Methodology and Computing in Applied Probability最新文献

We give an upper bound on the conditional error of Quadratic Discriminant Analysis (QDA), conditioned on parameter estimates. In the case of maximum likelihood estimation (MLE), our bound recovers the well-known Chernoff and Bhattacharyya bounds in the infinite sample limit. We perform an empirical assessment of the behaviour of our bound in a finite sample MLE setting, demonstrating good agreement with the out-of-sample error, in contrast with the simpler but uninformative estimated error, which exhibits unnatural behaviour with respect to the sample size. Furthermore, our conditional error bound is applicable whenever the QDA decision function employs parameter estimates that differ from the true parameters, including regularised QDA.

We present a new approach to the problem of characterizing and choosing equivalent martingale pricing measures for a contingent claim, in a finite-state incomplete market. This is the entropy segmentation method achieved by means of convex programming, thanks to which we divide the claim no-arbitrage prices interval into two halves, the buyer’s and the seller’s prices at successive entropy levels. Classical buyer’s and seller’s prices arise when the entropy level approaches 0. Next, we apply Fenchel duality to these primal programs to characterize the hedging positions, unifying in the same expression the cases of super (resp. sub) replication (arising when the entropy approaches 0) and partial replication (when entropy tends to its maximal value). We finally apply linear programming to our hedging problem to find in a price slice of the dual feasible set an optimal partial replicating portfolio with minimal CVaR. We apply our methodology to a cliquet style guarantee, using Heston’s dynamic with parameters calibrated on EUROSTOXX50 index quoted prices of European calls. This way prices and hedging positions take into account the volatility risk.

This paper considers a risk sharing scheme of independent discrete losses that combines risk retention at individual level, risk transfer for too expensive losses and risk pooling for the middle layer. This ensures that pooled losses can be considered as being uniformly bounded. We study the no-sabotage requirement and diversification effects when the conditional mean risk-sharing rule is applied to allocate pooled losses. The no-sabotage requirement is equivalent to Efron’s monotonicity property for conditional expectations, which is known to hold under log-concavity. Elementary proofs of this result for discrete losses are provided for finite population pools. The no-sabotage requirement and diversification effects are then examined within large pools. It is shown that Efron’s monotonicity property holds asymptotically and that risk can be eliminated under fairly general conditions which are fulfilled in applications.

This paper considers a multidimensional risk model with cádlág investment return processes, in which there exists some dependence structure among claims and claim-arrival time. Specifically, if claims follow the subexponential distribution or the regular variation distribution, we obtain some precise asymptotic estimates for the finite-time ruin probabilities. In addition, some numerical simulations are presented to test the performance of the theoretical results.

The self-adjoint, positive Markov operator defined by the Pólya-Gamma Gibbs sampler (under a proper normal prior) is shown to be trace-class, which implies that all non-zero elements of its spectrum are eigenvalues. Consequently, the spectral gap is (1-lambda _*), where (lambda _* in [0,1)) is the second largest eigenvalue. A method of constructing an asymptotically valid confidence interval for an upper bound on (lambda _*) is developed by adapting the classical Monte Carlo technique of Qin et al. (Electron J Stat 13:1790–1812, 2019) to the Pólya-Gamma Gibbs sampler. The results are illustrated using the German credit data. It is also shown that, in general, uniform ergodicity does not imply the trace-class property, nor does the trace-class property imply uniform ergodicity.

In this paper, we study the optimal investment and proportional reinsurance problem for an insurer with short-selling and borrowing constraints under the expected value premium principle. The claim process follows a Brownian risk model with a drift. The insurer’s surplus is allowed to invest in one risk-free asset and one risky asset. By using the dynamic programming approach and solving the corresponding boundary-value problems, the optimization objective of maximizing the probability of drawup before drowdown is considered initially. The optimal strategy and the corresponding value function are derived through solving the Hamilton-Jacobi-Bellman (HJB) equation. Moreover, numerical examples are performed to illustrate the effects of model parameters on the optimal strategy. In addition, we verify the optimality of the strategies obtained from the dynamic programming principle by Euler method.

The theory of linear recurrence sequences is applied to obtain an explicit formula for the ultimate ruin probability in a discrete-time risk process. It is assumed that the claims distribution is arbitrary but has finite support (varvec{{0,1,ldots ,m+1}}), for some integer (varvec{mge 1}). The method requires finding the zeroes of an m degree polynomial and solving a system of m linear equations. An approximation is derived and some numerical results and plots are provided as examples.

In the present we provide the definition of the new concept of the General Non-Homogeneous Markov System (G-NHMS) and establish the expected population structure of a NHMS in the various states. These results will be the basis to build on the new concepts and the basic theorems of what follows.We then establish the set of all possible expected relative distributions of the initial number of memberships at time t and all possible expected relative distributions of the expansion memberships at time t. We call this set the general expected relative population structure in the states of a G-NHMS. We then proceed by providing the new definitions of weak ergodicity in a G-NHMS and weak ergodicity with a geometrical rate of convergence. We then prove the Theorem 4 which is a new building block in the theory of G-NHMS. We also prove a similar theorem under the assumption that relative expansion of the population vanishes at infinity.We then provide a generalization of the coupling theorem for populations. We proceed then to study the asymptotic behavior of a G-NHMS when the input policy consists of independent non-homogeneous Poisson variates for each time interval (left( t-1,tright] ). It is founded in Theorem 7 that it displays a kind of weak ergodicity behavior, that is, it converges at each step to the row of a stable matrix. This row is independent of the initial distribution and of the asymptotic input policy unlike the results in previous works. Hence it generalizes the result in that works. Finally we illustrate our results numerically for a manpower system with three states.

This paper investigates the asymptotic analysis of the hitting time of Markov-type jump processes (i.e., semi-Markov, Markov, in continuous or discrete time) with a small probability of entering a non-empty terminal subset. This means that absorption is a rare event. The mean hitting time function of all four type processes obeyed the same equation. We obtain unified results of asymptotic approximation in a series scheme or, equivalently, a functional type of mean hitting time.

In this paper, we establish the complete f-moment convergence and the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of extended negatively dependent (END, for short) random variables under sub-linear expectations, which extend and improve corresponding ones in sub-linear expectation space. As applications of the main results, the complete consistency and strong consistency of weighted estimators in nonparametric regression models under sub-linear expectations are also obtained.