Is Brownian Motion Really Necessary?

Risk management is an essential component of effective campus recreational sport programming. However, research on risk management policies, procedures, and practices within recreational sports is both limited and outdated. As such, this study provides an overview of current risk management policies, procedures, and practices across North American collegiate recreational sports departments. Specifically, the results emphasize current training practices for both part-time and full-time staff, practices in establishing and updating emergency action plans, and contents covered in emergency action plans. The results suggest that while the vast majority of recreational sport programs have established risk management procedures and engage in risk management training, they diverge in best practices related to emergency action plans and in terms of contents covered in such plans. Applications of these results for practitioners and institutions are emphasized, including discussions around best practices in risk management.

Simulation of Brownian motion at first-passage times

We introduce a suitable backward stochastic differential equation (BSDE) to represent the value of an optimal control problem with partial observation for a controlled stochastic equation driven by Brownian motion. Our model is general enough to include cases with latent factors in mathematical finance. By a standard reformulation based on the reference probability method, it also includes the classical model where the observation process is affected by a Brownian motion (even in presence of a correlated noise), a case where a BSDE representation of the value was not available so far. This approach based on BSDEs allows for greater generality beyond the Markovian case, in particular our model may include path-dependence in the coefficients (both with respect to the state and the control), and does not require any nondegeneracy condition on the controlled equation. We use a randomization method, previously adopted only for cases of full observation, and consisting in a first step, in replacing the control by an exogenous process independent of the driving noise and in formulating an auxiliary (“randomized”) control problem where optimization is performed over changes of equivalent probability measures affecting the characteristics of the exogenous process. Our first main result is to prove the equivalence between the original partially observed control problem and the randomized problem. In a second step, we prove that the latter can be associated by duality to a BSDE, which then characterizes the value of the original problem as well.

The study of exponential functionals of Brownian motion has recently attracted much attention, partly motivated by several problems in financial mathematics. Let be a linear Brownian motion starting from 0. Following Dufresne (1989), (1990), De Schepper and Goovaerts (1992) and De Schepper et al. (1992), we are interested in the process (for δ > 0), which stands for the discounted values of a continuous perpetuity payment. We characterize the upper class (in the sense of Paul Lévy) of X, as δ tends to zero, by an integral test. The law of the iterated logarithm is obtained as a straightforward consequence. The process exp(W(u))du is studied as well. The class of upper functions of Z is provided. An application to the lim inf behaviour of the winding clock of planar Brownian motion is presented.

The aim of this paper is to find optimal portfolio strategies for an n-stock extension of the stochastic volatility model proposed in Ref. [2]. It is a modification of [19], and characterizes the dependence by the use of a factor structure. The idea of a factor structure is that the diffusion components of the stocks contain one Brownian motion that is unique for each stock, and a few Brownian motions that all stocks share. Our model can obtain strong correlations between the returns for different stocks without affecting their marginal distributions. This was not possible in Ref. [19]. Further, the number of model parameters does not grow too fast as the number of stocks n grows. We use dynamic programming to solve Merton’s optimization problem for power utility, with utility drawn from terminal wealth. Optimal strategies for n stocks are obtained. We also discuss how to estimate the model from data. This is illustrated with an example.

Approximating payoffs and pricing formulas

The articles in this volume summarize the lectures given by the authors on February 1, 2005, at an open conference held at the Academie des Sciences, quai Conti, on Financial Mathematics, a field in constant development since the end of the 1970s. The aim is to make these articles accessible to a wide audience belonging to scientific horizons; indeed, this half-day conference was attended by many young postgraduates in disciplines such as economics, management, and applied mathematics, along with many members of the different sections of the Academie. Among the reasons to get interested in financial mathematics, the following is one: who has never wondered, looking at the financial pages of a newspaper, displaying the erratic evolutions of quotations on the Stock Exchange, if these were not “governed” by some models, likely to be probabilistic? This question was at the heart of the studies conducted by Louis Bachelier, particularly in his famous thesis (1900), and he answered the above question in terms of Brownian Motion. His remarkable results, however, remained in a kind of scientific limbo for almost 75 years, until Samuelson “corrected” Bachelier (in 1965) by replacing the Brownian Motion by its exponential, and the famous Black-Scholes formula began (in 1973) to play an essential role in the computation of option prices. Since the 1980s, we have witnessed the explosion of probabilistic models, along with financial products, each in turn becoming more and more complex. All this technology, which now forms an important part of financial engineering, exists only because some mathematical concepts both simple and universal allow building a “theory of the laws of markets,” based on principles such as the prices across time of an uncertain asset having the probabilistic structure of a fair game, that is to say of a martingale. From this concept, little by little was built the entire theory of stochastic processes, which is a pillar on which the mathematical theory of “arbitrage” was developed by Delbaen and Schachermayer. The six articles in this volume present the following main topics.

Construction is a critical sector of any economy in terms of value production, labor, and contributing to the gross national product. Managing risk is a relatively young area in Yemen’s construction sector, but it is gaining traction as building activity and competition rise. Construction firms mitigate risk by using a variety of risk management methods. Therefore, there is a need to assess these procedures in order to detect shortcomings. This research aims to establish the existing risk management strategies used in Yemeni building projects. Survey questionnaires were used to collect data. Respondents were drawn from Yemeni construction businesses. Risk management is not executed systematically, intentionally, or continuously, and most firms’ risk management procedures are reactive, semipermanent, informal, and unstructured, with no or few dedicated resources to address risks. This strategy is inconsistent with generally accepted risk management principles. Nonetheless, the findings suggest a general understanding of risk management and a willingness to learn from previous errors. The study of the findings suggests that risk identification approaches such as judgment and historical data are employed for risk analysis, and that the industry typically attempts to avoid or transfer risks in Yemeni building projects. The results shed light on the shortcomings of Yemen’s project management practices. To guarantee that construction projects obtain maximum value for money, project managers of big construction businesses in Yemen need a strong understanding of and training in globally accepted systematic risk management procedures. Finally, this study can help future stakeholders determine how to work together to manage risk.

A Mayfly algorithm for cardinality constrained portfolio optimization

This paper considers the implications of network effects and distinction of security measures in the risk management procedure. We compare three models in the risk management procedure: without network effects and general protection measures, without network effects but with general protection, and with both network effects and general protection measures. The paper details the impact in terms of security risks, investment levels, and benefits of security investment, in the three models. We show that the preferable way to conduct risk management procedure is to follow the latter of the three models.KeywordsNetwork EffectSecurity MeasureRisk MitigationSecurity RiskSuccessive AttackThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

<p>Since the fractional $ G $-Brownian motion (fGBm) generalizes the concepts of the standard Brownian motion, fractional Brownian motion, and $ G $-Brownian motion, while it can exhibit long-range dependence or antipersistence and feature the volatility uncertainty simultaneously, it can be a better alternative stochastic process in the financial applications. Thus, in this paper, some empirical studies for the financial applications of the fGBm were carried out, where the recent high-frequency data for some selected assets in the financial market are from the Oxford-Man Institute of Quantitative Finance Realized Library. There are two main empirical findings. One was that the H-$ G $-normal distributions associated with the fGBm are more suitable in describing the dynamics of daily returns and increments of log-volatility for these assets than the usual distributions, since they not only characterize the properties of skewness, excess kurtosis, and long-range dependence $ \left(\frac{1}{2} < H < 1\right) $ or antipersistence $ \left(0 < H < \frac{1}{2}\right) $, but also feature the volatility uncertainty. The other one was that the daily return and log-volatility both behave essentially as fGBm with different $ \underline{\sigma}^2 $ and $ \overline{\sigma}^2 $, but Hurst parameters $ H < \frac{1}{2} $, at any reasonable time scale. Then a generalized stochastic model for the dynamics of the assets called rough fractional stochastic volatility model driven by fGBm (RFSV-fGBm) was developed. Finally, some parameter estimates and numerical experiments for the RFSV-fGBm model were investigated and carried out.</p>

K. Ito invented his famous stochastic calculus on Brownian motion in the 1940s. In the 1960s and 1970s, the “Strasbourg school,” headed by P.A. Meyer, developed a modern theory of martingales, the general theory of stochastic processes, and stochastic calculus on semimartingales. It turned out soon that semimartingales constitute the largest class of right continuous adapted integrators with respect to which stochastic integrals of simple predictable integrands satisfy the theorem of dominated convergence in probability. Stochastic calculus on semimartingales not only became an important tool for modern probability theory and stochastic processes but also has broad applications to many branches of mathematics, physics, engineering, and mathematical finance.

Denote by αt(μ)the probability law ofAt(μ)= ∫0texp(2(Bs+μs))dsfor a Brownian motion {Bs,s≥ 0}. It is well known that αt(μ)is of interest in a number of domains, e.g. mathematical finance, diffusion processes in random environments, stochastic analysis on hyperbolic spaces and so on, but that it has complicated expressions. Recently, Dufresne obtained some remarkably simple expressions for αt(0)and αt(1), as well as an equally remarkable relationship between αt(μ)and αt(ν)for two different drifts μ and ν. In this paper, hinging on previous results about αt(μ), we give different proofs of Dufresne's results and present extensions of them for theprocesses{At(μ),t≥ 0}.

This paper addresses the problem of approximating random variables in terms of sums consisting of a real constant and of a stochastic integral with respect to a given semimartingale $X$. The criterion is minimization of $\\mathbf{L}^2$–distance, or "least-squares". This problem has a straightforward and well-known solution when $X$ is a Brownian motion or, more generally, a square-integrable martingale, with respect to the underlying probability measure $P$. We address the general, semimartingale case by means of a duality approach; the adjoint variables in this duality are signed measures, absolutely continuous with respect to $P$, under which $X$ behaves like a martingale. It is shown that this duality is useful, in that the value of an appropriately formulated dual problem can be computed fairly easily; that it "has no gap" (i.e., the values of the primal and dual problems coincide); that the signed measure which is optimal for the dual problem can be easily identified whenever it exists; and that the duality is also "strong", in the sense that one can then identify the optimal stochastic integral for the primal problem. In so doing, the theory presented here both simplifies and extends the extant work on the subject. It has also natural connections and interpretations in terms of the theory of "variance-optimal" and "mean-variance efficient" portfolios in Mathematical Finance, pioneered by H. Markowitz and then greatly extended by H. Föllmer, D. Sondermann and most notably M. Schweizer.

Optimal portfolios with regime switching and value-at-risk constraint