Weak Convergence Of Stochastic Processes Research Articles

Diffusion Approximations of the Ruin Probability for the Insurer–Reinsurer Model Driven by a Renewal Process

We introduce here a diffusion-type approximation of the ruin probability both in finite and infinite time for a two-dimensional risk process, where claims and premiums are shared with a predetermined proportion. This type of process is often called the insurer–reinsurer model. We assume that the flow of claims is governed by a general renewal process. A simple ruin probability formula for the model is known only in infinite time for the special case of the Poisson process and exponentially distributed claims. Therefore, there is a need for simple analytical approximations. In the literature, in the infinite-time case, for the Poisson process, a De Vylder-type approximation has already been introduced. The idea of the diffusion approximation presented here is based on the weak convergence of stochastic processes, which enables one to replace the original risk process with a Brownian motion with drift. By applying this idea to the insurer–reinsurer model, we obtain simple ruin probability approximations for both finite and infinite time. We check the usefulness of the approximations by studying several claim amount distributions and comparing the results with the De Vylder-type approximation and Monte Carlo simulations. All the results show that the proposed approximations are promising and often yield small relative errors.

Extended weak convergence and utility maximisation with proportional transaction costs

In this paper, we study utility maximisation with proportional transaction costs. Assuming extended weak convergence of the underlying processes, we prove the convergence of the time-0 values of the corresponding utility maximisation problems. Moreover, we establish a limit theorem for the optimal trading strategies. The proofs are based on the extended weak convergence theory developed in Aldous (Weak Convergence of Stochastic Processes for Processes Viewed in the Strasbourg Manner, 1981) and on the Meyer–Zheng topology introduced in Meyer and Zheng (Ann. Inst. Henri Poincare Probab. Stat. 20:353–372, 1984).

Weak convergence of stochastic processes from spaces $F_\psi(\Omega)$

This paper is devoted to the investigation of conditions for the eak convergence in the space $C(T)$ of the stochastic processes from the space $\mathbf{F}_\psi(\Omega)$. Using this conditions the limit theorem for stochastic processes from the space $\mathbf{F}_\psi(\Omega)$ has been obtained. This theorem can be utilized for gaining the given approximation accuracy and reliability of integrals depending on parameter by Monte Carlo method.

A Generally Efficient Targeted Minimum Loss Based Estimator based on the Highly Adaptive Lasso

Suppose we observe n $n$ independent and identically distributed observations of a finite dimensional bounded random variable. This article is concerned with the construction of an efficient targeted minimum loss-based estimator (TMLE) of a pathwise differentiable target parameter of the data distribution based on a realistic statistical model. The only smoothness condition we will enforce on the statistical model is that the nuisance parameters of the data distribution that are needed to evaluate the canonical gradient of the pathwise derivative of the target parameter are multivariate real valued cadlag functions (right-continuous and left-hand limits, (G. Neuhaus. On weak convergence of stochastic processes with multidimensional time parameter. Ann Stat 1971;42:1285-1295.) and have a finite supremum and (sectional) variation norm. Each nuisance parameter is defined as a minimizer of the expectation of a loss function over over all functions it its parameter space. For each nuisance parameter, we propose a new minimum loss based estimator that minimizes the loss-specific empirical risk over the functions in its parameter space under the additional constraint that the variation norm of the function is bounded by a set constant. The constant is selected with cross-validation. We show such an MLE can be represented as the minimizer of the empirical risk over linear combinations of indicator basis functions under the constraint that the sum of the absolute value of the coefficients is bounded by the constant: i.e., the variation norm corresponds with this L1 $L_1$-norm of the vector of coefficients. We will refer to this estimator as the highly adaptive Lasso (HAL)-estimator. We prove that for all models the HAL-estimator converges to the true nuisance parameter value at a rate that is faster than n-1/4 $n^{-1/4}$ w.r.t. square-root of the loss-based dissimilarity. We also show that if this HAL-estimator is included in the library of an ensemble super-learner, then the super-learner will at minimal achieve the rate of convergence of the HAL, but, by previous results, it will actually be asymptotically equivalent with the oracle (i.e., in some sense best) estimator in the library. Subsequently, we establish that a one-step TMLE using such a super-learner as initial estimator for each of the nuisance parameters is asymptotically efficient at any data generating distribution in the model, under weak structural conditions on the target parameter mapping and model and a strong positivity assumption (e.g., the canonical gradient is uniformly bounded). We demonstrate our general theorem by constructing such a one-step TMLE of the average causal effect in a nonparametric model, and establishing that it is asymptotically efficient.

Hedging of game options with the presence of transaction costs

We study the problem of super-replication for game options under proportional transaction costs. We consider a multidimensional continuous time model, in which the discounted stock price process satisfies the conditional full support property. We show that the super-replication price is the cheapest cost of a trivial super-replication strategy. This result is an extension of previous papers (see [Statist. Decisions 27 (2009) 357–369] and [Ann. Appl. Probab. 18 (2008) 491–520]) which considered only European options. In these papers the authors showed that with the presence of proportional transaction costs the super-replication price of a European option is given in terms of the concave envelope of the payoff function. In the present work we prove that for game options the super-replication price is given by a game variant analog of the standard concave envelope term. The treatment of game options is more complicated and requires additional tools. We combine the theory of consistent price systems together with the theory of extended weak convergence which was developed in [Weak convergence of stochastic processes for processes viewed in the strasbourg manner (1981) Preprint]. The second theory is essential in dealing with hedging which involves stopping times, like in the case of game options.

Tail exponent estimation via broadband log density-quantile regression

Heavy tail probability distributions are important in many scientific disciplines such as hydrology, geology, and physics and therefore feature heavily in statistical practice. Rather than specifying a family of heavy-tailed distributions for a given application, it is more common to use a nonparametric approach, where the distributions are classified according to the tail behavior. Through the use of the logarithm of Parzen's density-quantile function, this work proposes a consistent, flexible estimator of the tail exponent. The approach we develop is based on a Fourier series estimator and allows for separate estimates of the left and right tail exponents. The theoretical properties for the tail exponent estimator are determined, and we also provide some results of independent interest that may be used to establish weak convergence of stochastic processes. We assess the practical performance of the method by exploring its finite sample properties in simulation studies. The overall performance is competitive with classical tail index estimators, and, in contrast, with these our method obtains somewhat better results in the case of lighter heavy-tailed distributions.

Reinforced weak convergence of stochastic processes

We consider a sequence of stochastic processes X n on C [ 0 , 1 ] converging weakly to X and call it polynomially convergent, if E F ( X n ) → E F ( X ) for continuous functionals F of polynomial growth. We present a sufficient moment conditions on X n for polynomial convergence and provide several examples, e.g. discrete excursions and depth first path associated to Galton–Watson trees. This concept leads to a new approach to moments of functionals of rooted trees such as height and path length.

Diffusion Approximation and Optimal Stochastic Control

In this paper a stochastic control model is studied that admits a diffusion approximation. In the prelimit model the disturbances are given by noise processes of various types: additive stationary noise, rapidly oscillating processes, and discontinuous processes with large intensity for jumps of small size. We show that a feedback control that satisfies a Lipschitz condition and is $\delta$-optimal for the limit model remains $\delta$-optimal also in the prelimit model. The method of proof uses the technique of weak convergence of stochastic processes. The result that is obtained extends a previous work by the authors, where the limit model is deterministic.

Weak convergence for the row sums of a triangular array of empirical processes under bracketing conditions

We study the weak convergence for the row sums of a triangular array of empirical processes under bracketing conditions involving majorizing measures. As an application, we consider the weak convergence of stochastic processes of the form a n −1 ∑ j=1 n f(X j,t) −c n(t):t∈T , n⩾1, where { X j } j=1 ∞ is a sequence of i.i.d.r.v.s with values in the measurable space (S, S) , f( · ,t):S→ R is a measurable function for each t∈ T, { a n } is an arbitrary sequence of real numbers and c n ( t) is a real number, for each t∈ T and each n⩾1. We also consider the weak convergence of processes of the form ∑ j=1 n f j(X j,t):t∈T , n⩾1, where { X j } j=1 ∞ is a sequence of independent r.v.s with values in the measurable space (S j, S j) , and f j( · ,t):S j→ R is a measurable function for each t∈ T. Instead of measuring the size of the brackets using the strong or weak L p norm, we use a distance inherent to the process. We present applications to the weak convergence of stochastic processes satisfying certain Lipschitz conditions.

Laplace and the origin of the Ornstein-Uhlenbeck process

It is shown that a second-order partial differential equation, found by Laplace in 1810, is the Fokker-Planck equation for a one-dimensional Ornstein-Uhlenbeck process. It is argued that Laplace's reasoning, although not rigorous, can be entirely justified by using the modern theory of weak convergence of stochastic processes. The solutions to the differential equation found by Laplace and others, using expansions in terms of Hermite polynomials, are discussed.

Some remarks on the uniform weak convergence of stochastic processes

It is known that the uniform weak convergence of a sequence of stochastic processes is equivalent to the finite dimensional convergence, plus an asymptotic equicontinuity condition with respect to any pseudometric which makes the parameter space totally bounded. Here, we see that in this characterization, it is possible to take an intrinsic metric of the process. This is applied to obtain necessary and sufficient conditions for the weak convergence of local U-processes.

Weak convergence of stochastic processes indexed by smooth functions

We give some easy methods to check sufficient conditions for the weak convergence of stochastic processes indexed by smooth functions. The main condition is a moment condition on the increments of the process. We apply this to empirical processes and U-processes in the independent identically distributed case. We also consider empirical processes under different types of dependence conditions.

Memorial: Nikolai Nikolaevitch Chentsov

Memorial: Nikolai Nikolaevitch Chentsov

A remark on the weak convergence of processes in the Skorohod topology

A Skorohod representation type theorem is proved for the weak convergence of stochastic processes in the Skorohod topology. This allows the “time changes” arising from the Skorohod topology to be considered as stochastic processes. While thenth time change processA t n is not adapted to thenth filtration (ℱ t n )t⩾0, it is possible to choose the processesA n such that they are adapted to , where , where γ n is a sequence of constants decreasing to 0 asn tends to ∞.

Inference for Near-Integrated Time Series with Infinite Variance

Abstract An autoregressive time series is said to be near-integrated (nearly nonstationary) if some of its characteristic roots are close to the unit circle. Statistical inference for the least squares estimators of near-integrated AR(1) models has been under rigorous study recently both in the statistics and econometric literatures. Although classical asymptotics are no longer available, through the study of weak convergence of stochastic processes, one can establish the asymptotic theories in terms of simple diffusion processes or Brownian motions. Such results rely heavily on the finiteness of the variance of the noise. When this finite variance condition fails, whereas many physical and economic phenomena are believed to be generated by an infinite variance noise sequence, the aforementioned asymptotics are not applicable. In this article, a unified theory concerning near-integrated autoregressive time series with infinite variance is developed. In particular, when the noise sequence {e t } belongs to...

Testing for the Constancy of Parameters over Time

Abstract Tests are proposed for detecting possible changes in parameters when the observations are obtained sequentially in time. While deriving the tests the alternative one has in mind specifies the parameter process as a martingale. The distribution theory of these tests relies on the large-sample results; that is, only the limiting null distributions are known (except in very special cases). The main tool in establishing these limiting distributions is weak convergence of stochastic processes. Suppose that we have vector-valued observations x 1, …, x n obtained sequentially in time (or ordered in some other linear fashion). Their joint distribution is described by determining the initial distribution for x 1 and the conditional distribution for each x k given the past up to x k–1. Suppose further that these distributions depend on a p-dimensional parameter vector θ. At least locally (i.e., in a short time period) this may be more or less legitimate. In the long run, however, the possibility of some ch...

On the weak convergence of stochastic processes with embedded point processes

We consider a stochastic process in continuous time and two point processes on the real line, all jointly stationary. We show that under a certain mixing condition the values of the process at the points of the second point process converge weakly under the Palm distribution with respect to the first point process, and we identify the limit. This result is a supplement to two other known results which are mentioned below.

On the weak convergence of stochastic processes with embedded point processes

We consider a stochastic process in continuous time and two point processes on the real line, all jointly stationary. We show that under a certain mixing condition the values of the process at the points of the second point process converge weakly under the Palm distribution with respect to the first point process, and we identify the limit. This result is a supplement to two other known results which are mentioned below.

On the weak convergence of alternating processes

Abstract The weak convergence of stochastic processes whose law of motion changes, is considered. The law of motion consists of two parts, the first depends on the processes' present value and the other is the input process. It is shown that if the input process is a compound Poisson process that converges weakly to Brownian motion with drift, then the alternating process converges to a diffusion process with similar law of motion.

Necessary and sufficient conditions of weak convergence of stochastic processes

In this case we shall say that an s.p. {~ (t), t ~ T} is defined in (x, ~). Let an extended sequence of s. p.'s {~a (t), t T}, c~ ~ A according to some directed set A and an s. p. { ~ (t),t ~ T} be defined in (x, ~). We denote the induced measures on (x, e), corresponding to the s. p.'s just mentioned by Pa, a ~ A, and ~, respectively. The weak convergence of the s.p. ~ ~ ~, c~ ~ A is determined as the weak convergence of the induced measures ~a =~ P, c~ ~ A, i.e., as the fulfillment of the following limiting relation: for all f ~ C O (~), where C O ($) is the set of bounded and continuous complex-valued functionais on ~. We denote by ~ the space of the probability measures on (~, e), provided with a topology of weak convergence. In applications of theorems on weak convergence ~ o~ ~ ~, ~ ~ A, it is highly convenient to verify the condition, expressed in terms of the convergence of finite-dimension al distributions of s. p.'s { ~ (t), t ~ T}, ~ A. Precisely speaking, we have, on account of the following two conditions, which may be sufficient or necessary and sufficient for the convergence ~c~ ~ ~, c~ ~ A : 1 ~ for an arbitrary finite set T, ~ T joint distributions {~c~(t), t ~ T'}, c~ ~A converge to the distribution { ~ (t), t ~ T'} ; 2~ from each partial sequence {p ~,}a,e A ~ {U ~}a ~ A it is possible to extract a sequence converging in