A lower bound for the transition density function of a stochastic differential equation

In this paper, we study the strong rate of convergence for the Euler-Maruyama approximation of a class of one-dimensional stochastic differential equations involving the local time (SDELT) of the unknown process, corresponding to divergence form operator with a discontinuous coefficients at zero. We use a space transform in order to remove the local time $L_{t}^{0}$ from the stochastic differential equation of type, \begin{equation*} dX_{t}=\sigma(X_{t})dB_{t}+\psi(X_{t})dt+\beta dL_{t}^{0}(X). \end{equation*} Here $B$ is a standard one-dimensional Brownian motion. $\sigma$ and $\psi$ are a bounded measurable functions, and $\beta\in(-1,1)$ . We provide the approximation of Euler-Maruyama for the stochastic differential equation without local time. After that the approximation can be transformed back, giving an approximation of Euler-Maruyama $X_{t}^{n}$ to the solution of the original SDELT, and we provide the rate of strong convergence Error $=\mathbb{E}[\vert X_{T}^{\mathrm{n}}-X_{T}\vert]$ .

One-dimensional Ito stochastic differential equations and stochastic differential equations with local time that depend on a small parameter and have irregular coefficients (for example, the unlimited drift coefficient or the non-Lipschitz diffusion coefficient) have been considered. The available results concerning the conditions for a mutually univocal correspondence between stochastic Ito equations and stochastic equations with local time were generalized. The weak convergence of the solutions of those equations at $\varepsilon \to 0$ were analyzed. The form of the coefficients for the limiting process was obtained. The necessary and sufficient conditions for the weak convergence of the solutions of those equations to the limit random process were proved.

This article discusses an alternative method for estimating marginal probability densities of the solution to stochastic differential equations (SDEs). Two algorithms for calculating the numerical–statistical projection estimate for distributions of solutions to SDEs using Legendre polynomials are proposed. The root-mean-square error of this estimate is studied as a function of the projection expansion length, while the step of a numerical method for solving SDE and the sample size for expansion coefficients are fixed. The proposed technique is successfully verified on three one-dimensional SDEs that have stationary solutions with given one-dimensional distributions and exponential correlation functions. A comparative analysis of the proposed method for calculating the numerical–statistical projection estimate and the method for constructing the histogram is carried out.

Stochastic averaging for estimating a ship roll in random longitudinal or oblique waves

We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability distribution that the sum of two one-dimensional Brownian motions will exit from the unit interval before time t, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffusion of the Brownian motion is sufficiently large.

In this paper we discuss existence and uniqueness for a one-dimensional time inhomogeneous stochastic differential equation directed by an $\mathbb F$-semimartingale $M$ and a finite cubic variation process $\xi$ which has the structure $Q + R$ where $Q$ is a finite quadratic variation process and $R$ is {\it{strongly predictable}} in some technical sense: that condition implies in particular that $R$ is \textit{weak Dirichlet}, and it is fulfilled, for instance, when $R$ is independent of $M$. The method is based on a transformation which reduces the {{\it diffusion}} coefficient multiplying $\xi$ to 1. We use generalized Ito and Ito-Wentzell type formulae. A similar method allows to discuss existence and uniqueness theorem when $\xi$ is a Holder continuous process and $\sigma$ is only Holder in space. Using an Ito formula for {\it{reversible}} semimartingales we also show existence of a solution when $\xi$ is a Brownian motion and $\sigma$ is only continuous.

We obtain the limit theorem for the sequence of stochastic flows generated by the stochastic ordinary differential equations dx/dt = F$sub:n$esub:(t, x, ω)n =l, 2,.... Under some regularity and mixing conditions on the sequence F$sub:n$esub: n = l,2,..., it is shown that the associated flows converge weakly to a Brownian motion in the diffeomorphisms group and that the latter is generated by an Ito's stochastic differential equation. The strong convergence is also established: It covers the approximation theorems on stochastic flows studied by Malliavin. Ikeda Watanabe. Bismut and Shu.

Optimal Control of Stochastic Systems

In this article, we consider a coefficient stability problem for one-dimensional stochastic differential equations driven by an $$\alpha$$ -stable process with $$\alpha \in (1,2)$$ . More precisely, we find an upper bound for the $$L^{\alpha -1}(\varOmega ,{\mathbb {P}})$$ distance between two solutions in terms of the $$L^{\alpha }\left( {\mathbb {R}},\mu^{\alpha }_{x_0}\right)$$ distance of the coefficients for an appropriate measure $$\mu^{\alpha} _{x_0}$$ which characterizes symmetric stable laws and depends on the initial value of the stochastic differential equation. We obtain this result using the method introduced by Komatsu (Proc Jpn Acad Ser A Math Sci 58(8):353–356, 1982) which is used in the proof of uniqueness of solutions together with an upper bound for the transition density function of the solution of the stochastic differential equation obtained by Kulik (The parametrix method and the weak solution to an SDE driven by an $$\alpha$$ -stable noise. arXiv:1412.8732 , 2014).

In this paper we prove the existence and uniqueness of a solution for a one-dimensional quasilinear stochastic differential equation of the Skorohod type, with a non-linear boundary condition of the form . The method of proof uses the Girsanov transformation and some basic exponential estimates. When the coefficients of the equation are deterministic we show that the solution is weakly differentiable on the Wiener space and possesses a continuous modification.

Fractional stochastic differential equation (FSDE)-based random processes are used in a wide spectrum of scientific disciplines. However, in the majority of cases, explicit solutions for these FSDEs do not exist and approximation schemes have to be applied. In this paper, we study one-dimensional stochastic differential equations (SDEs) driven by stochastic process with Hölder continuous paths of order 1/2<γ<1. Using the Lamperti transformation, we construct a backward approximation scheme for the transformed SDE. The inverse transformation provides an approximation scheme for the original SDE which converges at the rate h2γ, where h is a time step size of a uniform partition of the time interval under consideration. This approximation scheme covers wider class of FSDEs and demonstrates higher convergence rate than previous schemes by other authors in the field.

On the Fourier structure of the zero set of fractional Brownian motion

In this paper, the model formulated incorporated stochastic variables such as bank loans and deposits as well as some deterministic variables: cash available, depreciation, capital expenditure, tax and costs, comprising variable costs and fixed costs. This paper assumes that the dynamics of bank loans and deposits at time t follow a geometric Brownian motion, therefore, it satisfies certain stochastic differential equations (SDEs) formulated on some probability space. On the other hand, the growth rate μL(t) in loan at time t, growth rate μD(t) in deposit at time t, and the variable cost η(t) at time t are assumed to be driven by mean-reverting Ornstein-Uhlenbeck processes. The SDEs of the dynamics of bank loans, growth rate in loans, bank deposits, growth rate in deposits and variable cost arising from the model were solved by means of the It&Ocirc Lemma. Discrete time approximations of the exact solutions of the SDEs were derived and used in a Monte Carlos simulation software.

A simple path transformation is described which connects one-dimensional Brownian motion with the radial part of three-dimensional Brownian motion. This provides simple proofs of various path decompositions for these processes described by David Williams.

We prove the comparison theorems for scalar stochastic differential equations in the case of different diffusion coefficients. Conditions are given of stability with probability 1 with respect to the trivial solution to stochastic differential equations with random coefficients. The results remain valid for deterministic analogs of stochastic differential equations with symmetric integrals.