On discrete time hedging errors in a fractional Black-Scholes model

In this paper we investigate asymptotic behavior of error of a discrete time hedging strategy in a fractional Black-Scholes model in the sense of Wick-Ito-Skorohod integration.The rate of convergence of the hedging error due to discrete-time trading when the true strategy is known for the trader,is investigated.The result provides new statistical tools to study and detect the effect of the long-memory and the Hurst parameter for the error of discrete time hedging.     

Two extensions for fitting discrete time term structure models with normally distributed factors

This paper provides extensions to procedures for the implementation of two well‐known term structure models. In the first part, a misleading implication given in two textbooks concerning the ability to fit a Ho–Lee type term structure tree through trial and error is corrected, and it is shown that the tree can be fitted precisely with a simple and easily programmable formula. In the second part, a previously published result that obtains the drift for a single‐factor discrete time Heath–Jarrow–Morton model is extended to a multi‐factor world. In both cases numerical examples are provided.     

Variance optimal hedging for continuous time additive processes and applications

For a large class of vanilla contingent claims, we establish an explicit Föllmer–Schweizer decomposition when the underlying is an exponential of an additive process. This allows to provide an efficient algorithm for solving the mean variance hedging problem. Applications to models derived from the electricity market are performed.     

Risk model with fuzzy random individual claim amount

In this paper, we consider a risk model in which individual claim amount is assumed to be a fuzzy random variable and the claim number process is characterized as a Poisson process. The mean chance of the ultimate ruin is researched. Particularly, the expressions of the mean chance of the ultimate ruin are obtained for zero initial surplus and arbitrary initial surplus if individual claim amount is an exponentially distributed fuzzy random variable. The results obtained in this paper coincide with those in stochastic case when the fuzzy random variables degenerate to random variables. Finally, two numerical examples are presented.     

An exact minimum variance filter for a class of discrete time systems with random parameter perturbations

An exact, closed-form minimum variance filter is designed for a class of discrete time uncertain systems which allows for both multiplicative and additive noise sources. The multiplicative noise model includes a popular class of models (Cox-Ingersoll-Ross type models) in econometrics. The parameters of the system under consideration which describe the state transition are assumed to be subject to stochastic uncertainties. The problem addressed is the design of a filter that minimizes the trace of the estimation error variance. Sensitivity of the new filter to the size of parameter uncertainty, in terms of the variance of parameter perturbations, is also considered. We refer to the new filter as the ‘perturbed Kalman filter’ (PKF) since it reduces to the traditional (or unperturbed) Kalman filter as the size of stochastic perturbation approaches zero. We also consider a related approximate filtering heuristic for univariate time series and we refer to filter based on this heuristic as approximate perturbed Kalman filter (APKF). We test the performance of our new filters on three simulated numerical examples and compare the results with unperturbed Kalman filter that ignores the uncertainty in the transition equation. Through numerical examples, PKF and APKF are shown to outperform the traditional (or unperturbed) Kalman filter in terms of the size of the estimation error when stochastic uncertainties are present, even when the size of stochastic uncertainty is inaccurately identified.     

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.     

A generalized linear model with smoothing effects for claims reserving

In this paper, we continue the development of the ideas introduced in England and Verrall (2001) by suggesting the use of a reparameterized version of the generalized linear model (GLM) which is frequently used in stochastic claims reserving. This model enables us to smooth the origin, development and calendar year parameters in a similar way as is often done in practice, but still keep the GLM structure. Specifically, we use this model structure in order to obtain reserve estimates and to systemize the model selection procedure that arises in the smoothing process. Moreover, we provide a bootstrap procedure to achieve a full predictive distribution.     

Asymptotic estimates of Gerber-Shiu functions in the renewal risk model with exponential claims

This paper continues to study the asymptotic behavior of Gerber-Shiu expected discounted penalty functions in the renewal risk model as the initial capital becomes large. Under the assumption that the claim-size distribution is exponential, we establish an explicit asymptotic formula. Some straightforward consequences of this formula match existing results in the field.     

Equilibrium investment strategy for DC pension plan with default risk and return of premiums clauses under CEV model

This paper considers an optimal investment problem for a defined contribution (DC) pension plan with default risk in a mean–variance framework. In the DC plan, contributions are supposed to be a predetermined amount of money as premiums and the pension funds are allowed to be invested in a financial market which consists of a risk-free asset, a defaultable bond and a risky asset satisfied a constant elasticity of variance (CEV) model. Notice that a part of pension members could die during the accumulation phase, and their premiums should be withdrawn. Thus, we consider the return of premiums clauses by an actuarial method and assume that the surviving members will share the difference between the return and the accumulation equally. Taking account of the pension fund size and the volatility of the accumulation, a mean–variance criterion as the investment objective for the DC plan can be formulated, and the original optimization problem can be decomposed into two sub-problems: a post-default case and a pre-default case. By applying a game theoretic framework, the equilibrium investment strategies and the corresponding equilibrium value functions can be obtained explicitly. Economic interpretations are given in the numerical simulation, which is presented to illustrate our results.     

No-arbitrage with multiple-priors in discrete time

In a discrete time and multiple-priors setting, we propose a new characterisation of the condition of quasi-sure no-arbitrage which has become a standard assumption. We show that it is equivalent to the existence of a subclass of priors having the same polar sets as the initial class and such that the uni-prior no-arbitrage holds true for all priors in this subset. This characterisation shows that it is indeed a well-chosen condition being equivalent to several previously used alternative notions of no-arbitrage and allowing the proof of important results in mathematical finance. We also revisit the geometric and quantitative no-arbitrage conditions and explicit two important examples where all these concepts are illustrated.     

Ruin problems for a discrete time risk model with random interest rate

In this paper, we study a discrete time risk model with random interest rate. The convergence of the discounted surplus process is proved by using martingale techniques, an expression of ruin probability is obtained, and bounds for ruin probability are included. In the second part of the paper, the distribution of surplus immediately after ruin, the distribution of surplus just before ruin, the joint distribution of the surplus immediately before and after ruin, and the distribution of ruin time are discussed.     

An exact formula for the mean number of claims served by a one-channel queuing system with rejections on a given time interval

An exact formula for the mean number of claims served by a one-channel queuing system (QS) with rejections on a given time interval is obtained, which makes it possible, in particular, to calculate exactly the QS efficiency indexes on an arbitrary time interval.     

Asymptotics of discounted aggregate claims for renewal risk model with risky investment

Under the assumption that the claim size is subexponentially distributed and the insurance surplus is totally invested in risky asset, a simple asymptotic relation of tail probability of discounted aggregate claims for renewal risk model within finite horizon is obtained. The result extends the corresponding conclusions of related references.     

A first–passage time random walk distribution with five transition probabilities: a generalization of the shifted inverse trinomial

In this paper a univariate discrete distribution, denoted by GIT, is proposed as a generalization of the shifted inverse trinomial distribution, and is formulated as a first-passage time distribution of a modified random walk on the half-plane with five transition probabilities. In contrast, the inverse trinomial arises as a random walk on the real line with three transition probabilities. The probability mass function (pmf) is expressible in terms of the Gauss hypergeometric function and this offers computational advantage due to its recurrence formula. The descending factorial moment is also obtained. The GIT contains twenty-two possible distributions in total. Special cases include the binomial, negative binomial, shifted negative binomial, shifted inverse binomial or, equivalently, lost-games, and shifted inverse trinomial distributions. A subclass GIT_3,1 is a particular member of Kemp’s class of convolution of pseudo-binomial variables and its properties such as reproductivity, formulation, pmf, moments, index of dispersion, and approximations are studied in detail. Compound or generalized (stopped sum) distributions provide inflated models. The inflated GIT_3,1 extends Minkova’s inflated-parameter binomial and negative binomial. A bivariate model which has the GIT as a marginal distribution is also proposed.     

Bachelier model with stopping time and its insurance application

A modification of a classical Bachelier model by letting a stock price absorb at zero is revisited. Alternative proofs are given to derive option pricing formulas under the modified Bachelier model and numerical comparison with the Black–Scholes formula is provided. Quantile hedging methodology is developed for both classical and modified Bachelier models and application to pricing the pure endowment with fixed guarantee life insurance contracts is demonstrated, both theoretically and by means of a numerical example.     

Evaluation and default time for companies with uncertain cash flows

In this study, we propose a modelling framework for evaluating companies financed by random liabilities, such as insurance companies or commercial banks. In this approach, earnings and costs are driven by double exponential jump–diffusion processes and bankruptcy is declared when the income falls below a default threshold, which is proportional to the charges. A change of numeraire, under the Esscher risk neutral measure, is used to reduce the dimension. A closed form expression for the value of equity is obtained in terms of the expected present value operators, with and without disinvestment delay. In both cases, we determine the default threshold that maximizes the shareholder’s equity. Subsequently, the probabilities of default are obtained by inverting the Laplace transform of the bankruptcy time. In numerical applications of the proposed model, we apply a procedure for calibration based on market and accounting data to explain the behaviour of shares for two real-world examples of insurance companies.     

Asymptotic finite-time ruin probability for a bidimensional renewal risk model with constant interest force and dependent subexponential claims

This paper considers a bidimensional renewal risk model with constant interest force and dependent subexponential claims. Under the assumption that the claim size vectors form a sequence of independent and identically distributed random vectors following a common bivariate Farlie–Gumbel–Morgenstern distribution, we derive for the finite-time ruin probability an explicit asymptotic formula.     

Existence and Multiplicity for Elliptic Problems with Quadratic Growth in the Gradient

We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.     

随机效应模型中方差分量的经验Bayes检验问题

给出了双向分类随机效应模型中方差分量的Bayes检验的判决函数,利用核估计的方法,构造了相应的经验Bayes(EB)检验的判决函数．在适当的条件下证明了EB判决函数是渐近最优的且有收敛速度．给出了模型的特例和推广．最后,举出一个满足定理条件的例子．