Abstract The valuation of equity swap under the condition of risk neutral is similar to the forward interest rate swap with the same period. Therefore, its valuation formula is consistent to interest rate swap model in the traditional methods. But it is not the same as in pricing the cross-currency equity swap. The dymanic prices of foreign stock index and exchange rate, and the correlation coefficients between exchange rates and foreign assets also affect the swap rate. In this paper, we extend Chance and Rich(1998)¡¦s valuation formula of equity swaps, and apply Amin (1991)¡BAmin and Bodurtha(1995)¡BLin(1997)¡¦s dymanic prices of assets in discrete time period. To derive the risk neutral valuation formula of equity swap, it uses the method of transfer probability measure. This study finds the expected return of foreign stock index in the no arbitrage condition, in addition equal to foreign forward interest rate with the same period , must be add a correction term to reflect the exchange rate risk and the transfer of forward martingale measure. This paper also derives the pricing formula of equity swaptions¡Bcaps¡Bfloors¡Bvariable notional principal and blended equity swap. Finally, we find the volatility of foreign forward interest rate is the most important factor of pricing the swap rate from numerical simulation. And if the correlation of the volatility of exchange rate and foreign stock index¡Bthe correlation of the volatility of exchange rate and foreign forward interest rate are negative, the swap rate will be higher.

This work is concerned with sampling and computation of rare events in molecular systems. In particular, we present new methods for sampling the canonical ensemble corresponding to the Boltzmann-Gibbs probability measure. We combine an equation for controlling the kinetic energy of the system with a random noise to derive a highly degenerate diffusion (i.e. a diffusion equation where diffusion happens only along one or few degrees of freedom of the system). Next the concept of hypoellipticity is used to show that the corresponding Fokker-Planck equation of the highly degenerate diffusion is well-posed, hence we prove that the solution of the highly degenerate diffusion is ergodic with respect to the Boltzmann-Gibbs measure. We find that the new method is more efficient for computation of dynamical averages such as autocorrelation functions than the commonly used Langevin dynamics, especially in systems with many degrees of freedom. Finally we study the computation of free energy using an adaptive method which is based on the adaptive biasing force technique.

Investment-linked insurance in Taiwan has been listed for almost a decade since 2001. In 2002, after the big sales of the investment-linked insurance, the domestic insurance companies also joined the market. For the investment-linked insurance, the policyholders retain the protection of the life insurance as well as share the earnings of the investment. Since the main investment instruments of the investment-linked insurance are mutual funds, it is important to study how to optimally allocate the portfolio. This research consider the returns of the mutual funds under tree models assumption. The objective is to find the optimal portfolio which has minimum variance and attained a given expected return level. The problem is also known as mean-variance portfolio problem. In the empirical work, we study eleven daily mutual fund price data from Sep. 2007 to Nov. 2008. Using the data of the first 12 months, we first establish initial tree price models, then update the parameters of the tree model by the EWMAmethod. The optimal trading strategies of the mean-variance portfolio are investigated under this model setting. We class the mutual funds into three categories: equity funds, balanced funds and bond funds. Different combination of these three kinds of funds are considered to find the optimal trading strategy respectively. The results showed that the realized returns using this optimal trading strategy in practice is close to the pre-specified expected return level.

Darbe gaunama, jog silpno matų konvergavimo erdvės X prasme išlpaukia charakteringųjų transformacijų konvergavimas ir atvirkščiai, jeigu charakteringosios transformacijos konverguoja į funkcijas tolydžias nuliniame taške, tai iš čia išplaukia matų silpnas konvergavimas erdvės X prasme. / It is obtained, that the weak convergence in the sense of X implies the convergence of characteristic transforms, and, on the contrary, if the characteristic transforms converge weakly to the functions contiuous at zero, the from this the weak convergence in the sense of X for the probability measures fallows.

Darbe įrodyta diskreti ribinė teorema bendrųjų Dirichlė eilučių poklasiui meromorfinių funkcijų erdvėje. Pateiktas ribinio mato išreikštinis pavidalas. / The discrete limit theorem for general Dirichlet series in the space of meromorphic functions was proved in this paper. Expressed shape of limit measue was provided.

Rusų matematikas S. M. Voroninas įrodė, kad vienos funkcijos pagalba galima aproksimuoti norimu tikslumu tam tikros srities kompleksinėse aibėse bet kurią analizinę funkciją. Tačiau neįrodė 1 teoremos analogo Dirichlė L funkcijoms. Darbo tikslas pateikti šios teoremos pilną įrodymą. / Russian mathematician S.M. Voronin proved, that any function can be approximated to the desired accuracy by one function in a specific sets in complex plane. But failed to theorem 1 analogue Dirichlet L-functions. The aim of this to provide a complete proof of the theorem.

Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 332–350, July–September, 2006.

In this thesis, we are concerned with the modifications of the stochastic processes and the random probability measures. First chapter is devoted to modifications of the stochastic process to the space of continuous functions, modifications of submartingale to the set of right-continuous with finite left-hand limits functions and separable modifications of stochastic process. In the second chapter is the attention on the regularization of random probability measure in Markov kernel focused. In particular, we work with random probability measures on the Borel subset of the Polish space, or Radon separable topological space.

Magistro darbe nagrinėjamas Oilerio sandaugų reikšmių pasiskirstymas analizinių funkcijų erdvėje. Taip pat gaunamas išreikštinis ribinio mato pavidalas. / In the Master work, we investigate the value-distribution of Euler products in the space of analytic functions. Also, we give an explicit form of the limit measure.

Tegul s=σ+it yra kompleksinis kintamasis. Oilerio sandaugos yra apibrėžiamos pagal pirminius skaičius, taip pat yra reikalaujama, kad funkcija L(s) tenkintų papidomas sąlygas. Mes įrodome diskrečią ribinę teoremą tikimybinių matų silpno konvergavimo prasme kompleksinėje plokštumoje C Oilerio sandaugoms. / Let s=σ+it be a complex variable. The Euler products L(s) is defined by the prime number. If the function L(s) satisfies some additional hypotheses. In the Master work we prove the discrete limit theorem in the sense of weakly convergent probability measures for the Euler products on the complex plane.