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Annulus SLE partition functions and martingale-observables

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In this talk, I will introduce a version of conformal field theory (CFT) and explain its implementations to SLE theory in a doubly connected domain. The basic fields in these implementations are one-parameter family of Gaussian free fields whose boundary conditions are given by a weighted combination of Dirichlet boundary condition and excursion-reflected one. After explaining basic notions in CFT such as OPE families of central charge modifications of the Gaussian free field and presenting certain equations including a version of Eguchi-Ooguri and Ward’s equations, I will outline the relation between CFT and SLE theory. As an application, I will explain how to apply the method of screening to find Euler integral type solutions to the parabolic partial differential equations for the annulus SLE partition functions introduced by Zhan and present a class of SLE martingale-observables associated with these solutions. This is based on joint work with Nam-Gyu Kang and Hee-Joon Tak.

Blow-up results for space-time fractional dynamics

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Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type, in $(d+1)$ dimensions, where $\nu > 0, \beta \in (0, 1)$, $\alpha \in (0,2]$. The operator $\partial^\beta_t$ is the Caputo fractional derivative while $-(-\Delta)^{\alpha / 2} $ is the generator of an isotropic $\alpha$-stable Lévy process and $I_t^{1 - \beta}$ is the Riesz fractional integral operator. The forcing noise denoted by $\stackrel{\cdot}{F}(t, x)$ is a Gaussian noise. These equations might be used as a model for materials with random thermal memory. We derive non-existence (blow-up) of global random field solutions under some additional conditions, most notably on $b$, $\sigma$ and the initial condition. Our results complement those of P. Chow in “P.-L. Chow. Unbounded positive solutions of nonlinear parabolic Itô equations. Commun. Stoch. Anal., 3(2)(2009), 211—222.” and “P.-L. Chow. Explosive solutions of stochastic reaction-diffusion equations in mean $l_{p}$-norm. J. Differential Equations, 250(5) (2011), 2567—2580.” and Foondun and Parshad “M. Foondun and R. Parshad, On non-existence of global solutions to a class of stochastic heat equations. Proc. Amer. Math. Soc. 143 (2015), no. 9, 4085—4094”, among others. The results presented are our recent joint work with Sunday Asogwa, Mohammud Foondun, Wei Liu, and Jebessa Mijena.

Two-curve Green’s function for 2-SLE

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A 2-SLE$_ \kappa $, $\kappa\in (0,8)$, is a pair of random curves $(\eta_1,\eta_2)$ in a simply connected domain $D$ connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal SLE$_ \kappa$ curve in a complement domain. We prove that, for the exponent $\alpha=\frac{(12 - \kappa) (\kappa + 4)}{8 \kappa}$, for any $z_0 \in D$, the limit $\lim_{r \to 0^+} r^{-\alpha} \mathbb{P}[\mbox{dist}(\eta_j, z_0) < r,j=1,2]$ converges to a positive number, called the two-curve Green’s function. To prove the convergence, we transform the original problem into the study of a two-dimensional diffusion process, and use orthogonal polynomials to derive its transition density and invariant density.

Mean-field anticipated BSDEs driven by fractional Brownian motion and related stochastic control problem

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In this talk, we introduce a new type of BSDEs, we call it mean-field anticipated backward stochastic differential equations (MF-BSDEs, for short) driven by a fractional Brownian motion with Hurst parameter $H > 1/2$. We will show that it’s possible to prove the existence and uniqueness of this new type of BSDEs using two different approaches. Then, we will present a comparison theorem for such BSDEs. Finally, as an application of this type of equations, a related stochastic optimal control problem is studied. This is a joint work with Yufeng Shi and Jiaqiang Wen : Institute for Financial Studies and School of Mathematics, Shandong University, Jinan 250100, China.

Berry-Esseen bounds in the Breuer-Major central limit theorem

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The Breuer-Major theorem provides sufficient conditions in order that a normalized sum of non-linear functionals of Gaussian random fields exhibits Gaussian fluctuation. Such a result has far-reaching applications in statistical inference of Gaussian models. In this talk, I will be focusing on the rate of convergence in the total variation distance of the Breuer-Major theorem. To this end, we apply Malliavin calculus (stochastic calculus of variation) techniques, Stein’s method for normal approximations, and Gebelein’s inequality for functionals of correlated Gaussian fields. Based on joint work with I. Nourdin and G. Peccati.

A Lévy regime-switching temperature dynamics model for weather derivatives

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Weather is a key production factor in agricultural crop production but at the same time, the most significant and least controllable source of peril in agriculture. These effects of weather on agricultural crop production have triggered a widespread support for weather derivatives as a means of mitigating the risk associated with climate change on agriculture. However, these products are faced with basis risk as a result of poor design and modeling of the underlying weather variable (temperature). In other to circumvent this problem, a novel time-varying mean-reversion Lévy regime-switching model is used to model the dynamics of the deseasonalized temperature dynamics. Using plots and test statistics, it is observed that the residuals of the deseasonalized temperature data are not normally distributed. To model the non-normality in the residuals, we propose to use the hyperbolic distribution to capture the semi-heavy tails and skewness in the empirical distributions of the residuals for the shifted regime. The proposed regime-switching model has a mean reverting heteroskedastic process in the base regime and a Lévy process in the shifted regime. By using the expectation-maximization algorithm, the parameters of the proposed model are estimated. The proposed model is flexible as it modelled the deseasonalized temperature data accurately. (Samuel Asante Gyamerah, Philip Ngare and Dennis Ikpe)

Poincare type inequalities via 1-dimensional Malliavin calculus

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We will review briefly 3 types of operators which are mapping spaces of real-valued functions which are defined on the real line equipped with standard normal probability measure. Those are the derivative, divergence and Ornstein-Uklenbeck operators. There are simple formulas that describe the relationships between those operators. Using those formulas the proofs of the following will be presented: 1. Poincare inequality : The variance of a function of N(0,1) is dominated by the second moment of its derivative. 2. An upper bound to the Wasserstein distance between the distribution of a function of N(0,1) (the function has mean 0 and standard deviation 1) and N(0,1) itself. This upper bound is (up to a constant) the multiplication of the L4 norm of the function derivative and the L4 norm of the function 2nd derivative. The material is based on Nourdin and Peccati book.

Variance swaps on time-changed Markov processes

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We prove that a variance swap has the same price as a co-terminal European-style contract, when the underlying is a Markov process, time-changed by a general continuous stochastic clock, which is allowed to have general correlation with the driving Markov process, which is allowed to have state-dependent jump distributions. The European contract’s payoff function satisfies an ordinary integro-differential equation, which depends only on the dynamics of the Markov process, not on the clock. In some examples, the payoff function that prices the variance swap can be computed explicitly. Joint work with Peter Carr and Matt Lorig.

Ordinal patterns in clusters of extremes of regularly varying time series

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The purpose is to investigate temporal clusters of extremes defined as subsequent exceedances of high thresholds in a stationary time series. Two meaningful features of these clusters are the probability distribution of the cluster size and the ordinal patterns within a cluster. The latter have been introduced in order to handle data sets with several thousand data points appearing in medicine, biology, finance and computer science. Since these patterns take only the ordinal structure of consecutive data points into account, the method is robust under monotone transformations and measurement errors. We verify the existence of the corresponding limit distributions in the framework of regularly varying time series, develop non-parametric estimators and show and their asymptotic normality under appropriate mixing conditions. (This is joint work with Marco Oesting.)

Localization of Gaussian disordered systems at low temperature

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The fundamental premise of statistical mechanics is that a physical system’s state is random according to some probability measure, which is determined by the various forces of interaction between the system’s constituent particles. In the “disordered” setting, these interactions are also random (meant to capture the effect of a random medium), meaning the probability measure is itself a random object. This setting includes several of the models most widely studied by mathematical physicists, such as the Random Energy Model, the Sherrington–Kirkpatrick spin glass, and directed polymers. The most intriguing part of their phase diagrams occurs at low temperature, when the measure concentrates, or “freezes”, on energetically favorable states. In general, quantifying this phenomenon is especially challenging, in large part due to the extra layer of randomness created by the disorder. This talk will describe recent progress on this question, leading us to some conjectures on further open problems. (Joint work with Sourav Chatterjee)

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