Large deviations for small noise diffusions with discontinuous statistics

This paper proves the large deviation principle for a class of non-degenerate small noise diffusions with discontinuous drift and with state-dependent diffusion matrix. The proof is based on a variational representation for functionals of strong solutions of stochastic differential equations and on weak convergence methods. Received: 26 May 1998 / Revised version: 24 February 1999     

A simple constructive approach to quadratic BSDEs with or without delay

This paper provides a simple approach for the consideration of quadratic BSDEs with bounded terminal conditions. Using solely probabilistic arguments, we retrieve the existence and uniqueness result derived via PDE-based methods by Kobylanski (2000) [14]. This approach is related to the study of quadratic BSDEs presented by Tevzadze (2008) [19]. Our argumentation, as in Tevzadze (2008) [19], highly relies on the theory of BMO martingales which was used for the first time for BSDEs by Hu et al. (2005) [12]. However, we avoid in our method any fixed point argument and use Malliavin calculus to overcome the difficulty. Our new scheme of proof allows also to extend the class of quadratic BSDEs, for which there exists a unique solution: we incorporate delayed quadratic BSDEs, whose driver depends on the recent past of the Y$Y$ component of the solution. When the delay vanishes, we verify that the solution of a delayed quadratic BSDE converges to the solution of the corresponding classical non-delayed quadratic BSDE.     

An inverse first-passage problem for one-dimensional diffusions with random starting point

We consider an inverse first-passage time (FPT) problem for a homogeneous one-dimensional diffusion X(t), starting from a random position η. Let S(t) be an assigned boundary, such that P(η≥S(0))=1, and F an assigned distribution function. The problem consists of finding the distribution of η such that the FPT of X(t) below S(t) has distribution F. We obtain some generalizations of the results of Jackson et al., 2009, which refer to the case when X(t) is Brownian motion and S(t) is a straight line across the origin.     

Effort associated with a class of random optimization methods

When differentiability is not assumed random procedures can be successfully used to estimate the extreme values of a given function. For a class of such algorithms we treat the problem of estimating the mean effort.Research partially supported by CNPq-Brazil.     

Interacting Brownian particles and the Wigner law

Summary In this paper, we study interacting diffusing particles governed by the stochastic differential equationsdX _j (t)= _n dB _j (t) –D _jØ_n(X ₁,...,X _n)dt,j=1, 2,...,n. Here theB _jare independent Brownian motions in ^d , and Ø _n (X ₁,...,X _n)= _{n ij} V(X _iX _j) + _ni U(X ₁). The potentialV has a singularity at 0 strong enough to keep the particles apart, and the potentialU serves to keep the particles from escaping to infinity. Our interest is in the behaviour as the number of particles increases without limit, which we study through the empirical measure process. We prove tightness of these processes in the case ofd=1,V(x)=–log|x|,U(x)=x ²/2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure-valued process exists. This process is deterministic, and converges to the Wigner law ast. Some information on the rates of convergence is derived, and the case of a Cauchy initial distribution is analysed completely.Supported by SERC grant number GR/H 00444     

On the time inhomogeneous skew Brownian motion

This paper is devoted to the construction of a solution for the “Inhomogeneous skew Brownian motion” equation, which first appeared in a seminal paper by Sophie Weinryb, and recently, studied by Étoré and Martinez. Our method is based on the use of the Balayage formula. At the end of this paper we study a limit theorem of solutions.     

Viability for differential equations driven by fractional Brownian motion

In this paper we prove a viability result for multidimensional, time dependent, stochastic differential equations driven by fractional Brownian motion with Hurst parameter , using pathwise approach. The sufficient condition is also an alternative global existence result for the fractional differential equations with restrictions on the state.     

Local independence of fractional Brownian motion

Let _σ(t,t′)${\sigma }_{(t,t^{\prime })}$ be the sigma-algebra generated by the differences _Xs−_Xs′$X_s-X_{s^{\prime }}$ with s,s^′∈(t,t^′)$s,s^{\prime }\in (t,t^{\prime })$, where _{(Xt)−∞<t<∞}${\left(X_t\right)}_{-\infty <t<\infty }$ is the fractional Brownian motion with Hurst index H∈(0,1)$H\in (0,1)$. We prove that for any two distinct timepoints _t1$t_1$ and _t2$t_2$ the sigma-algebras _{σ(t1−ε,t1+ε)}${\sigma }_{(t_1-\epsilon ,t_1+\epsilon )}$ and _{σ(t2−ε,t2+ε)}${\sigma }_{(t_2-\epsilon ,t_2+\epsilon )}$ are asymptotically independent as ε↘0$\epsilon \searrow 0$. We show the independence in the strong sense that Shannon’s mutual information between the two σ$\sigma$-algebras tends to zero as ε↘0$\epsilon \searrow 0$. Some generalizations and quantitative estimates are also provided.     

Uniqueness of Nelsons diffusions

Let ℒ≔Δ/2+(∇φ/φ) ·∇ be a generalized Schr?dinger operator or generator of Nelsons diffusion, defined on C ^∞ ₀(D) where φ is a continuous and strictly positive function on an open domain D⊂ℝ ^d such that ∇φ∈L _loc ²(D). Some results are given about the two questions below: (i) Whether does ℒ generate a unique semigroup in L ¹(D, φ² dx)? (ii) Whether the semigroup determined by ℒ is strong Feller? Received: 21 October 1997 / Revised version: 3 September 1998     

Self-attracting diffusions: Case of the constant interaction

Summary. In this paper we study a self-attracting diffusion in the case of a constant self-attraction and for dimension larger than two. We prove that this process converges almost surely. Received: 27 March 1995 / In revised form: 22 May 1996     

Regularity of the Local Time for the d-dimensional Fractional Brownian Motion with N-parameters

ABSTRACT

We give the Wiener–Ito? chaotic decomposition for the local time of the d-dimensional fractional Brownian motion with N-parameters and study its smoothness in the Sobolev–Watanabe spaces.     

Constructing hierarchical Archimedean copulas with Lévy subordinators

A probabilistic interpretation for hierarchical Archimedean copulas based on Lévy subordinators is given. Independent exponential random variables are divided by group-specific Lévy subordinators which are evaluated at a common random time. The resulting random vector has a hierarchical Archimedean survival copula. This approach suggests an efficient sampling algorithm and allows one to easily construct several new parametric families of hierarchical Archimedean copulas.     

On strong solutions for positive definite jump diffusions

We show the existence of unique global strong solutions of a class of stochastic differential equations on the cone of symmetric positive definite matrices. Our result includes affine diffusion processes and therefore extends considerably the known statements concerning Wishart processes, which have recently been extensively employed in financial mathematics.Moreover, we consider stochastic differential equations where the diffusion coefficient is given by the αth positive semidefinite power of the process itself with 0.5<α<1 and obtain existence conditions for them. In the case of a diffusion coefficient which is linear in the process we likewise get a positive definite analogue of the univariate GARCH diffusions.     

Laplace approximation of transition densities posed as Brownian expectations

We construct the Laplace approximation of the Lebesgue density for a discrete partial observation of a multi-dimensional stochastic differential equation. This approximation may be computed integrating systems of ordinary differential equations. The construction of the Laplace approximation begins with the definition of the point of minimum energy. We show how such a point can be defined in the Cameron–Martin space as a maximum a posteriori estimate of the underlying Brownian motion given the observation of a finite-dimensional functional. The definition of the MAP estimator is possible via a renormalization of the densities of piecewise linear approximations of the Brownian motion. Using the renormalized Brownian density the Laplace approximation of the integral over all Brownian paths can be defined. The developed theory provides a method for performing approximate maximum likelihood estimation.     

On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn

Let B₁, B₂, ... be a sequence of independent, identically distributed random variables, letX₀ be a random variable that is independent ofB_n forn?1, let ρ be a constant such that 0<ρ<1 and letX₁,X₂, ... be another sequence of random variables that are defined recursively by the relationshipsX_n=ρX_n-1+B_n. It can be shown that the sequence of random variablesX₁,X₂, ... converges in law to a random variableX if and only ifE[log⁺¦B₁¦]<∞. In this paper we let {B(t):0≦t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0?t<∞} that stands in the same relationship to the stochastic process {B(t):0?t<∞} as the sequence of random variablesX₁,X₂,...stands toB₁,B₂,.... It is shown thatX(t) converges in law to a random variableX ast →+∞ if and only ifE[log⁺¦B(1)¦]<∞ in which caseX has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals.     

Potential analysis for a class of diffusion equations: A Gaussian bounds approach

We axiomatically develop a potential analysis for a general class of hypoelliptic diffusion equations under the following basic assumptions: doubling condition and segment property for an underlying distance and Gaussian bounds of the fundamental solution. Our analysis is principally aimed to obtain regularity criteria and uniform boundary estimates for the Perron-Wiener solution to the Dirichlet problem. As an example of application, we also derive an exterior cone criterion of boundary regularity and scale-invariant Harnack inequality and Hölder estimate for an important class of operators in non-divergence form with Hölder continuous coefficients, modeled on Hörmander vector fields.     

A multivariate dispersion ordering based on quantiles more widely separated

A multivariate dispersion ordering based on quantiles more widely separated is defined. This new multivariate dispersion ordering is a generalization of the classic univariate version. If we vary the ordering of the components in the multivariate random variable then the comparison could not be possible. We provide a characterization using a multivariate expansion function. The relationship among various multivariate orderings is also considered. Finally, several examples illustrate the method of this paper.