Related Stochastic Differential Equations Research Articles

Relaxation and Noise-Driven Oscillations in a Model of Mitotic Spindle Dynamics

During cell division, the mitotic spindle moves dynamically through the cell to position the chromosomes and determine the ultimate spatial position of the two daughter cells. These movements have been attributed to the action of cortical force generators which pull on the astral microtubules to position the spindle, as well as pushing events by these same microtubules against the cell cortex and plasma membrane. Attachment and detachment of cortical force generators working antagonistically against centring forces of microtubules have been modelled previously (Grill et al. in Phys Rev Lett 94:108104, 2005) via stochastic simulations and mean-field Fokker–Planck equations (describing random motion of force generators) to predict oscillations of a spindle pole in one spatial dimension. Using systematic asymptotic methods, we reduce the Fokker–Planck system to a set of ordinary differential equations (ODEs), consistent with a set proposed by Grill et al., which can provide accurate predictions of the conditions for the Fokker–Planck system to exhibit oscillations. In the limit of small restoring forces, we derive an algebraic prediction of the amplitude of spindle-pole oscillations and demonstrate the relaxation structure of nonlinear oscillations. We also show how noise-induced oscillations can arise in stochastic simulations for conditions in which the mean-field Fokker–Planck system predicts stability, but for which the period can be estimated directly by the ODE model and the amplitude by a related stochastic differential equation that incorporates random binding kinetics.

Casimir preserving stochastic Lie–Poisson integrators

AbstractCasimir preserving integrators for stochastic Lie–Poisson equations with Stratonovich noise are developed, extending Runge–Kutta Munthe-Kaas methods. The underlying Lie–Poisson structure is preserved along stochastic trajectories. A related stochastic differential equation on the Lie algebra is derived. The solution of this differential equation updates the evolution of the Lie–Poisson dynamics using the exponential map. The constructed numerical method conserves Casimir-invariants exactly, which is important for long time integration. This is illustrated numerically for the case of the stochastic heavy top and the stochastic sine-Euler equations.

Multifractal properties of sample paths of ground state-transformed jump processes

We consider a class of L\'evy-type processes with unbounded coefficients, arising as Doob $h$-transforms of Feynman-Kac type representations of non-local Schr\"odinger operators, where the function $h$ is chosen to be the ground state of such an operator. First, we show the existence of a c\`adl\`ag version of the so-obtained ground state-transformed processes. Next, we prove that they satisfy a related stochastic differential equation with jumps. Making use of this SDE, we then derive and prove the multifractal spectrum of local H\"older exponents of sample paths of ground state-transformed processes.

Second-order asymptotics for the block counting process in a class of regularly varying ${\Lambda}$-coalescents

Consider a standard ${\Lambda }$-coalescent that comes down from infinity. Such a coalescent starts from a configuration consisting of infinitely many blocks at time $0$, but its number of blocks $N_t$ is a finite random variable at each positive time $t$. Berestycki et al. [Ann. Probab. 38 (2010) 207-233] found the first-order approximation $v$ for the process $N$ at small times. This is a deterministic function satisfying $N_t/v_t\to1$ as $t\to0$. The present paper reports on the first progress in the study of the second-order asymptotics for $N$ at small times. We show that, if the driving measure $\Lambda$ has a density near zero which behaves as $x^{-\beta}$ with $\beta\in(0,1)$, then the process $(\varepsilon^{-1/(1+\beta)}(N_{\varepsilon t}/v_{\varepsilon t}-1))_{t\ge0}$ converges in law as $\varepsilon\to0$ in the Skorokhod space to a totally skewed $(1+\beta)$-stable process. Moreover, this process is a unique solution of a related stochastic differential equation of Ornstein-Uhlenbeck type, with a completely asymmetric stable L\'{e}vy noise.

Skew Ornstein–Uhlenbeck processes and their financial applications

In this paper, we investigate a special class of skew diffusions: skew Ornstein–Uhlenbeck (abbr. OU) processes, whose scale and speed densities are both piecewise functions. The existence and uniqueness of solutions regarding the related stochastic differential equations (abbr. SDEs) with local time are established, as well as the construction through time changes. Afterwards, we concentrate on three computing issues including the explicit expressions of transition densities, the cumulative distributions and Laplace transforms of the first hitting times for skew OU processes. With the hypothesis on asset dynamics, two financial instances in the field of credit risk are illustrated at the end of this paper.

Green function of the double-fractional Fokker-Planck equation: Path integral and stochastic differential equations

The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.

A New Numerical Scheme for a Class of Reflected Stochastic Differential Equations

We propose a new numerical scheme for a class of one-dimensional reflected stochastic differential equations (SDEs) by virtue of their explicit solutions, which enables us to carry out the simulation of this class of reflected SDEs by simulating some related SDEs without reflections. The new scheme yields the same order of convergence as the scheme for the SDEs without reflections.

A new numerical scheme for a class of reflected stochastic differential equations

We propose a new numerical scheme for a class of one-dimensional reflected stochastic differential equations (SDEs) by virtue of their explicit solutions, which enables us to carry out the simulation of this class of reflected SDEs by simulating some related SDEs without reflections. The new scheme yields the same order of convergence as the scheme for SDEs without reflections.

One-dimensional parabolic diffraction equations: pointwise estimates and discretization of related stochastic differential equations with weighted local times

In this paper we consider one-dimensional partial differential equations of parabolic type involving a divergence form operator with a discontinuous coefficient and a compatibility transmission condition. We prove existence and uniqueness result by stochastic methods which also allow us to develop a low complexity Monte Carlo numerical resolution method. We get accurate pointwise estimates for the derivatives of the solutionfrom which we get sharp convergence rate estimates for our stochastic numerical method.

Non-Gaussian distribution for stock returns and related stochastic differential equation

Daily returns of stock prices are observed to have heavy-tailed and non-central distribution. In this paper, we adopt the type VII and IV family of Pearson System to express the daily returns of stock prices. Furthermore, we consider related stochastic differential equation whose stationary distribution are the type VII or IV of Pearson system, and estimate the parameters of stochastic differential equation by maximum likelihood method, where the transition probability density function and local linearization method are used. This paper is the basic research for the risk management.

Properties at infinity of diffusion semigroups and stochastic flows via weak uniform covers

A unified treatment is given of some results of H. Donnelly, P. Li and L. Schwartz concerning the behaviour of heat semigroups on open manifolds with given compactifications, on one hand, and the relationship with the behaviour at infinity of solutions of related stochastic differential equations on the other. A principal tool is the use of certain covers of the manifold: which also gives a non-explosion test. As a corollary we obtain known results about the behaviour of Brownian motions on a complete Riemannian manifold with Ricci curvature decaying at most quadratically in the distance function.

Singular stochastic control for diffusions and sde with discontinuous

In this paper, we continue to study a diffusion-type, finite-fuel singular stochastic control problem and the related stochastic differential equations with discontinuous paths and reflecting boundary conditions as denned in the previous work of the author [15]. The measurable dependence of the solution with respect to the initial state and the underlying probability measures (with Prohorov metric) is derived. Also, the approximation of certain complete class of controls by those with continuous paths is proved to be possible in a weak sense. With the help of these results, we prove the Dynamic Programming Principle (Bellman principle) rigorously and show that the value function is the viscosity solution of certain Hamilton-Jacobi-Bellman equation

Discontinuous reflection, and a class of singular stochastic control problems for diffusions

We study two kinds of Discontinuous Reflecting Problem (DRP for short), defined by Chaleyat-Maurel et al. [3] and Dupuis and Ishii [5] (reduced to the one-dimensional case) and the related Stochastic Differential Equations with Discontinuous Paths and Reflecting Boundary Conditions (SDEDR for short). We compare the properties of the solutions to the two DRP's as well as the two SDEDR's. Some comparison theorems, either for the solutions of different kinds of SDEDR's or for the same kind of SDEDR but with different data, are derived. As an application, we consider a class of finite-fuel singular stochastic control problems for diffusions. With the new approach, the complete class of admissible controls can now be obtained as a direct consequence of our comparison theorems without any extra conditions. The existence of the optimal control (in a wider sense) is derived