Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations

New weak and strong existence and weak and strong uniqueness results for the solutions of multi-dimensional stochastic McKean–Vlasov equation are established under relaxed regularity conditions. Weak existence requires a non-degeneracy of diffusion and no more than a linear growth of both coefficients in the state variable. Weak and strong uniqueness are established under the restricted assumption of diffusion, yet without any regularity of the drift; this part is based on the analysis of the total variation metric

Boundary noncrossings of additive Wiener fields

Let {W (i) (t), t a a"e(+)}, i = 1, 2, be two Wiener processes, and let W (3) = {W (3)(t), t a a"e (+) (2) } be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary noncrossing probability P (f) = P{W (1)(t (1)) + W (2)(t (2)) + W (3)(t) + f(t) a parts per thousand currency sign u(t), t a a"e (+) (2) }, where f, u : a"e (+) (2) -> a"e are two general measurable functions. We further show that, for large trend functions gamma f > 0, asymptotically, as gamma -> a, P (gamma f) is equivalent to , where is the projection of f onto some closed convex set of the reproducing kernel Hilbert space of the field W(t) = W (1)(t (1)) + W (2)(t (2)) + W (3)(t). It turns out that our approach is also applicable for the additive Brownian pillow

Positivity of solution of nonhomogeneous stochastic differential equation with non-Lipschitz diffusion

We give a sufficient condition on coefficients of a nonhomogeneous stochastic differential equation with non-Lipschitz diffusion for a solution starting from arbitrary nonrandom positive point to stay positive. Some examples of application of the condition mentioned above are considered

Another approach to the problem of the ruin probability estimate for risk process with investments

An exponential estimate of ruin probability for an insurance company which invests all its capital in risk assets is found. The process which describes the risky assets is assumed to follow a geometrical Brownian motion. Insurance premium flow depends on the value of reserves of the insurance company. The problem is solved by reduction of the generalized risk process to the classical risk process without investments

The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions

The paper is devoted to the problem of quantile hedging of contingent claims in the framework of a model defined by the finite number of independent Brownian and fractional Brownian motions. The maximal success probability depending on initial capital is estimated

Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process

The existence and uniqueness of solution of stochastic differential equation driven by standard Brownian motion and fractional Brownian motion with Hurst parameter H belongs (3/4, 1) is established

Наближення дробових iнтегралiв гельдерових функцiй

The paper is devoted to the rate of convergence of integral sums of two different types to fractional integrals. The first theorem proves the H¨older property of fractional integrals of functions from various integral spaces. Then we estimate the rate of convergence of the integral sums of two types corresponding to the H¨older functions, to the respective fractional integrals. We illustrate the obtained results by several figures. Pages of the article in the issue: 18 - 25 Language of the article: EnglishСтаттю присвячено дослiдженню швидкостi збiжностi iнтегральних сум двох типiв до дробового iнтегралу. В першiй теоремi доведено гельдерiвську властивiсть дробових iнтегралiв вiд функцiй з рiзних iнтегральних просторiв. Потiм ми оцiнюємо швидкiсть збiжностi iнтегральних сум, побудованих за гельдерiвськими функцiями, до вiдповiдних дробових iнтегралiв. Отриманi результати проiлюстровано декiлькома чисельними прикладами

Approximation of fractional Brownian motion by martingales

We study the problem of optimal approximation of a fractional Brownian motion by martingales. We prove that there exist a unique martingale closest to fractional Brownian motion in a specific sense. It shown that this martingale has a specific form. Numerical results concerning the approximation problem are given

Boundary non-crossing probabilities for fractional Brownian motion with trend

In this paper, we investigate the boundary non-crossing probabilities of a fractional Brownian motion considering some general deterministic trend function. We derive bounds for non-crossing probabilities and discuss the case of a large trend function. As a by-product, we solve a minimization problem related to the norm of the trend function