On large deviations for the parabolic Anderson model

The focus of this article is on the different behavior of large deviations of random functionals associated with the parabolic Anderson model above the mean versus large deviations below the mean. The functionals we treat are the solution u(x, t) to the spatially discrete parabolic Anderson model and a functional A _n which is used in analyzing the a.s. Lyapunov exponent for u(x, t). Both satisfy a “law of large numbers”, with ${\lim_{t\to \infty} \frac{1}{t} \log u(x,t)=\lambda (\kappa)}$ and ${\lim_{n\to \infty} \frac{A_n}{n}=\alpha}$ . We then think of αn and λ(κ)t as being the mean of the respective quantities A _n and log u(t, x). Typically, the large deviations for such functionals exhibits a strong asymmetry; large deviations above the mean take on a different order of magnitude from large deviations below the mean. We develop robust techniques to quantify and explain the differences.     

On the small time large deviations of diffusion processes on configuration spaces

In this paper, we establish a small time large deviation principle for diffusion processes on configuration spaces.     

Extended large deviations

LetX ₁,X ₂,..., be a sequence ofi.i.d. random variables with a moment generating function finite in a neighborhood of 0. Further, for each integern1, letS _n denote the sum of the firstn terms in this sequence. We study the extended large deviation of such sums, meaning,P{S _n >n _n }, where _n is any sequence converging to infinity. We also derive functional extended large deviation theorems and then apply them to obtain functional versions of the Erdös-Rényi strong law of large numbers.Research partially supported by an NSF Grant.     

Precise large deviations for a customer-based individual risk model

In this paper,we propose a customer-based individual risk model,in which potential claims by customers are described as i.i.d.heavy-tailed random variables,but different insurance policy holders are allowed to have different probabilities to make actual claims.Some precise large deviation results for the prospective-loss process are derived under certain mild assumptions,with emphasis on the case of heavy-tailed distribution function class ERV(extended regular variation).Lundberg type limiting results on the finite time ruin probabilities are also investigated.     

Nonconventional large deviations theorems

We obtain large deviations theorems for both discrete time expressions of the form $\sum _{n=1}^NF\big (X(q_1(n)),\ldots ,X(q_\ell (n))\big )$ and similar expressions of the form $\int _0^TF\big ( X(q_1(t)),\ldots , X(q_\ell (t))\big )dt$ in continuous time. Here $X(n),n\ge 0$ or $X(t), t\ge 0$ is a Markov process satisfying Doeblin’s condition, $F$ is a bounded continuous function and $q_i(n)=in$ for $i\le k$ while for $i>k$ they are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when $q_i$ ’s are polynomials of increasing degrees. Applications to some types of dynamical systems such as mixing subshifts of finite type and hyperbolic and expanding transformations will be obtained, as well.     

Precise large deviations for consistently varying-tailed distributions in the compound renewal risk model

We investigate precise large deviations for heavy-tailed random sums. We prove a general asymptotic relation in the compound renewal risk model for consistently varying-tailed distributions. This model was introduced in [Q. Tang, C. Su, T. Jiang, and J.S. Zang, Large deviation for heavy-tailed random sums in compound renewal model, Stat. Probab. Lett., 52:91–100, 2001] as a more practical risk model. The proof is based on the inequality found in [D. Fuk and S.V. Nagaev, Probability for sums of independent random variables, Theory Probab. Appl., 16:600–675, 1971].     

等价类空间中离散时间选举一排它混合模型的遍历性

利用李雅谱诺夫函数首先证明了等价类空间中离散时间非紧邻选举模型是正常返的,且首次击中D0时刻的阶为15/14,其次给出了等价类空间中离散时间非紧邻选举模型与排它过程的混合模型遍历性的一个判别准则,从而推广和改进了紧邻情形的相应结果.     

带干扰的双险种风险模型下的大偏差问题

考虑了一类带干扰的双险种风险模型,得到了索陪额分布属于εRV族时,索赔盈利过程的两个偏差结果.     

Poisson approximation for large deviations

Upper and lower bounds are given for P(S ≤ k), 0 ≤ k ≤ ES, where S is a sum of indicator variables with a special structure, which appears, for example, in subgraph counts in random graphs. in typical cases, these bounds are close to the corresponding probabilities for a Poisson distribution with the same mean as S. There are no corresponding general bounds for P(S ≥ k), k > ES, but some partial results are given.     

Asymptotic arbitrage and large deviations

Typical models of mathematical finance admit equivalent martingale measures up to any finite time horizon but not globally, and this means that arbitrage opportunities arise in the long run. In this paper, we derive explicit estimates for asymptotic arbitrage, and we show how they are related to large deviation estimates for the market price of risk. As a case study we consider a geometric Ornstein–Uhlenbeck process. In this setting we also compute the optimal trading strategies and the resulting optimal growth rates of expected utility for all HARA utilities.     

等价类空间中离散时间选举-排它混合模型的遍历性

利用李雅谱诺夫函数首先证明了等价类空间中离散时间非紧邻选举模型是正常返的,且首次击中D0时刻的阶为15/14,其次给出了等价类空间中离散时间非紧邻选举模型与排它过程的混合模型遍历性的一个判别准则,从而推广和改进了紧邻情形的相应结果.     

Probabilities of large deviations for L-statistics

Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR, Vilnius University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 30, No. 3, pp. 479–488, July–September, 1990.     

Probabilities of large deviations for martingales

Vilnius University. Published in Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 30, No. 4, pp. 784–795, October–December, 1990.     

Sample path large and moderate deviations for risk model with delayed claims

Sample path large and moderate deviation principles for Markov modulated risk models with delayed claims are proved by the exponential martingale method. As applications, asymptotic estimates and exponential bounds of the ruin probability are also studied.     

关于大偏差概率的一个界

研究得到了关于随机和S(t)=∑N(t)i=1Xi,t≥0大偏差的幂的一个界,其中(N(t))t≥0是一族非负整值随机变量,(Xn)n∈N是独立同分布的随机变量,其共同的分布函数是F与(N(t))t≥0独立.本结论是在假设分布函数F的右尾属于ERV族的情况下得到的.