Universal L^s -rate-optimality of L^r-optimal quantizers by dilatation and contraction

Let $r, s>0$. For a given probability measure $P$ on $\mathbb{R}^d$, let $(\alpha_n)_{n \geq 1}$ be a sequence of (asymptotically) $L^r(P)$- optimal quantizers. For all $\mu \in \mathbb{R}^d$ and for every $\theta >0$, one defines the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ by : $\forall n \geq 1, \alpha_n^{\theta, \mu} = \mu + \theta(\alpha_n - \mu) = \{\mu + \theta(a- \mu), a \in \alpha_n \}$. In this paper, we are interested in the asymptotics of the $L^s$-quantization error induced by the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$. We show that for a wide family of distributions, the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ is $L^s$-rate-optimal. For the Gaussian and the exponential distributions, one shows how to choose the parameter $\theta$ such that $(\alpha_n^{\theta, \mu})_{n \geq 1}$ satisfies the empirical measure theorem and probably be asymptotically $L^s$-optimal.Comment: 26 page

Conditional hitting time estimation in a nonlinear filtering model by the Brownian bridge method

The model consists of a signal process $X$ which is a general Brownian diffusion process and an observation process $Y$, also a diffusion process, which is supposed to be correlated to the signal process. We suppose that the process $Y$ is observed from time 0 to $s>0$ at discrete times and aim to estimate, conditionally on these observations, the probability that the non-observed process $X$ crosses a fixed barrier after a given time $t>s$. We formulate this problem as a usual nonlinear filtering problem and use optimal quantization and Monte Carlo simulations techniques to estimate the involved quantities

An application to credit risk of a hybrid Monte Carlo-Optimal quantization method

In this paper we use a hybrid Monte Carlo-Optimal quantization method to approximate the conditional survival probabilities of a firm, given a structural model for its credit defaul, under partial information. We consider the case when the firm's value is a non-observable stochastic process $(V_t)_{t \geq 0}$ and inverstors in the market have access to a process $(S_t)_{t \geq 0}$, whose value at each time t is related to $(V_s, s \leq t)$. We are interested in the computation of the conditional survival probabilities of the firm given the "investor information". As a application, we analyse the shape of the credit spread curve for zero coupon bonds in two examples.credit risk, structural approach, survival probabilities, partial information, filtering, optimal quantization, Monte Carlo method.

Product Markovian quantization of an R^d -valued Euler scheme of a diffusion process with applications to finance

We introduce a new approach to quantize the Euler scheme of an $\mathbb{R}^d$-valued diffusion process. This method is based on a Markovian and componentwise product quantization and allows us, from a numerical point of view, to speak of {\em fast online quantization} in dimension greater than one since the product quantization of the Euler scheme of the diffusion process and its companion weights and transition probabilities may be computed quite instantaneously. We show that the resulting quantization process is a Markov chain, then, we compute the associated companion weights and transition probabilities from (semi-) closed formulas. From the analytical point of view, we show that the induced quantization errors at the $k$-th discretization step $t_k$ is a cumulative of the marginal quantization error up to time $t_k$. Numerical experiments are performed for the pricing of a Basket call option, for the pricing of a European call option in a Heston model and for the approximation of the solution of backward stochastic differential equations to show the performances of the method

Killing Us Softly 4

This week’s speaker, Ms. Sagna, emphasized the objectification of women in the media through advertisements. She showed a video entitled “Killing Us Softly 4” where speaker and former model Jean Kilbourne spoke about what it means to be a woman in American culture today. Jean started collecting ads in the late 1960’s, inspired by her involvement with the feminist movement, her interest in media, and her experiences as a model

Effectiveness of Remedial Mathematics Supplemental Instruction: a Community College Study.

The purpose of this mixed method study was to determine if there is a relationship between characteristics of supplemental instructors’ personal traits, teaching skills, subject matter, constructive/active, collaborative learning, effective communication, and their practices (as judged by students) and student success in their remedial mathematics course. The college Provost was contacted by email to request an authorization to conduct this study in his college. Once approved, the investigator contacted face to face his colleagues to ask their students to participate in the study because they enrolled in a remedial algebra class where the instructor is assisted by a supplemental instructor (SI leader). Sixteen algebra classes were selected, and each were assisted by supplemental instructors. Students’ scores on the pre-test (at beginning of the semester) and post-test (at the end of the semester) were collected to gauge their achievement on both tests. Students completed a questionnaire that asked about their perceptions about their supplemental instructors’ personal traits, teaching skills, subject matter, constructive/active, collaborative learning, effective communication, and their practices throughout the semester. Students’ mean scores difference on the post-test were higher in 62.5% of the sections than on the pre-test. The evaluation of achievement on both tests, the responses to the questionnaire and comments from students showed that SI leader’s characteristics associated to effective communication/active learning, teaching skills, and personal traits could be contributor to score achievements. The linear regression in the study shows that the three factors did not significantly predict the post-test score. However, the pre-test did significantly predict the post-test score in a remedial Math 20 at the end of the semester (Beta = .47, t (197) = 6.56, p <.05). In addition, the comments in the questionnaire found that students acknowledged their supplemental instructor role in the classroom and during the SI’s weekly sessions

Morphological alternation and event delimitation in Eegimaa

It is rare for a language to be able to use noun class markers in the nominal domain to categorise entities, and at the same time, use these same linguistic markers to categorise events from the verbal domain. Such a system can be found in Eegimaa and some other related Atlantic languages spoken in the Basse-Casamance area of Southern Senegal, where non-finite verbs and the events they refer to are classified using several different noun class prefixes. In these languages, the use of individual noun class markers as nonfinite verb classificatory markers is lexically determined. But, there are also instances where different noun class prefixes can alternate on verbal stems. Whenever these alternations are attested, one of the alternants must be e-, and the other can be any prefix attested on non-finite verbs, including class prefixes ga- and ba- which are studied here. I show that in these alternations, class marker e- is used to express event delimitation by expressing features such as individuation and telicity which, in the typological literature, have been associated with properties of high transitivity. However, when other prefixes like ga- and ba- alternate with e-, they express values of non-individuation and atelicity which are placed on the lower end of the transitivity scale

Universal LsL^s-rate-optimality of LrL^r-optimal quantizers by dilatation and contraction

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