# Mishura Yu. S.

### Rate of convergence in the Euler scheme for stochastic differential equations with non-Lipschitz diffusion and Poisson measure

Mishura Yu. S., Zubchenko V. P.

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 40-60

We study the rate of convergence and some other properties of the Euler scheme for stochastic differential equations with the non-Lipschitz diffusion and the Poisson measure.

### Rate of convergence of the price of European option on a market for which the jump of stock price is uniformly distributed over an interval

Mishura Yu. S., Soloveiko O. M.

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1075–1086

We consider a model of the market such that a jump of share price is uniformly distributed on some symmetric interval and establish the rate of convergence of fair prices of European options by using the theorem on asymptotic decompositions of distribution function for the sum of independent identically distributed random variables. We show that, in the prelimit model, there exists a martingale measure on the market such that the rate of convergence of prices of European options to the Black - Scholes price is of order 1/*n* ^{1/2}.

### Estimation of the ruin probability of an insurance company operating on a BS-market

Androshchuk M. O., Mishura Yu. S.

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1443–1453

We obtain an estimate for the ruin probability of an insurance company that invests a part of its capital in stocks and puts the rest of the capital in a bank account. An insurance premium is established depending on the capital of the insurance company.

### Weak convergence of integral functionals of random walks weakly convergent to fractional Brownian motion

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1040–1046

We consider a random walk that converges weakly to a fractional Brownian motion with Hurst index *H* > 1/2. We construct an integral-type functional of this random walk and prove that it converges weakly to an integral constructed on the basis of the fractional Brownian motion.

### Generalized two-parameter Lebesgue-Stieltjes integrals and their applications to fractional Brownian fields

Il'chenko S. A., Mishura Yu. S.

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 435–450

We consider two-parameter fractional integrals and Weyl, Liouville, and Marchaut derivatives and substantiate some of their properties. We introduce the notion of generalized two-parameter Lebesgue-Stieltjes integral and present its properties and computational formulas for the case of differentiable functions. The main properties of two-parameter fractional integrals and derivatives of Hölder functions are considered. As a separate case, we study generalized two-parameter Lebesgue-Stieltjes integrals for an integrator of bounded variation. We prove that, for Hölder functions, the integrals indicated can be calculated as the limits of integral sums. As an example, generalized two-parameter integrals of fractional Brownian fields are considered.

### Euler Approximations of Solutions of Abstract Equations and Their Applications in the Theory of Semigroups

Mishura Yu. S., Shevchenko H. M.

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 399-410

Using the Euler approximations of solutions of abstract differential equations, we obtain new approximation formulas for *C* _{0}-semigroups and evolution operators.

### Differentiability of Fractional Integrals Whose Kernels Contain Fractional Brownian Motions

Krvavich Yu. V., Mishura Yu. S.

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 30-40

We prove the stochastic Fubini theorem for Wiener integrals with respect to fractional Brownian motions. By using this theorem, we establish conditions for the mean-square and pathwise differentiability of fractional integrals whose kernels contain fractional Brownian motions.

### The problem of extension for two-parameter kernels

Lavrent'ev A. S., Mishura Yu. S.

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1206–1212

We solve the problem of construction of a two-parame to either a multiplicative or a coordinatewise two-parameter semigroup. The construction is carried out on the basis of the “initial family of kernels.”

### Martingale field transformations under a change of probability measure

Ukr. Mat. Zh. - 1989. - 41, № 7. - pp. 923-929

### Exponential estimates for two-parameter martingales

Ukr. Mat. Zh. - 1987. - 39, № 3. - pp. 353–358

### Ito's formula for two-parameter stochastic integrals with respect to martingale measures

Ukr. Mat. Zh. - 1984. - 36, № 4. - pp. 456 – 461

### Some limit theorems for Scharf-Stieltjes stochastic integrals

Ukr. Mat. Zh. - 1980. - 32, № 3. - pp. 340 - 347