explain this phenomenon as an adaptive facultative response favoured by selection.An organism's environment is more adverse if some quality such as resources, physical structure, climate, competitors, parasites or predators changes in such a way as to decrease darwinian fitness. Environmental adversity is species-specific, e.g. high temperature may be adverse for some organisms, but not for thermophilic bacteria. As an example of the uncertainty or unpredictability of the environment [15], feeding in a patchy area, where some places are rich in food and others barren, results in greater uncertainty of nutritional stat

Abstract:
Cost-benefit analysis (CBA) is controversial for environmental issues, but is nevertheless employed by many governments and private organizations for making environmental decisions. Controversy centers on the practice of economic discounting in CBA for decisions that have substantial long-term consequences, as do most environmental decisions. Customarily, economic discounting has been calculated at a constant exponential rate, a practice that weights the present heavily in comparison with the future. Recent analyses of economic data show that the assumption of constant exponential discounting should be modified to take into account large uncertainties in long-term discount rates. A proper treatment of this uncertainty requires that we consider returns over a plausible range of assumptions about future discounting rates. When returns are averaged in this way, the schemes with the most severe discounting have a negligible effect on the average after a long period of time has elapsed. This re-examination of economic uncertainty provides support for policies that prevent or mitigate environmental damage. We examine these effects for three examples: a stylized renewable resource, management of a long-lived species (Atlantic Right Whales), and lake eutrophication.

Abstract:
Optimal uncertainty quantification (OUQ) is a framework for numerical extreme-case analysis of stochastic systems with imperfect knowledge of the underlying probability distribution. This paper presents sufficient conditions under which an OUQ problem can be reformulated as a finite-dimensional convex optimization problem, for which efficient numerical solutions can be obtained. The sufficient conditions include that the objective function is piecewise concave and the constraints are piecewise convex. In particular, we show that piecewise concave objective functions may appear in applications where the objective is defined by the optimal value of a parameterized linear program.

Abstract:
We propose a rigorous framework for Uncertainty Quantification (UQ) in which the UQ objectives and the assumptions/information set are brought to the forefront. This framework, which we call \emph{Optimal Uncertainty Quantification} (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions they have finite-dimensional reductions. As an application, we develop \emph{Optimal Concentration Inequalities} (OCI) of Hoeffding and McDiarmid type. Surprisingly, these results show that uncertainties in input parameters, which propagate to output uncertainties in the classical sensitivity analysis paradigm, may fail to do so if the transfer functions (or probability distributions) are imperfectly known. We show how, for hierarchical structures, this phenomenon may lead to the non-propagation of uncertainties or information across scales. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact and on the seismic safety assessment of truss structures, suggesting the feasibility of the framework for important complex systems. The introduction of this paper provides both an overview of the paper and a self-contained mini-tutorial about basic concepts and issues of UQ.