Stochastics and Dynamics

This article deals with the Skorokhod embedding problem in bounded time for the Brownian motion with drift X_t = κt + W_t: Given a probability measure μ we aim at finding a stopping time τ such that the law of X_τ is μ, and τ is almost surely smaller than some given fixed time horizon T > 0. We provide necessary and sufficient conditions on the distribution μ for the existence of such bounded stopping times.

We show that for a large class of stochastic flows the spatial derivative grows at most exponentially fast even if one takes the supremum over a bounded set of initial points. We derive explicit bounds on the growth rates that depend on the local characteristics of the flow and the box dimension of the set.

In this paper we establish precise estimates for the supremum norm for the solution of a dynamical system driven by a Hölder continuous function of order between ⅓ and ½ using the techniques of fractional calculus. As an application we deduce the existence of moments for the solutions to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈(⅓, ½) and we obtain an estimate for the supremum norm of the Malliavin derivative.

An asymptotic stability analysis of numerical methods used for simulating stochastic differential equations with additive noise is presented. The initial part of the paper is intended to provide a clear definition and discussion of stability concepts for additive noise equation derived from the principles of stability analysis based on the theory of random dynamical systems. The numerical stability analysis presented in the second part of the paper is based on the semi-linear test equation dX(t) = (AX(t) + f(X(t))) dt + σ dW(t), the drift of which satisfies a contractive one-sided Lipschitz condition, such that the test equation allows for a pathwise stable stationary solution. The θ-Maruyama method as well as linear implicit and two exponential Euler schemes are analysed for this class of test equations in terms of the existence of a pathwise stable stationary solution. The latter methods are specifically developed for semi-linear problems as they arise from spatial approximations of stochastic partial differential equations.

We study an optimal consumption and investment problem in a possibly incomplete market with general, not necessarily convex, stochastic constraints. We provide explicit solutions for investors with exponential, logarithmic as well as power utility and show that they are unique if the constraints are convex. Our approach is based on martingale methods that rely on results on the existence and uniqueness of solutions to BSDEs with drivers of quadratic growth.

It is shown that a stochastic partial differential equation of the reaction–diffusion type on time-varying domains obtained by a temporally continuous dependent spatially diffeomorphic transformation of a reference domain, which is bounded with a smooth boundary, generates a "partial-random" dynamical system, which has a pathwise nonautonomous pullback attractor.

Here we present a simple stochastic threshold model consisting of a deterministic slowly decaying term and a fast stochastic noise term. The process shows a pseudo-resonance, in the sense that for small and large intensities of the noise the signal is irregular and the distribution of threshold crossings is broad, while for a tuned intermediate value of noise intensity, the signal becomes quasi-periodic and the distribution of threshold crossings is narrow. The mechanism captured by the model might be relevant for explaining apparent quasi-periodicity of observed climatic variations where no internal or external periodicities can be identified.

Brownian motion regularizes ODEs, in the sense that non-well-posed ODEs become well-posed in the strong probabilitic sense by the addition of a non-degenerate noise. A version of such regularization property for single paths of Brownian motion has been given by A. M. Davie. We discuss some fragments of this deep result.

We study a coherent version of the entropic risk measure, both in the law-invariant case and in a situation of model ambiguity. In particular, we discuss its behavior under the pooling of independent risks and its connection with a classical and a robust large deviations bound.

Multiscale stochastic partial differential equations arise as models for various complex systems. An averaging principle for a class of stochastic partial differential equations with slow and fast time scales is established. Under suitable conditions, it is shown that the slow component converges to an effective dynamical system in the mean-square uniform sense.

We investigate a random differential equation with random delay. First the non-autonomous case is considered. We show the existence and uniqueness of a solution that generates a cocycle. In particular, the existence of an attractor is proved. Secondly we look at the random case. We pay special attention to the measurability. This allows us to prove that the solution to the random differential equation generates a random dynamical system. The existence result of the attractor can be carried over to the random case.

Enlargement of filtrations is a classical topic in the general theory of stochastic processes. This theory has been applied to stochastic finance in order to analyze models with insider information. In this paper we study initial enlargement in a Markov chain market model, introduced by Norberg. In the enlarged filtration, several things can happen: some of the jumps times can be accessible or predictable, but in the original filtration all the jumps times are totally inaccessible. But even if the jumps times change to accessible or predictable, the insider does not necessarily have arbitrage possibilities.

We give a survey about the different approaches to model financial markets with inside information in continuous time. In particular, we consider the Karatzas–Pikovsky, Kyle–Back and the weak Kyle–Back approach. These three types of modeling are based on the enlargement of filtration problem, which we explain with some examples and use it for these three modeling approaches.

This paper is a survey of recent results on the dynamics of Stochastic Burgers equation (SBE) and two-dimensional Stochastic Navier–Stokes Equations (SNSE) driven by affine linear noise. Both classes of stochastic partial differential equations are commonly used in modeling fluid dynamics phenomena. For both the SBE and the SNSE, we establish the local stable manifold theorem for hyperbolic stationary solutions, the local invariant manifold theorem and the global invariant flag theorem for ergodic stationary solutions. The analysis is based on infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle [22] (cf. [20, 21]). The results in this paper are based on joint work of the author with T. S. Zhang and H. Zhao ([17–19]).

We study the motion of overdamped Brownian particles in a periodic potential with a temporally oscillating amplitude. First we investigate diffusive motion in the untitled potential. Furthermore, if a constant force is applied, the oscillating potential induces a synchronized motion. The deterministic dynamics becomes in resonance with the potential oscillations. This dynamics gives rise to a transport with extremely low dispersion. We distinguish slow and fast oscillatory driving and give analytical expressions for the mean velocity and effective diffusion.

In this paper, a semilinear elliptic PDE with rapidly oscillating coefficients is homogenized. The novelty of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some portion of ℝ^d.

Our fully probabilistic method is based on the connection between PDEs and BSDEs with random terminal time and the weak convergence of a class of diffusion processes.

In this paper, we study first exit times from a bounded domain of a gradient dynamical system Ẏ_t = -∇U(Y_t) perturbed by a small multiplicative Lévy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular, we determine the asymptotics of the first exit time of solutions of Itô, Stratonovich and Marcus canonical SDEs.

We show how gradient estimates for transition semigroups can be used to establish exponential mixing for a class of Markov processes in infinite dimensions. We concentrate on semilinear systems driven by cylindrical α-stable noises introduced in [18], α ∈ (0,2). We first prove that if the nonlinearity is bounded, then the system is ergodic and strong mixing. Then we show that the system is exponentially mixing provided that the nonlinearity, or its Lipschitz constant, are sufficiently small.

In this paper, we introduce a new approach to rough and stochastic partial differential equations (RPDEs and SPDEs): we consider general Banach spaces as state spaces and — for the sake of simplicity — finite dimensional sources of noise, either rough or stochastic. By means of a time-dependent transformation of state space and rough path theory, we are able to construct unique solutions of the respective R- and SPDEs. As a consequence of our construction, we can apply the pool of results of rough path theory, in particular we can obtain strong and weak numerical schemes of high order converging to the solution process.

With the help of the theory of multi-parameter martingales we prove almost everywhere convergence of the Fejér means of two-dimensional Walsh–Fourier series of f ∈ L log L.

The study of random dynamical systems involves understanding the evolution of state variables that contain uncertainties and that are usually hidden, or not directly observable. Therefore, state variables have to be estimated and updated based on system models using information from observational data, which themselves are noisy, in the sense that they contain uncertainties and disturbances due to imperfections in observational devices and disturbances in the environment within which data are being collected. The development of efficient data assimilation methods for integrating observational data in predicting the evolution of random state variables is thus an important aspect in the study of random dynamical systems. In this paper, we consider a particle filtering approach to nonlinear filtering in multiscale dynamical systems. Particle filtering methods [1–3] utilizes ensembles of particles to represent the conditional density of state variables using particle positions, distributed over a sample space. The distribution of an ensemble of particles is updated using observational data to obtain the best representation of the conditional density of the state variables of interest. On the other hand, homogenization theory [4, 5], allows us to estimate the coarse-grained (slow) dynamics of a multiscale system on a larger timescale without having to explicitly study the fast variable evolution on a small timescale. The results of filter convergence presented in [6] shows the convergence of the filter of the actual state variable to a homogenized solution to the original multiscale system, and thus we develop a particle filtering scheme for multiscale random dynamical systems that utilizes this convergence result. This particle filtering method is called the Homogenized Hybird Particle Filter, and it incorporates a multiscale computation scheme, the Heterogeneous Multiscale Method developed in [7], with the novel branching particle filter described in [8–10]. By incorporating a multiscale scheme based on homogenization of the original system, estimation of the coarse-grained dynamics using observational data is performed over a larger timescale, thus resulting in computational time and cost reduction in terms of the evolution of the state variables as well as functional evaluations for the filtering aspect. We describe the theory behind this combined scheme and its general algorithm, concluded with an application to the Lorenz-96 [11] atmospheric model that mimics midlatitude geophysical dynamics with microscopic convective processes.