# 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.

### Existence of solutions of abstract volterra equations in a banach space and its subsets

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 648-657

We consider a criterion and sufficient conditions for the existence of a solution of the equation $$Z_t x = \frac{{t^{n - 1} x}}{{\left( {n - 1} \right)!}} + \int\limits_0^t {a\left( {t - s} \right)AZ_s xds} $$ in a Banach space *X*. We determine a resolvent of the Volterra equation by differentiating the considered solution on subsets of *X*. We consider the notion of "incomplete" resolvent and its properties. We also weaken the Priiss conditions on the smoothness of the kernel *a* in the case where A generates a *C* _{0}-semigroup and the resolvent is considered on *D(A)*.

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

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 804–809

We solve the problem of finding the optimal switching time for two alternative strategies at the financial market in the case where a random process*X* _{ 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 process*Y* _{ t } generated by the process*X* _{ t } so that the average investor's assets are maximized at the final time, i.e.,*EX* _{ T }.

### Method of successive approximations for abstract volterra equations in a banach space

Mishura Yu. S., Tomilov Yu. V.

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 376–382

We apply the method of successive approximations to abstract Volterra equations of the form*x=f+a*Ax*, where*A* is a closed linear operator. The assumption is made that a kernel*a* is continuous but is not necessarily of bounded variation.

### 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.”

### The Hille-Yosida theorem for resolvent operators of multiparameter semigroups

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

Ukr. Mat. Zh. - 1996. - 48, № 1. - pp. 57-65

We consider multiparameter semigroups of two types (multiplicative and coordinatewise) and resolvent operators associated with such semigroups. We prove an alternative version of the Hille-Yosida theorem in terms of resolvent operators. For simplicity of presentation, we give statements and proofs for two-parameter semigroups.

### Two-parameter Lévy processes: ItÔ formula, semigroups, and generators

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 952–961

We consider random Lévy fields, i.e., stationary fields continuous in probability and having independent increments. We prove that the trajectories of such fields have at most one jump on every line parallel to the axes. We derive an expression for the ItÔ change of variables for Lévy fields. We also consider semigroups generated by Lévy fields and their generators.

### Existence and properties of local times for Markov random fields

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 56–63

Random fields that have the “coordinatewise” Markov property are considered. The notions of an excessive function, a potential, and a continuous additive functional are introduced. Sufficient conditions for the existence of local time as a special form of continuous additive functional are formulated, and the uniqueness of this time to within a multiplicative constant is proved.

### 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