The optimal strategy for insurance company under the influence of terminal value

This paper considers a model of an insurance company which is allowed to invest a risky asset and to purchase proportional reinsurance. The objective is to find the policy which maximizes the expected total discounted dividend pay-out until the time of bankruptcy and the terminal value of the company under liquidity constraint. We find the solution of this problem via solving the problem with zero terminal value. We also analyze the influence of terminal value on the optimal policy.     

UNIQUENESS OF THE MILD SOLUTION OF SEMILINEAR STOCHASTIC EVOLUTION EQUATION IN HILBERT SPACE

In this paper, we will consider following initial value problem of semilinear stochastic evolution equation in Hilbert Space: where W(t) is a wiener process in H, H and Y are two real separable Hilbert Spaces, A is an infinitesimal generator of a strongly continuous semigroup s(t) on Y,f(t, y): [0, T]×Y→Y, and G(t, y): [0, T]×Y→L(H, Y), y_0: Ω→Y is a ramdom variable of square integrable. We apply theory of the semigroup and obtain two conclusions of uniqueness of the mild solution of (1) which include the corresponding results in [4].     

Dynamic optimal strategy for monitoring disease recurrence

Surveillance to detect cancer recurrence is an important part of care for cancer survivors.In this paper we discuss the design of optimal strategies for early detection of disease recurrence based on each patient’s distinct biomarker trajectory and periodically updated risk estimated in the setting of a prospective cohort study.We adopt a latent class joint model which considers a longitudinal biomarker process and an event process jointly,to address heterogeneity of patients and disease,to discover distinct biomarker trajectory patterns,to classify patients into different risk groups,and to predict the risk of disease recurrence.The model is used to develop a monitoring strategy that dynamically modifies the monitoring intervals according to patients’ current risk derived from periodically updated biomarker measurements and other indicators of disease spread.The optimal biomarker assessment time is derived using a utility function.We develop an algorithm to apply the proposed strategy to monitoring of new patients after initial treatment.We illustrate the models and the derivation of the optimal strategy using simulated data from monitoring prostate cancer recurrence over a 5-year period.     

FORMATION OF NECROTIC CORES IN THE GROWTH OF TUMORS： ANALYTIC RESULTS

In this article, the author studies the mechanism of formation of necrotic cores in the growth of tumors by using rigorous analysis of a mathematical model. The model modifies a corresponding tumor growth model proposed by Byrne and Chaplain in 1996, in the case where no inhibitors exist. The modification is made such that both necrotic tumors and nonnecrotic tumors can be considered in a joint way. It is proved that if the nutrient supply is below a threshold value, then there is not dormant tumor, and all evolutionary tumors will finally vanish. If instead the nutrient supply is above this threshold value then there is a unique dormant tumor which can either be necrotic or nonnecrotic, depending on the level of the nutrient supply and the level of dead-cell dissolution rate, and all evolutionary tumors will converge to this dormant tumor. It is also proved that, in the second case, if the dormant tumor is necrotic then an evolutionary tumor will form a necrotic core at a finite time, and if the dormant tumor is nonnecrotic then an evolutionary tumor will also be nonnecrotic from a finite time.     

Age-dependent branching processes in random environments

We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ0,ξ1,...) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξn) on R , and reproduce independently new particles according to a probability law p(ξn) on N. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean EξZ(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.     

On pricing of corporate securities in the case of jump-diffusion

Structural models of credit risk are known to present vanishing spreads at very short maturitiesThis shortcoming, which is due to the diffusive behavior assumed for asset values, can be circumvented by considering discontinuities of the jump type in their evolution over timeIn this paper, we extend the pricing model for corporate bond and determine the default probability in jump-diffusion model to address this issueTo make the problem clearly,we first investigate the case that the firm value follows a geometric Brownian motion under similar assumptions to those in Black and Scholes(1973), Briys and de Varenne(1997), i.e, the default barrier is KD(t, T) and the recovery rate is(1- ω), where D(t, T) is the price of zero coupon default free bond and ω is a constant(0 ω≤ 1)By changing the numeraire, we obtain the closed-form solution for both the price of bond and default probabilityFurther, we consider the case of jump-diffusion and suppose that a firm will go bankruptcy if its value Vt ≤ KD(t, T)and at the same time, the bondholder will receive(1- ω)Vt KBy introducing the Green function of PDE with absorbing boundary and converting the problem to an II-type Volterra integral equation, we get the closed-form expressions in series form for bond price and corresponding default probabilityNumerical results are presented to show the impact of different parameters to credit spread of bond.     

A 2.79 COMPETITIVE ON-LINE ALGORITHM FOR TWO PROCESSOR REAL-TIME SYSTEMS WITH UNIFORM VALUE DENSITY

We consider the problem of competitive on-liae scheduling in two processor real-timesystems. In our model all tasks have common value density. Each task has a release time. an exe-cution time and a deadline. The scheduler is given no information about a task until it is released.And the value will be achieved if and only if the task is completed by its deadlirte. Moreover. wesuppose that migratbn is not allowed. The goal of the scheduler is to obtain as much value as pos-sible. In this paper we show that the competitive multipber of Safe-Risky-fixed Algorithm irt [4]is really 3 and presents a modified algorithm (Safe-Risky-unfixed Algorithm) that achieves a com-petitive mukiplier of 2. 79.     

Asymptotical Behaviors for Neumann Boundary Problem with Singular Data

In this paper, we will analyze the blow-up behaviors for solutions to the Laplacian equation with exponential Neumann boundary condition. In particular, the boundary value is with a kind of singular data. We show a Brezis–Merle type concentration-compactness theorem, calculate the blow up value at the blow-up point, and give a point-wise estimate for the profile of the solution sequence at the blow-up point.     

A numerical algorithm for determination of a control parameter in two-dimensional parabolic inverse problems

Numerical solution of the parabolic partial differential equations with an unknown parameter play a very important role in engineering applications. In this study we present a high order scheme for determining unknown control parameter and unknown solution of two-dimensional parabolic inverse problem with overspecialization at a point in the spatial domain. In this approach, a compact fourth-order scheme is used to discretize spatial derivatives of equation and reduces the problem to a system of ordinary differential equations(ODEs).Then we apply a fourth order boundary value method to the solution of resulting system of ODEs. So the proposed method has fourth order of accuracy in both space and time components and is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are presented and some comparisons are made with several well-known finite difference schemes in the literature.Also we will investigate the effect of noise in data on the approximate solutions.     

Optimal stopping times for solutions of nonlinear stochastic differential equations and their application to one problem of financial mathematics

We solve the problem of finding the optimal switching time for two alternative strategies at the financial market in the case where a random processX _t ,t ∈ [0, T], describing an investor's assets satisfies a nonlinear stochastic differential equation. We determine this switching time τ∈[0,T] as the optimal stopping time for a certain processY _t generated by the processX _t so that the average investor's assets are maximized at the final time, i.e.,EX _T . Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 804–809, June, 1999.     

Asymptotic Behavior of the Maximum of Sums of I.I.D. Random Variables along Monotone Blocks

Let {X_i, Y_i}_i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y₁ = y) = 0 for all y. Put M_n(j) = max_0≤k≤n-j (X_k+1 + ... X_k+j)I_k,j, where I_k,k+j = I{Y_k+1 < ⋯ < Y_k+j} denotes the indicator function for the event in brackets, 1 ≤ j ≤ n. Let L_n be the largest index l ≤ n for which I_k,k+l = 1 for some k = 0, 1, ..., n - l. The strong law of large numbers for “the maximal gain over the longest increasing runs,” i.e., for M_n(L_n) has been recently derived for the case where X₁ has a finite moment of order 3 + ε, ε > 0. Assuming that X₁ has a finite mean, we prove for any a = 0, 1, ..., that the s.l.l.n. for M(L_n - a) is equivalent to EX ₁ ^3+a I{X₁ > 0} < ∞. We derive also some new results for the a.s. asymptotics of L_n. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 179–189.     

Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence

We establish an estimate for the rate of convergence of a solution of an ordinary stochastic differential equation of order p ≥ 2 with a small parameter in the coefficient of the leading derivative to a solution of a stochastic equation of order p − 1 in the metric ρ(X, Y) = (sup_0≤t≤T M|X(t) − Y(t)|²)^1/2 __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1587–1601, December, 2006.     

On the Countable Tightness of Product Spaces

In this paper, we discuss the countable tightness of products of spaces which are quotient simages of locally separable metric spaces, or k-spaces with a star-countable k-network. The main result is that the following conditions are equivalent： （1） b = ω1; （2） t（Sω×Sω1） 〉 ω; （3） For any pair （X, Y）, which are k-spaces with a point-countable k-network consisting of cosmic subspaces, t（X×Y）≤ω if and only if one of X, Y is first countable or both X, Y are locally cosmic spaces. Many results on the k-space property of products of spaces with certain k-networks could be deduced from the above theorem.     

Filtration of components of processes of random evolution

The problem of estimation of a nonobservable component θ_t for a two-dimensional process (θ_t, ξ_t) of random evolution (θ _t,ξ_t);x_t, 0≤t≤T, is investigated on the basis of observations of ξ_s. s≤t, where x _t is a homogeneous Markov process with infinitesimal operator Q. Applications to stochastic models of a (B,S)-market of securities is described under conditions of incomplete market. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1701–1705, December, 1998.     

Asymptotic expansions in limit theorems for stochastic processes. II

. For a certain class of families of stochastic processes η^ε(t), 0≤t≤T, constructed starting from sums of independent random variables, limit theorems for expectations of functionals F(η^ε[0,T]) are proved of the form

The optimal stopping problem concerned with ultimate maximum of a Lévy process

For a Lévy process X = (X _t )₀≤t<∞ we consider the time θ = inf{t ≥ 0: sup _s≤t X _s = sup _s≥0 X _s }. We study an optimal approximation of the time θ using the information available at the current instant. A Lévy process being a combination of a Brownian motion with a drift and a Poisson process is considered as an example.     

Convergence of distributions of integral functionals of extremal random functions

We study the convergence of distributions of integral functionals of random processes of the formU _n (t)=b _n (Z _n (t)-a _n G(t)),t⃛T, where {X=X(t), t⃛T} is a random process,X _n ,n≥1, are independent copies ofX, andZ _n (t)=max_1≤k≤n X _k (t). Ukrainian State Academy of Light Industry, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1201–1209, September, 1999.     

Nonautonomous Stochastic Search in Global Optimization

We present a general method how to prove convergence of a sequence of random variables generated by a nonautonomous scheme of the form X _t =T _t (X _t−1,Y _t ), where Y _t represents randomness, used as an approximation of the set of solutions of the global optimization problem with a continuous cost function. We show some of its applications.     

Extensions of the measurable choice theorem by means of forcing

Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions: LetT be the closed unit interval [0,1] and letm be the usual Lebesgue measure defined on the Borel subsets ofT. Theorem1. LetS⊂T×T be a Borel set such that for alltεT,S _t ^def={x|(t,x)εS} is countable and non-empty. Then there exists a countable series of Lebesgue-measurable functionsf _n: T→T such thatS _t={f_n(t)|nεω} for alltε[0,1],W _x={y|(x,y)εW} is uncountable. Then there exists a functionh:[0,1]×[0,1]→W with the following properties: (a) for each xε[0,1], the functionh(x,·) is one-one and ontoW _x and is Borel measurable; (b) for eachy, h(·, y) is Lebesgue measurable; (c) the functionh is Lebesgue measurable.