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We study Markov chains on ℤm,m≥ 2, that behave like a standard symmetric random walk outside of the hyperplane (membrane)H= {0} × ℤm−1. The exit probabilities from the membrane (penetration probabilities) H are periodic and also depend on the incoming direction toH, what makes the membrane H two-sided. Moreover, sliding along the membrane is allowed. We show that the natural scaling limit of such Markov chains is a m-dimensional diffusion whose first coordinate is a skew Brownian motion and the otherm− 1 coordinates is a Brownian motion with a singular drift controlled by the local time of the first coordinate at 0. In the proof we utilize a martingale characterization of the Walsh Brownian motion and determine the effective permeability and slide direction. Eventually, a similar convergence theorem is established for the one-sided membrane without slides and random iid penetration probabilities.

In this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters $$H H < 1 2 . Here, the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a “local time variational calculus” argument.

UDC 519.21 In this paper we solve a selection problem for multidimensional SDE $d X^{\epsilon}(t)=a(X^{\epsilon}(t))\, d t + \epsilon\sigma(X^{\epsilon}(t))\, d W(t),$ where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane $H.$It is assumed that $X^{\epsilon}(0)=x^0\in H,$ the drift $a(x)$ has a Hoelder asymptotics as $x$ approaches $H,$ and the limit ODE $d X(t)=a(X(t))\, d t$ does not have a unique solution.We show that if the drift pushes the solution away from $H,$ then the limit process with certain probabilities selects some extremal solutions to the limit ODE. If the drift attracts the solution to $H,$ then the limit process satisfies an ODE with some averaged coefficients. To prove the last result we formulate an averaging principle, which is quite general and new.

We study the asymptotic behavior of a real-valued diffusion whose nonregular drift is given as a sum of a dissipative term and a bounded measurable one. We prove that two trajectories of that diffu...

Let $\xi_1$, $\xi_2,\ldots$ be i.i.d. random variables of zero mean and finite variance and $\eta_1$, $\eta_2,\ldots$ positive i.i.d. random variables whose distribution belongs to the domain of attraction of an $\alpha$-stable distribution, $\alpha\in (0,1)$. The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump $\xi_k$ occurs; if the present position of the Markov chain is nonpositive, then its next position is $\eta_j$. We prove a functional limit theorem for this Markov chain under Donsker's scaling. The weak limit is a nonnegative process $(X(t))_{t\geq 0}$ satisfying a stochastic equation ${\rm d}X(t)={\rm d}W(t)+ {\rm d}U_\alpha(L_X^{(0)}(t))$, where $W$ is a Brownian motion, $U_\alpha$ is an $\alpha$-stable subordinator which is independent of $W$, and $L_X^{(0)}$ is a local time of $X$ at $0$. Also, we explain that $X$ is a Feller Brownian motion with a `jump-type' exit from $0$., Comment: accepted for publication in Bernoulli, 25 pages

We revisit the zero-noise Peano selection problem for Lévy-driven stochastic differential equation considered in [Pilipenko and Proske, Statist. Probab. Lett., 132:62–73, 2018] and show that the selection phenomenon pertains in the multiplicative noise setting and is robust with respect to certain perturbations of the irregular drift and of the small jumps of the noise.

We study the limit behavior of differential equations with non-Lipschitz coefficients that are perturbed by a small self-similar noise. It is proved that the limiting process is equal to the maximal solution or minimal solution with certain probabilities p + and p − = 1 − p + , respectively. We propose a space–time transformation that reduces the investigation of the original problem to the study of the exact growth rate of a solution to a certain SDE with self-similar noise. This problem is interesting in itself. Moreover, the probabilities p + and p − coincide with probabilities that the solution of the transformed equation converges to + ∞ or − ∞ as t → ∞ , respectively.

Let (ξ1, η1), (ξ2, η2),… be a sequence of i.i.d. two-dimensional random vectors. In the earlier article Iksanov and Pilipenko (2014) weak convergence in the J1-topology on the Skorokhod space of \(n^{-1/2}\underset {0\leq k\leq [n\cdot ]}{\max }\,(\xi _{1}+\ldots +\xi _{k}+\eta _{k+1})\) was proved under the assumption that contributions of \(\underset {0\leq k\leq n}{\max }\,(\xi _{1}+\ldots +\xi _{k})\) and \(\underset {1\leq k\leq n}{\max }\,\eta _{k}\) to the limit are comparable and that n−1/2(ξ1+… + ξ[n⋅]) is attracted to a Brownian motion. In the present paper, we continue this line of research and investigate a more complicated situation when ξ1+… + ξ[n⋅], properly normalized without centering, is attracted to a centered stable Levy process, a process with jumps. As a consequence, weak convergence normally holds in the M1-topology. We also provide sufficient conditions for the J1-convergence. For completeness, less interesting situations are discussed when one of the sequences \(\underset {0\leq k\leq n}{\max }\,(\xi _{1}+\ldots +\xi _{k})\) and \(\underset {1\leq k\leq n}{\max }\,\eta _{k}\) dominates the other. An application of our main results to divergent perpetuities with positive entries is given.

The linear functional strategy for the regularization of inverse problems is considered. For selecting the regularization parameter therein, we propose the heuristic quasi-optimality principle and some modifications including the smoothness of the linear functionals. We prove convergence rates for the linear functional strategy with these heuristic rules taking into account the smoothness of the solution and the functionals and imposing a structural condition on the noise. Furthermore, we study these noise conditions in both a deterministic and stochastic setup and verify that for mildly-ill-posed problems and Gaussian noise, these conditions are satisfied almost surely, where on the contrary, in the severely-ill-posed case and in a similar setup, the corresponding noise condition fails to hold. Moreover, we propose an aggregation method for adaptively optimizing the parameter choice rule by making use of improved rates for linear functionals. Numerical results indicate that this method yields better results than the standard heuristic rule.

We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state. We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Similar result was obtained by Raimond and Schapira, their proof was based on the Ray-Knight type theorems. We propose a new method based on a study of the Radon-Nikodym density of the ERW distribution with respect to the distribution of a symmetric random walk.

We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space.

Let ( ξ 1 , η 1 ) , ( ξ 2 , η 2 ) , … be a sequence of i.i.d. two-dimensional random vectors. We prove a functional limit theorem for the maximum of a perturbed random walk max 0 ≤ k ≤ n ( ξ 1 + ⋯ + ξ k + η k + 1 ) in a situation where its asymptotics is affected by both max 0 ≤ k ≤ n ( ξ 1 + ⋯ + ξ k ) and max 1 ≤ k ≤ n η k to a comparable extent. This solves an open problem that we learned from the paper “Renorming divergent perpetuities” by P. Hitczenko and J. Wesolowski.

We construct a stochastic flow generated by an SDE with L\'evy noise and a drift coefficient being a function of bounded variation on R. It is proved that this flow is non-coalescing and Sobolev differentiable with respect to initial data. The representation for the derivative is given., Comment: 10 pages

The problem on identification of a limit of an ordinary differential equation with discontinuous drift that perturbed by a zero-noise is considered in multidimensional case. This problem is a classical subject of stochastic analysis, see, for example, [6, 29, 11, 20]. However the multidimensional case was poorly investigated. We assume that the drift coefficient has a jump discontinuity along a hyperplane and is Lipschitz continuous in the upper and lower half-spaces. It appears that the behavior of the limit process depends on signs of the normal component of the drift at the upper and lower half-spaces in a neighborhood of the hyperplane, all cases are considered. Electronic version of an article published as Pilipenko, Andrey, and Frank Norbert Proske. "On a selection problem for small noise perturbation in the multidimensional case." Stochastics and Dynamics (2018): 1850045. https://doi.org/10.1142/S0219493718500454 © World Scientific Publishing Company https://www.worldscientific.com/worldscinet/sd

Local perturbations of a Brownian motion are considered. As a limit we obtain a non-Markov process that behaves as a reflected Brownian motion on the positive half line until its local time at zero reaches some exponential level, then changes a sign and behaves as a reflected Brownian motion on the negative half line until some stopping time, etc., 9 pages

We consider a $d$-dimensional SDE with an identity diffusion matrix and a drift vector being a vector function of bounded variation. We give a representation for the derivative of the solution with respect to the initial data., 15 pages

We consider a stochastic flow on $\mathds{R}$ generated by an SDE with its drift being a function of bounded variation. We show that the flow is differentiable with respect to the initial conditions. Asymptotic properties of the flow are studied., 19 pages

General statement of the problem of testing two hypotheses Let the trajectory \(x(\cdot)\) of a stochastic process \(\{X(t), t\in[0,T]\}\) be observed.

Consider a complete filtration \({\mathcal{F}}_t\), t∈[0,T]} and an m-dimensional Wiener process {W(t), t ∈ [0,T]} with respect to it. By definition, a stochastic differential equation (SDE) is an equation of the form with X 0=ξ, where ξ is an \({\mathcal{F}}_0\)-measurable random vector, \(b=b(t,x): [0,T]\times \mathbb{R}^n\rightarrow \mathbb{R}^n\), and \(\sigma=\sigma(t,x): [0,T]\times \mathbb{R}^n\rightarrow \mathbb{R}^{n\times m}\) are measurable functions.

The following theorem shows that all the finite-dimensional distributions for the process with independent increments on

The optimal stopping problem can be considered for nearly any stochastic process, and its formulation will be similar in each case. But its solution will be relatively simple only for a few processes. One class of such processes consists of discrete-time Markov chains.

In this chapter we consider random functions with the phase space being either real line \(\Bbb{R}\) or complex plane \(\Bbb{C}\).

Let \({\Bbb T} \subset \mathbb{R}\), \((\Omega, {\rm{F}}, \{{\rm{F}}_t\}_{t\in {\Bbb T}}, {\mathsf{P}})\) be a probability space with complete filtration. Let \(X = \{X(t), t\in{\Bbb T}\}\) be an adapted stochastic process taking values in some metric space \(({\Bbb X}, {\frak X})\), which sometimes is called the phase space of the process X.

If a random function is stochastically continuous (continuous with probability one, continuous in the L p sense) at every point of the parametric set

Let \(\{W(t),t\in \mathbb{R}^+\}\) be a Wiener process, and \(\{g(t),{\mathcal{F}}_t^W,t\in \mathbb{R}^+\}\) a stochastic process (recall that the previous notation means that g is adapted to a natural filtration \(\{{\mathcal{F}}_t^W\}\) of the Wiener process). Let \({\mathcal{F}}^W=\sigma\{W_t, t\geq 0\}.\) A process g is said to belong to the class \(\hat{\mathcal{L}}_2([a,b])\) if it is measurable and \({{\mathsf{E}}}\int_a^b g^2(s) d s

In this chapter we consider complex-valued stochastic processes.Let us recall the definition of covariance for complex-valued random variables X, Y

Let phase space \(\Bbb{X}\) of a random sequence \(\{X_n,\,n\in\Bbb{Z}^+\}\) be enumerable. The sequence \(\{X_n,\,n\in\Bbb{Z}^+\}\) is called a Markov chain if $$\forall n\in\Bbb{N}\ \forall i_1,\ldots,i_n,i_{n+1}\in \Bbb{X} \ \forall t_1\leq\cdots\leq t_n\leq t_{n+1}\in\Bbb{Z}_+:$$ $$\mathsf{P}(X_{t_{n+1}}=i_{n+1} / X_{t_{1}}=i_{1},\ldots,X_{t_{n}}=i_{n})=\mathsf{P}(X_{t_{n+1}}=i_{n+1} / X_{t_{n}}=i_{n}).$$

Below in this chapter we denote the discrete-time processes as X n , M n , and so on (with the lower index as time) and continuous-time processes, as before, are denoted as X(t), M(t), and so on (the time index is inside the parentheses).

Consider a model of a financial market with a finite number of periods (i.e., of the moments of time) at which it is possible to trade, consume, spend, or receive money or other valuables. The model consists of the following components.

Hereinafter the mapping X(t, ) is denoted as X(t). According to commonly accepted terminology it is a random element taking values in X. The definition introduced above is obviously equivalent to the following one.

In this chapter, we consider random elements taking values in metric spaces and their distributions. The definition of a random element taking values in \({{\Bbb X}}\) involves the predefined \(\sigma\)-algebra \({{\rm X}}\) of subsets of \({{\Bbb X}}\). The following statement shows that in a separable metric space, in fact, the unique natural choice for the \(\sigma\)-algebra \({{\rm X}}\) is the Borel \(\sigma\)-algebra \({{\rm B}}({{\Bbb X}})\).

Mathematical foundations of investigating of the risk process in insurance were created by Swedish mathematician Filip Lundberg in 1903–1909. For a long time this theory had been developed by mostly Nordic mathematicians, such as Cramer, Segerdal, Teklind, and others. Later on risk theory started to develop not only with connection to insurance but also as the method of solving different problems in actuarial and financial mathematics, econometrics. In the second half of the twentieth century the applied area of risk theory was expanded significantly.

We consider a Brownian motion on the plane with semipermeable membranes on n rays that have a common endpoint in the origin. We obtain the necessary and sufficient conditions for the process to reach the origin and we show that the probability of hitting the origin is equal to zero or one.