1,720,999 research outputs found
Équations aux dérivées partielles stochastiques avec bruit de Lévy
No full textIn this thesis, we develop a stochastic calculus for the space-time Lévy white noise introduced in [1] as an alternative for the Gaussian white noise perturbing an stochastic partial differential equation (SPDE). We give a new proof for the Itô formula for some integral processes related to this Lévy white noise. Then, we consider a general non-linear SPDE on R_+* R driven by this Lévy white noise and we show that this equation has a unique random-field solution. Using Rosenthal's inequality, we develop a maximal inequality for the moments of order p≥2 of the stochastic integral with respect to this noise. Based on this inequality, we show that the stochastic wave equation equation has a unique solution, which is weakly intermittent in the sense of [2, 3]. Finally, we develop a Malliavin calculus with respect to the compensated Poisson random measure associated to the Lévy white noise. Under certain conditions, we show that the solution is Malliavin differentiable and its Malliavin derivative satisfies an integral equation.
[1] Integration with respect to Lévy colored noise, with application to SPDEs: Stochastics An International Journal of Probability and Stochastic Processes , 87, 363-381.
[2] Intermittence and nonlinear parabolic stochastic partial differential equations. Electronic Journal of Probability. Vol 21, 548-568.
[3] Analysis of stochastic partial differential equations. CBMS Regional Conference Series in Mathematics, Vol 119. American Mathematical Society
SPDEs with Infinite-Variance Lévy Noise
No full textThis thesis is devoted to the study of the existence and uniqueness of solutions for stochastic partial differential equations (SPDEs) driven by Lévy noise. The main contributions of this work are contained in the recent publications [32] and [5]. Article [32] focuses on a stochastic wave equation with multiplicative Lévy noise. We establish the existence and uniqueness of a random field solution, relying only on the integrability of the Lévy measure on the region |z| ≤ 1. Furthermore, we show that this solution has finite moments up to a certain stopping time, which depends on a bounded region of space. Article [5] studies a broader class of SPDEs driven by heavy-tailed Lévy noise, which includes the Parabolic Anderson Model (PAM) and the Hyperbolic Anderson Model (HAM). Specifically, we demonstrate the existence of solutions for SPDEs driven by symmetric α-stable Lévy noise. Using the Lepage representation of the noise and techniques borrowed from the theory of multiple stable integrals, we construct a solution that has a series representation which depends only on the points of the jump measure associated with the noise
Équations aux dérivées partielles stochastiques avec bruit de Lévy
No full textIn this thesis, we develop a stochastic calculus for the space-time Lévy white noise introduced in [1] as an alternative for the Gaussian white noise perturbing an stochastic partial differential equation (SPDE). We give a new proof for the Itô formula for some integral processes related to this Lévy white noise. Then, we consider a general non-linear SPDE on R_+* R driven by this Lévy white noise and we show that this equation has a unique random-field solution. Using Rosenthal's inequality, we develop a maximal inequality for the moments of order p≥2 of the stochastic integral with respect to this noise. Based on this inequality, we show that the stochastic wave equation equation has a unique solution, which is weakly intermittent in the sense of [2, 3]. Finally, we develop a Malliavin calculus with respect to the compensated Poisson random measure associated to the Lévy white noise. Under certain conditions, we show that the solution is Malliavin differentiable and its Malliavin derivative satisfies an integral equation.\ud
[1] Integration with respect to Lévy colored noise, with application to SPDEs: Stochastics An International Journal of Probability and Stochastic Processes , 87, 363-381.\ud
[2] Intermittence and nonlinear parabolic stochastic partial differential equations. Electronic Journal of Probability. Vol 21, 548-568.\ud
[3] Analysis of stochastic partial differential equations. CBMS Regional Conference Series in Mathematics, Vol 119. American Mathematical Society
On New Constructive Tools in Bayesian Nonparametric Inference
No full textThe Bayesian nonparametric inference requires the construction of priors on infinite dimensional spaces such as the space of cumulative distribution functions and the space of cumulative hazard functions. Well-known priors on the space of cumulative distribution functions are the Dirichlet process, the two-parameter Poisson-Dirichlet process and the beta-Stacy process. On the other hand, the beta process is a popular prior on the space of cumulative hazard functions. This thesis is divided into three parts. In the first part, we tackle the problem of sampling from the above mentioned processes. Sampling from these processes plays a crucial role in many applications in Bayesian nonparametric inference. However, having exact samples from these processes is impossible. The existing algorithms are either slow or very complex and may be difficult to apply for many users. We derive new approximation techniques for simulating the above processes. These new approximations provide simple, yet efficient, procedures for simulating these important processes. We compare the efficiency of the new approximations to several other well-known approximations and demonstrate a significant improvement. In the second part, we develop explicit expressions for calculating the Kolmogorov, Levy and Cramer-von Mises distances between the Dirichlet process and its base measure. The derived expressions of each distance are used to select the concentration parameter of a Dirichlet process. We also propose a Bayesain goodness of fit test for simple and composite hypotheses for non-censored and censored observations. Illustrative examples and simulation results are included. Finally, we describe the relationship between the frequentist and Bayesian nonparametric statistics. We show that, when the concentration parameter is large, the two-parameter Poisson-Dirichlet process and its corresponding quantile process share many asymptotic pr operties with the frequentist empirical process and the frequentist quantile process. Some of these properties are the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem
Smoothness with Respect to Noise Parameters for Parabolic/hyperbolic Anderson Model with Regular or Rough Noise in Space
No full textIn this work, we study the continuity in law of the solutions of two linear multiplicative SPDEs, namely, the parabolic Anderson model (PAM) and the hyperbolic Anderson model (HAM). The forcing term under investigation is examined in two cases: (i) the regular noise, with the spatial covariance given by the Riesz kernel of order α ∈ (0, d) in spatial dimension d ≥ 1; (ii) the rough noise, which is a fractional noise in space with Hurst index H < 1/2 in spatial dimension d = 1. In both cases, the noise is assumed to be colored in time. The similar problem for the case of the white noise in time was considered in [13, 23].
For the initial condition, we consider two scenarios: (a) constant initial condition; (b) initial condition given by a signed Borel measure on Rd. In the case of constant initial condition, we prove that the solution is continuous in law in the space C([0,T]×Rd) of continuous functions, with respect to the spatial parameter (α or H) of the noise. In the case of general initial condition (given by a measure), the weak convergence of the solution with respect to spatial parameter of the noise is obtained in the space C([t0, T ] × Rd). The solution is understood in Skorohod sense, using Malliavin Calculus, a stochastic calculus of variations theory that proves beneficial in exploring various aspects in the theory of SPDEs.
The results corresponding to cases (a) and (b) above are contained in the recent article [7], and respectively, in the preprint [30]
Continuity in Law with Respect to the Spatial Hurst Index of the Solutions to Some Linear SPDEs
No full textIn this thesis, we study the linear stochastic heat and wave equations with zero initial conditions, driven by a Gaussian noise, which is fractional in space with Hurst index
H ∈ (0, 1), and is either white in time (i.e. fractional in time with index H_0 = 1/2) or fractional in time with index H_0 > 1/2. We prove that the solution of each of these equations is continuous in law in the space C([0,T] × R) of continuous functions, with respect to the index H. This result has already been proved in the recent article [15] for the case H_0 = 1/2, and we extend it here to the case H_0 > 1/2
A Study of Parabolic and Hyperbolic Anderson Models Driven by Fractional Brownian Sheet with Spatial Hurst Index in (0,1)
No full textThe goal of this thesis is to present a comprehensive study of the parabolic and hyperbolic Anderson models with constant initial condition, driven by a Gaussian noise which is fractional in space with index H > 1/2 or H 1/2. As a preliminary step, we study the linear stochastic heat and wave equations with the same type of noise. In the case H_0 > 1/2 and H < 1/2, we present a new result, regarding the solution of the parabolic Anderson model with general initial condition given by a measure
Analysis of Longitudinal Data with Missing Responses Adjusted by Inverse Probability Weights
No full textWe propose a new method for analyzing longitudinal data which contain responses
that are missing at random. This method consists in solving the generalized estimating
equation (GEE) of [7] in which the incomplete responses are replaced by values
adjusted using the inverse probability weights proposed in [14]. We show that the
root estimator is consistent and asymptotically normal, essentially under some conditions on the marginal distribution and the surrogate correlation matrix as those
presented in [12] in the case of complete data, and under minimal assumptions on
the missingness probabilities. This method is applied to a real-life dataset taken from
[10], which examines the incidence of respiratory disease in a sample of 250 pre-school age Indonesian children which were examined every 3 months for 18 months, using as covariates the age, gender, and vitamin A deficiency
Hedge Funds and Survival Analysis
No full textUsing data from Hedge Fund Research, Inc. (HFR), this study adapts and expands
on existing methods in survival analysis in an attempt to investigate whether hedge
funds mortality can be predicted on the basis of certain hedge funds characteristics.
The main idea is to determine the characteristics which contribute the most to the
survival and failure probabilities of hedge funds and interpret them. We establish hazard
models with time-independent covariates, as well as time-varying covariates to interpret
the selected hedge funds characteristics. Our results show that size, age, performance,
strategy, annual audit, fund offshore and fund denomination are the characteristics that
best explain hedge fund failure. We find that 1% increase in performance decreases
the hazard by 3.3%, the small size and the less than 5 years old hedge funds are the
most likely to die and the event-driven strategy is the best to use as compare to others.
The risk of death is 0.668 times lower for funds who indicated that an annual audit
is performed as compared to the funds who did not indicated that an annual audit is
performed. The risk of death for the offshore hedge funds is 1.059 times higher than the
non-offshore hedge funds
Longitudinal Data Analysis Using Generalized Linear Model with Missing Responses
No full textLongitudinal studies rely on data collected at several occasions from a set of selected individuals. The purpose of these studies is to use a regression-type model to express a response variable as a function of explanatory variables, or covariates. In this thesis, we use marginal models for the analysis of such data, which, coupled with the method of estimating equations, provide estimators of the main regression parameter. When some of the responses are missing or there is error in the recorded covariates, the original estimating equation may be biased. We use techniques available in the literature to modify it and regain the unbiasedness property. We prove the asymptotic normality of the regression estimator obtained under these more realistic circumstances, and provide theoretical and numerical examples to illustrate this approach
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