Forward-backward stochastic differential equations and quasilinear parabolic PDEs

This paper studies, under some natural monotonicity conditions, the theory (existence and uniqueness, a priori estimate, continuous dependence on a parameter) of forward–backward stochastic differential equations and their connection with quasilinear parabolic partial differential equations. We use a purely probabilistic approach, and allow the forward equation to be degenerate. Received: 12 May 1997 / Revised version: 10 January 1999     

On the cohomology of flows of stochastic and random differential equations

We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise. Received: 25 May 1999 / Revised version: 19 October 2000 / Published online: 26 April 2001     

Stationary backward stochastic differential equations and associated partial differential equations

By replacing the final condition for backward stochastic differential equations (in short: BSDEs) by a stationarity condition on the solution process we introduce a new class of BSDEs. In a natural manner we associate to such BSDEs the periodic solution of second order partial differential equations with periodic structure. Received: 11 October 1996 / Revised version: 15 February 1999     

Finding adapted solutions of forward–backward stochastic differential equations: method of continuation

Summary. The notion of bridge is introduced for systems of coupled forward–backward stochastic differential equations (FBSDEs, for short). This notion helps us to unify the method of continuation in finding adapted solutions to such FBSDEs over any finite time durations. It is proved that if two FBSDEs are linked by a bridge, then they have the same unique solvability. Consequently, by constructing appropriate bridges, we obtain several classes of uniquely solvable FBSDEs. Received: 23 April 1996 / In revised form: 10 October 1996     

Some large deviation results for sparse random graphs

Summary. We obtain a large deviation principle (LDP) for the relative size of the largest connected component in a random graph with small edge probability. The rate function, which is not convex in general, is determined explicitly using a new technique. The proof yields an asymptotic formula for the probability that the random graph is connected. We also present an LDP and related result for the number of isolated vertices. Here we make use of a simple but apparently unknown characterisation, which is obtained by embedding the random graph in a random directed graph. The results demonstrate that, at this scaling, the properties `connected' and `contains no isolated vertices' are not asymptotically equivalent. (At the threshold probability they are asymptotically equivalent.) Received: 14 November 1996 / In revised form: 15 August 1997     

One-sided local large deviation and renewal theorems in the case of infinite mean

Summary. If {S _n ,n≧0} is an integer-valued random walk such that S _n /a _n converges in distribution to a stable law of index α∈ (0,1) as n→∞, then Gnedenko’s local limit theorem provides a useful estimate for P{S _n =r} for values of r such that r/a _n is bounded. The main point of this paper is to show that, under certain circumstances, there is another estimate which is valid when r/a _n → +∞, in other words to establish a large deviation local limit theorem. We also give an asymptotic bound for P{S _n =r} which is valid under weaker assumptions. This last result is then used in establishing some local versions of generalized renewal theorems. Received: 9 August 1995 / In revised form: 29 September 1996     

Stochastic functional differential equations on manifolds

In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters. Received: 6 July 1999 / Revised version: 19 April 2000 /?Published online: 14 June 2001     

Normal forms for stochastic differential equations

Summary.   We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dx _t = ∑ _{j =0} ^m f _j (x _t )∘dW _t ^j and dx _t =∑ _{j =0} ^m g _j (x _t )∘dW _t ^j in ℝ ^d with smooth coefficients satisfying f _j (0)=g _j (0)=0 are said to be smoothly equivalent if there is a smooth random diffeomorphism (coordinate transformation) h(ω) with h(ω,0)=0 and Dh(ω,0)=id which conjugates the corresponding local flows,

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     

Extending the Wong-Zakai theorem to reversible Markov processes

We show how to construct a canonical choice of stochastic area for paths of reversible Markov processes satisfying a weak H?lder condition, and hence demonstrate that the sample paths of such processes are rough paths in the sense of Lyons. We further prove that certain polygonal approximations to these paths and their areas converge in p-variation norm. As a corollary of this result and standard properties of rough paths, we are able to provide a significant generalization of the classical result of Wong-Zakai on the approximation of solutions to stochastic differential equations. Our results allow us to construct solutions to differential equations driven by reversible Markov processes of finite p-variation with p<4. Received May 18, 2001 / final version received April 3, 2001?Published online April 8, 2002     

Exact capacity results for stable processes

Exact results are proved for the capacity of pullbacks of analytic sets by stable processes. Received: 25 May 1988 / Revised version: 15 September 1997     

On linear, degenerate backward stochastic partial differential equations

In this paper we study the well-posedness and regularity of the adapted solutions to a class of linear, degenerate backward stochastic partial differential equations (BSPDE, for short). We establish new a priori estimates for the adapted solutions to BSPDEs in a general setting, based on which the existence, uniqueness, and regularity of adapted solutions are obtained. Also, we prove some comparison theorems and discuss their possible applications in mathematical finance. Received: 24 September 1997 / Revised version: 3 June 1998     

Asymptotic expansions in limit theorems for stochastic processes. II

. For a certain class of families of stochastic processes η^ε(t), 0≤t≤T, constructed starting from sums of independent random variables, limit theorems for expectations of functionals F(η^ε[0,T]) are proved of the form

Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: Stratonovitch's case

In this article we prove new results concerning the long-time behavior of random fields that are solutions in some sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state that these random fields eventually homogeneize with respect to the spatial variable and finally converge to a non-random global attractor which consists of two spatially and temporally homogeneous asymptotic states. More precisely, we prove that the random fields either stabilize exponentially rapidly with probability one around one of the asymptotic states, or that they set out to oscillate between them. In the first case we can also determine exactly the corresponding Lyapunov exponents. In the second case we prove that the random fields are in fact recurrent in that they can reach every point between the two asymptotic states in a finite time with probability one. In both cases we also interpret our results in terms of stability properties of the global attractor and we provide estimates for the average time that the random fields spend in small neighborhoods of the asymptotic states. Our methods of proof rest upon the use of a suitable regularization of the Brownian motion along with a related Wong-Zaka? approximation procedure. Received: 8 April 1997/Revised version: 30 January 1998     

Measure concentration for a class of random processes

Summary.   Let X={X _i } _{i =−∞} ^∞ be a stationary random process with a countable alphabet and distribution q. Let q ^∞(·|x _{− k} ⁰) denote the conditional distribution of X ^∞=(X ₁,X ₂,…,X _n ,…) given the k-length past:

On a class of stochastic partial differential equations related to turbulent transport

Summary. We consider the Cauchy problem for the mass density ρ of particles which diffuse in an incompressible fluid. The dynamical behaviour of ρ is modeled by a linear, uniformly parabolic differential equation containing a stochastic vector field. This vector field is interpreted as the velocity field of the fluid in a state of turbulence. Combining a contraction method with techniques from white noise analysis we prove an existence and uniqueness result for the solution ρ∈C ^1,2([0,T]×ℝ ^d ,(S)^*), which is a generalized random field. For a subclass of Cauchy problems we show that ρ actually is a classical random field, i.e. ρ(t,x) is an L ²-random variable for all time and space parameters (t,x)∈[0,T]×ℝ ^d . Received: 27 March 1995 / In revised form: 15 May 1997     

Successive approximations of solutions to stochastic functional differential equations

In the present paper, by means of the successive approximations method, the local or global existence and uniqueness theorems for a stochastic functional differential equation of the Ito type are proved.     

A super-Brownian motion with a locally infinite catalytic mass

Summary. A super-Brownian motion in with “hyperbolic” branching rate , is constructed, which symbolically could be described by the formal stochastic equation (with a space-time white noise ). Starting at this superprocess will never hit the catalytic center: There is an increasing sequence of Brownian stopping times strictly smaller than the hitting time of such that with probability one Dynkin's stopped measures vanish except for finitely many Received: 27 November 1995 / In revised form: 24 July 1996     

Rescaled contact processes converge to super-Brownian motion in two or more dimensions

We show that in dimensions two or more a sequence of long range contact processes suitably rescaled in space and time converges to a super-Brownian motion with drift. As a consequence of this result we can improve the results of Bramson, Durrett, and Swindle (1989) by replacing their order of magnitude estimates of how close the critical value is to 1 with sharp asymptotics. Received: 2 February 1998 / Revised version: 28 August 1998     

On the estimation of parameters for linear stochastic differential equations

The estimation of arbitrary number of parameters in linear stochastic differential equation (SDE) is investigated. The local asymptotic normality (LAN) of families of distributions corresponding to this SDE is established and the asymptotic efficiency of the maximum likelihood estimator (MLE) is obtained for the wide class of loss functions with polynomial majorants. As an example a single-degree of freedom mechanical system is considered. The results generalize [8], where all elements of the drift matrix are estimated and the asymptotic efficiency is proved only for the bounded loss functions. Received: 12 March 1997 / Revised version: 22 June 1998