Solutions Of SDEs Research Articles

SDE solutions in the space of smooth random variables

In this paper we analyze properties of a dual pair (G;G ) of spaces of smooth and generalized random variables on a L evy white noise space. We show that G L 2 ( ) which shares properties with a Fr echet algebra contains a larger class of solutions of It^ o equations driven by pure jump L evy processes. Further a characterization of ( G;G ) in terms of the S-transform is given. We propose (G;G ) as an attractive alternative to the Meyer-Watanabe test function and distribution space (D1;D1 ) (W) to study strong solutions of SDE's.

Local Hölder Continuity Property of the Densities of Solutions of SDEs with Singular Coefficients

We prove that the weak solution of a uniformly elliptic stochastic differential equation with locally smooth diffusion coefficient and Hölder continuous drift has a Hölder continuous density function. This result complements recent results of Fournier–Printems (Bernoulli 16(2):343–360, 2010), where the density is shown to exist if both coefficients are Hölder continuous, and exemplifies the role of the drift coefficient in the regularity of the density of a diffusion.

A note on exponential state feedback stabilizability by a Razumikhin type theorem of mild solutions of SDEs with delay

In this paper, we study the question of state feedback exponential stabilizability by using a Razumikhin type theorem of mild solutions of SDEs in infinite dimensions with finite delay. An example is presented to illustrate the theory.

Confined gluon from Minkowski space continuation of the PT-BFM SDE solution

Recent lattice studies exhibit infrared finite effective QCD charges. The corresponding gluon propagator in the Landau gauge is finite and nonzero, suggesting that the mechanism of dynamical gluon mass generation is in the operation. In this paper, the analytical continuation of the Euclidean (spacelike) pinch technique–background field method solution of the Schwinger–Dyson equation for the gluon propagator to the timelike region of q2 is found. We found the continuation numerically showing good agreement with a generalized Lehman representation for small Schwinger coupling. The associated non-positive spectral function has an unexpected behavior. Albeit the infrared Euclidean space solution naively suggests a like single-scale ‘massive’ propagator, the obtained spectrum of the gluon propagator does not correspond to the delta function at single scale q = m; instead more possible singularities are generated. The pattern depends on the details of the assumed Schwinger mechanism: for stronger coupling there are a few maxima and minima which appear at the scale Λ, while for perturbatively small Schwinger coupling, the spectral function shows two narrow peaks: particle and ghost excitation, which have mutually opposite signs.

A remark on the mean square distance between the solutions of fractional SDEs and Brownian SDEs

We study the family of solutions of differential equations driven by fractional Brownian motions when the Hurst parameter varies between 1/2 and 1. The drift and the diffusion coefficient may also vary in a family of differentiable functions. We prove that there exists a finite covering of this set of solutions by open balls of centred in some solutions of classical stochastic differential equations driven by a Brownian motion.

Exponential ergodicity and regularity for equations with Lévy noise

We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α -stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Liapunov’s function approach by Harris, and the second on Doeblin’s coupling argument in [8]. Irreducibility and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite dimensional equations, introduced in [27], with Hölder continuous drift and a general, non-degenerate, symmetric α -stable noise, and infinite dimensional parabolic systems, introduced in [29], with Lipschitz drift and cylindrical α -stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [28], with a different method.

The α-Dependence of Stochastic Differential Equations Driven by Variants of α-Stable Processes

In this article, we investigate two variants of α-stable processes, namely tempered stable subordinators and modified tempered stable process as well as their renormalization. We study the weak convergence in the Skorohod space and prove that they satisfy the uniform tightness condition. Finally, applications to the α-dependence of the solutions of SDEs driven by these processes are discussed.

Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions

We study smoothness of densities for the solutions of SDEs whose coefficients are smooth and nondegenerate only on an open domain $D$. We prove that a smooth density exists on $D$ and give upper bounds for this density. Under some additional conditions (mainly dealing with the growth of the coefficients and their derivatives), we formulate upper bounds that are suitable to obtain asymptotic estimates of the density for large values of the state variable ("tail" estimates). These results specify and extend some results by Kusuoka and Stroock [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985) 1--76], but our approach is substantially different and based on a technique to estimate the Fourier transform inspired from Fournier [Electron. J. Probab. 13 (2008) 135--156] and Bally [Integration by parts formula for locally smooth laws and applications to equations with jumps I (2007) The Royal Swedish Academy of Sciences]. This study is motivated by existing models for financial securities which rely on SDEs with non-Lipschitz coefficients. Indeed, we apply our results to a square root-type diffusion (CIR or CEV) with coefficients depending on the state variable, that is, a situation where standard techniques for density estimation based on Malliavin calculus do not apply. We establish the existence of a smooth density, for which we give exponential estimates and study the behavior at the origin (the singular point).

Hamilton-Jacobi-Bellman Equations Associated to Symmetric Stable Processes

We are concerned with an optimal stochastic control and stopping pro- blem of a jump diffusion process. The main interest of this paper lies in the case where the dynamics has infinite variance, especially in the case of solutions of SDEs driven by symmetric stable processes. We prove that the value function is a viscosity solution of the integro-differential variational inequality arising from the associated dynamic program- ming. We also establish comparison principles in the class of semi-continuous functions with polynomial growth of a given order. Mathematics Subject Classication 2000: 93E20, 49L25, 60J75.

Quadratic variations and estimation of the hurst index of the solution of SDE driven by a fractional Brownian motion

In this paper, we derive two strongly consistent Hurst index estimators from the limit behavior of the first- and second-order quadratic variations of the solution of an SDE driven by a fractional Brownian motion. We also show that these estimators retain their properties if the solution is replaced with its Milstein approximation.

The composite Milstein methods for the numerical solution of Ito stochastic differential equations

In this paper, we present the composite Milstein methods for the strong solution of Ito stochastic differential equations. These methods are a combination of semi-implicit and implicit Milstein methods. We give a criterion for choosing either the implicit or the semi-implicit scheme at each step of our numerical solution. The stability and convergence properties are investigated and discussed for the linear test equation. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. The stability properties of the composite Milstein methods are found to be more superior compared to those of the Milstein, the Euler and even better than the composite Euler method. This superiority in stability makes the methods a better candidate for the solution of stiff SDEs.

Absolute Continuity and Convergence in Variation for Distributions of Functionals of Poisson Point Measure

General sufficient conditions are given for absolute continuity and convergence in variation of the distributions of the functionals on the probability space generated by a Poisson point measure. The phase space of the Poisson point measure is supposed to be of the form ${\mathbb{R}}^{+}\times{\mathbb{U}}$ , and its intensity measure to equal dt Π(du). We introduce the family of time stretching transformations of the configurations of the point measure. Sufficient conditions for absolute continuity and convergence in variation are given in terms of the time stretching transformations and the relative differential operators. These conditions are applied to solutions of SDEs driven by Poisson point measures, including SDEs with non-constant jump rate.

Construction of strong solutions of SDE's via Malliavin calculus

In this paper we develop a new method for the construction of strong solutions of stochastic equations with discontinuous coefficients. We illustrate this approach by studying stochastic differential equations driven by the Wiener process. Using Malliavin calculus we derive the result of A.K. Zvonkin (1974) [31] for bounded and measurable drift coefficients as a special case of our analysis of SDE's. Moreover, our approach yields the important insight that the solutions obtained by Zvonkin are even Malliavin differentiable. The latter indicates that the “nature” of strong solutions of SDE's is tightly linked to the property of Malliavin differentiability. We also stress that our method does not involve a pathwise uniqueness argument but provides a direct construction of strong solutions.

Numerical solution of stochastic differential equations with Poisson and Lévy white noise

A fixed time step method is developed for integrating stochastic differential equations (SDE's) with Poisson white noise (PWN) and Lévy white noise (LWN). The method for integrating SDE's with PWN has the same structure as that proposed by Kim [Phys. Rev. E 76, 011109 (2007)], but is established by using different arguments. The integration of SDE's with LWN is based on a representation of Lévy processes by sums of scaled Brownian motions and compound Poisson processes. It is shown that the numerical solutions of SDE's with PWN and LWN converge weakly to the exact solutions of these equations, so that they can be used to estimate not only marginal properties but also distributions of functionals of the exact solutions. Numerical examples are used to demonstrate the applications and the accuracy of the proposed integration algorithms.

The composite Milstein methods for the numerical solution of Stratonovich stochastic differential equations

In this paper, we present two composite Milstein methods for the strong solution of Stratonovich stochastic differential equations driven by d-dimensional Wiener processes. The composite Milstein methods are a combination of semi-implicit and implicit Milstein methods. The criterion for choosing either the implicit or the semi-implicit method at each step of the numerical solution is given. The stability and convergence properties of the proposed methods are analyzed for the linear test equation. It is shown that the proposed methods converge to the exact solution in Stratonovich sense. In addition, the stability properties of our methods are found to be superior to those of the Milstein and the composite Euler methods. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. Hence, the proposed methods are a good candidate for the solution of stiff SDEs.

Explicit Representation of Strong Solutions of SDEs Driven by Infinite-Dimensional Lévy Processes

We develop a white noise framework for Levy processes on Hilbert spaces. As the main result of this paper, we then employ these white noise techniques to explicitly represent strong solutions of stochastic differential equations driven by a Hilbert-space-valued Levy process.

Remarks on uniqueness and strong solutions to deterministic and stochastic differential equations

Motivated by open problems of well posedness in fluid dynamics, two topics related to strong solutions of SDEs are discussed. The first one on stochastic flows for SDEs with non regular drift helps to solve a stochastic transport equation where the corresponding deterministic equation is not well posed. The second one is a concept of strong superposition solution motivated by problems where uniqueness is not true or not known.

A Large Deviation Principle for Stochastic Integrals

Assuming that {(Xn,Yn)} satisfies the large deviation principle with good rate function I♯, conditions are given under which the sequence of triples {(Xn,Yn,Xn⋅Yn)} satisfies the large deviation principle. An e-approximation to the stochastic integral is proven to be almost compact. As is well known from the contraction principle, we can derive the large deviation principle when applying continuous functions to sequences that satisfy the large deviation principle; the method showed here skips the contraction principle, uses almost compactness and can be used to derive a generalization of the work of Dembo and Zeitouni on exponential approximations. An application of the main result to stochastic differential equations is given, namely, a Freidlin-Wentzell theorem is obtained for a sequence of solutions of SDE’s.

Some results on strong solutions of SDEs with applications to interest rate models

In this work, we investigate SDEs whose coefficients may depend on the entire past of the solution process. We introduce different Lipschitz-type conditions on the coefficients. It turns out that for existence and uniqueness of a strong solution it suffices to have Lipschitz continuity in mean, in a sense to be made precise. We then investigate when it suffices to have local Lipschitz conditions. Furthermore we consider the case of drift coefficients which are locally Lipschitz in mean. Finally we show how these results can be applied to prove existence and uniqueness of solutions in interest rate term structure models.

On the radius of convergence of the logarithmic signature

It has recently been proved that a continuous path of bounded variation in R-d can be characterised in terms of its transform into a sequence of iterated integrals called the signature of the path. The signature takes its values in an algebra and always has a logarithm. In this paper we study the radius of convergence of the series corresponding to this logarithmic signature for the path. This convergence can be interpreted in control theory (in particular, the series can be used for effective computation of time invariant vector fields whose exponentiation yields the same diffeomorphism as a time inhomogeneous flow) and can provide efficient numerical approximations to solutions of SDEs. We give a simple lower bound for the radius of convergence of this series in terms of the length of the path. However, the main result of the paper is that the radius of convergence of the full log signature is finite for two wide classes of paths (and we conjecture that this holds for all paths different from straight lines).