Stochastic Analysis and Applications 2014

The strong convergence of Wong-Zakai approximations of the solution to the reflecting stochastic differential equations was studied in [2]. We continue the study and prove the strong convergence under weaker assumptions on the domain.

We describe symmetric diffusion operators where the spectral decomposition is given through a family of orthogonal polynomials. In dimension one, this reduces to the case of Hermite, Laguerre and Jacobi polynomials. In higher dimension, some basic examples arise from compact Lie groups. We give a complete description of the bounded sets on which such operators may live. We then provide in dimension \(2\) a classification of those sets when the polynomials are ordered according to their usual degrees.

We comment on old and new results related to the destruction of a random recursive tree (RRT), in which its edges are cut one after the other in a uniform random order. In particular, we study the number of steps needed to isolate or disconnect certain distinguished vertices when the size of the tree tends to infinity. New probabilistic explanations are given in terms of the so-called cut-tree and the tree of component sizes, which both encode different aspects of the destruction process. Finally, we establish the connection to Bernoulli bond percolation on large RRT’s and present recent results on the cluster sizes in the supercritical regime.

We use a simple \(N\)-player stochastic game with idiosyncratic and common noises to introduce the concept of Master Equation originally proposed by Lions in his lectures at the Collège de France. Controlling the limit \(N\rightarrow \infty \) of the explicit solution of the \(N\)-player game, we highlight the stochastic nature of the limit distributions of the states of the players due to the fact that the random environment does not average out in the limit, and we recast the Mean Field Game (MFG) paradigm in a set of coupled Stochastic Partial Differential Equations (SPDEs). The first one is a forward stochastic Kolmogorov equation giving the evolution of the conditional distributions of the states of the players given the common noise. The second is a form of stochastic Hamilton Jacobi Bellman (HJB) equation providing the solution of the optimization problem when the flow of conditional distributions is given. Being highly coupled, the system reads as an infinite dimensional Forward Backward Stochastic Differential Equation (FBSDE). Uniqueness of a solution and its Markov property lead to the representation of the solution of the backward equation (i.e. the value function of the stochastic HJB equation) as a deterministic function of the solution of the forward Kolmogorov equation, function which is usually called the decoupling field of the FBSDE. The (infinite dimensional) PDE satisfied by this decoupling field is identified with the master equation. We also show that this equation can be derived for other large populations equilibriums like those given by the optimal control of McKean-Vlasov stochastic differential equations. The paper is written more in the style of a review than a technical paper, and we spend more time motivating and explaining the probabilistic interpretation of the Master Equation, than identifying the most general set of assumptions under which our claims are true.

The problem of nonlinear filtering has engendered a surprising number of mathematical techniques for its treatment. A notable example is the change-of–probability-measure method introduced by Kallianpur and Striebel to derive the filtering equations and the Bayes-like formula that bears their names. More recent work, however, has generally preferred other methods. In this paper, we reconsider the change-of-measure approach to the derivation of the filtering equations and show that many of the technical conditions present in previous work can be relaxed. The filtering equations are established for general Markov signal processes that can be described by a martingale-problem formulation. Two specific applications are treated.

In this paper we extend the class of stochastic processes allowed to represent solutions of the Navier-Stokes equation on the two dimensional torus to certain non-Markovian processes which we call admissible. More precisely, using the variations of Ref. [3], we provide a criterion for the associated mean velocity field to solve this equation. Due to the fluctuations of the shift a new term of pressure appears which is of purely stochastic origin. We provide an alternative formulation of this least action principle by means of transformations of measure. Within this approach the action is a function of the law of the processes, while the variations are induced by some translations on the space of the divergence free vector fields. Due to the renormalization in the definition of the cylindrical Brownian motion, our action is only related to the relative entropy by an inequality. However we show that, if we cut the high frequency modes, this new approach provides a least action principle for the Navier-Stokes equation based on the relative entropy.

The dyadic method of Komlós, Major and Tusnády is a powerful way of constructing simultaneous normal approximations to a sequence of partial sums of i.i.d. random variables. We use a version of this KMT method to obtain order 1 approximation in a Vaserstein metric to solutions of vector SDEs under a mild non-degeneracy condition using an easily implemented numerical scheme.

Partial differential equations driven by rough paths are studied. We return to the investigations of [Caruana, Friz and Oberhauser: A (rough) pathwise approach to a class of non- linear SPDEs, Annales de l’Institut Henri Poincaré/Analyse Non Linéaire 2011, 28, pp. 27–46], motivated by the Lions–Souganidis theory of viscosity solutions for SPDEs. We continue and complement the previous (uniqueness) results with general existence and regularity statements. Much of this is transformed to questions for deterministic parabolic partial differential equations in viscosity sense. On a technical level, we establish a refined parabolic theorem of sums which may be useful in its own right.

The explicit results for the classical Merton optimal investment/consumption problem rely on the use of constant risk aversion parameters and exponential discounting. However, many studies have suggested that individual investors can have different risk aversions over time, and they discount future rewards less rapidly than exponentially. While state-dependent risk aversions and non-exponential type (e.g. hyperbolic) discountings align more with the real life behavior and household consumption data, they have tractability issues and make the problem time-inconsistent. We analyze the cases where these problems can be closely approximated by time-consistent ones. By asymptotic approximations, we are able to characterize the equilibrium strategies explicitly in terms of the corrections to solutions for the base problems with constant risk aversion and exponential discounting. We also explore the effects of hyperbolic discounting under proportional transaction costs.

Results by Cranston, Greven, and Feng-Yu Wang, on relationships between coupling and shift coupling, and harmonic functions and space time harmonic functions are reviewed. These lead to extensions of a result by Freire on the separate harmonicity of bounded harmonic functions on certain product manifolds. The extensions are to situations where a diffusion operator is decomposed into the sum of two other commuting diffusion operators. This is shown to arise for a class of foliated Riemannian manifolds with totally geodesic leaves. A form of skew product decomposition of Brownian motions on these foliated manifolds is obtained, as are gradient estimates in leaf directions. Relationships between stochastic completeness of the manifold itself and stochastic completeness of its leaves are established. Baudoin and Garafola’s “sub-Riemannian manifolds with transverse symmetries” are shown to be examples.

We study a mathematical consistency problem motivated by the interplay between local and global risk assessment in a large financial network. In analogy to the theory of Gibbs measures in Statistical Mechanics, we focus on the structure of global convex risk measures which are consistent with a given family of local conditional risk measures. Going beyond the locally law-invariant (and hence entropic) case studied in [11], we show that a global risk measure can be characterized by its behavior on a suitable boundary field. In particular, a global risk measure may not be uniquely determined by its local specification, and this can be seen as a source of “systemic risk”, in analogy to the appearance of phase transitions in the theory of Gibbs measures. The proof combines the spatial version [10] of Dynkin’s method for constructing the entrance boundary of a Markov process with the non-linear extension [14] of backwards martingale convergence.

The classical Loewner differential equation for simply connected domains is attracting new attention since Oded Schramm launched in 2000 the stochastic Loewner evolution (SLE) based on it. The Loewner equation itself has been extended to various canonical domains of multiple connectivity after the works by Y. Komatu in 1943 and 1950, but the Komatu-Loewner (K-L) equations have been derived rigorously only in the left derivative sense. In a recent work, Z.-Q. Chen, M. Fukushima and S. Rhode prove that the K-L equation for the standard slit domain is a genuine ODE by using a probabilistic method together with a PDE method, and that the right hand side of the equation admits an expression in terms of the complex Poisson kernel of the Brownian motion with darning (BMD). In the present paper, K-L equations for the annulus and circularly slit annili are investigated. For the annulus, we establish a K-L equation as a genuine ODE possessing a normalized Villat’s kernel on its right hand side by using a variant of the Carathéodory convergence theorem for annuli indicated by Komatu. This method is also used to obtain the same K-L equation in the right derivative sense on annulus for a more general family of growing hulls that satisfies a specific right continuity condition usually adopted in the SLE theory. Villat’s kernel is then identified with a BMD Schwarz kernel for the annulus. Finally we derive K-L equations for circularly slit annuli in terms of their normalized BMD Schwarz kernels, but only in the left derivative sense when at least one circular slit is present.

The Skorokhod reflection of a continuous semimartingale is unfolded, in a possibly skewed manner, into another continuous semimartingale on an enlarged probability space according to the excursion-theoretic methodology of [14]. This is done in terms of a skew version of the Tanaka equation, whose properties are studied in some detail. The result is used to construct a system of two diffusive particles with rank-based characteristics and skew-elastic collisions. Unfoldings of conventional reflections are also discussed, as are examples involving skew Brownian Motions and skew Bessel processes.

The purpose of this note is to survey some recent developments in the applications of Malliavin calculus combined with Stein’s method to derive central limit theorems for random variables on a finite sum of Wiener chaos. Starting from the fourth moment theorem by Nualart and Peccati [23], we will discuss several related topics such as conditions for the convergence in total variation, absolute continuity of probability laws and uniform convergence of densities under suitable non degeneracy assumptions. The fact that the random variables belong to a fixed Wiener chaos (or to a finite sum of Wiener chaos) will play a fundamental role in the results.

This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial differential equations. We start by a quick review of the Crandall-Ishii notion of viscosity solutions, so as to motivate the relevance of our definition in the path-dependent case. We focus on the wellposedness theory of such equations. In particular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, including the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12].

We consider the family of stochastic partial differential equations indexed by a parameter \(\varepsilon \in (0,1]\), \((t,x)\in (0,T]\times {\mathbb {R}^d}\) with suitable initial conditions. In this equation, \(L\) is a second-order partial differential operator with constant coefficients, \(\sigma \) and \(b\) are smooth functions and \(\dot{F}\) is a Gaussian noise, white in time and with a stationary correlation in space. Let \(p^\varepsilon _{t,x}\) denote the density of the law of \(u^\varepsilon (t,x)\) at a fixed point \((t,x)\in (0,T]\times {\mathbb {R}^d}\). We study the existence of \(\lim _{\varepsilon \downarrow 0} \varepsilon ^2\log p^\varepsilon _{t,x}(y)\) for a fixed \(y\in \mathbb {R}\). The results apply to classes of stochastic wave equations with \(d\in \{1,2,3\}\) and stochastic heat equations with \(d\ge 1\).