In this paper we prove necessary conditions for optimality of a stochastic control problem for a class of stochastic partial differential equations that is controlled through the boundary. This kind of problems can be interpreted as a stochastic control problem for an evolution system in an Hilbert space. The regularity of the solution of the adjoint equation, that is a backward stochastic equation in infinite dimension, plays a crucial role in the formulation of the maximum principle.
In this paper, by virtue of Malliavin calculus, we establish a relationship between backward doubly stochastic differential equations with random coefficients and quasilinear stochastic PDEs, and thus extend the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [14] to non-Markovian case.
We prove an existence and uniqueness result for two-obstacle problem for quasilinear Stochastic PDEs (DOSPDEs for short). The method is based on the probabilistic interpretation of the solution by using the backward doubly stochastic differential equations (BDSDEs for short).
We introduce a class of second order backward stochastic differential equations and show relations to fully non-linear parabolic PDEs. In particular, we provide a stochastic representation result for solutions of such PDEs and discuss Monte Carlo methods for their numerical treatment.
It is known that Markovian forward-backward stochastic differential equations provide nonlinear Feynman-Kac representation formulae for semilinear parabolic PDEs. We show that non-Markovian forward-backward stochastic differential equations provide nonlinear Feynman-Kac formulae for semilinear path-dependent PDEs. This extends the result proved in Ekren, Keller, Touzi, and Zhang [4] to the case with a possibly degenerate diffusion coefficient in the forward dynamics.
We study the long time behavior of the solution of a stochastic PDEs with random coefficients assuming that randomness arises in a different independent scale. We apply the obtained results to 2D- Navier--Stokes equations.
In the article random functions in modules over the octonion algebra and Cayley-Dickson algebras are investigated. For their study transition measures with values in the octonion algebra and Cayley-Dickson algebras are used. Stochastic integrals over these algebras are studied. They are applied to integration of stochastic PDEs. This approach permits subsequently to analyze and integrate PDEs of orders higher than two of different types including parabolic, elliptic and hyperbolic.
We prove a maximum principle for local solutions of quasilinear stochastic PDEs with obstacle (in short OSPDE). The proofs are based on a version of Itô's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.
Connections between a system of Forward-Backward SDEs and Backward Stochastic PDEs related to the utility maximiza- tion problem is established. Besides, we derive another version of FBSDE of the same problem and prove an existence of a solution
We prove an existence and uniqueness result for quasilinear Stochastic PDEs with Obstacle (in short OSPDE) under a weaker integrability condition on the coefficient and the barrier.
We examine the convergence in the Krylov--Bogolyubov averaging for nonlinear stochastic perturbations of linear PDEs with pure imaginary spectrum and show that if the involved effective equation is mixing, then the convergence is uniform in time.
In this paper we study the relationship between functional forward-backward stochastic systems and path-dependent PDEs. In the framework of functional Itô calculus, we introduce a path-dependent PDE and prove that its solution is uniquely determined by a functional forward-backward stochastic system.
We show that first order semilinear PDEs by stochastic perturbation are well-posedness for globally Holder continuous and bounded vector field, with an integrability condition on the divergence. This result extends the liner case presented in [2]. The proof is based on in the stochastic characteristics method and a version of the commuting Lemma.
We study the spatial homogenisation of parabolic linear stochastic PDEs exhibiting a two-scale structure both at the level of the linear operator and at the level of the Gaussian driving noise. We show that in some cases, in particular when the forcing is given by space-time white noise, it may happen that the homogenised SPDE is not what one would expect from existing results for PDEs with more regular forcing terms.
In this article, we propose a wellposedness theory for a class of second order backward doubly stochastic differential equation (2BDSDE). We prove existence and uniqueness of the solution under a Lipschitz type assumption on the generator, and we investigate the links between our 2BDSDEs and a class of parabolic fully nonLinear Stochastic PDes. Precisely, we show that the Markovian solution of 2BDSDEs provide a probabilistic interpretation of the classical and stochastic viscosity solution of fully nonlinear SPDEs.
We determine with positive probability the Hausdorff dimension of the level sets of a class of Navier-Stokes α-models at finite viscosity, forced by mildly rough Gaussian white noise.
We develop a novel theory of weak and strong stochastic integration for cylindrical martingale-valued measures taking values in the dual of a nuclear space. This is applied to develop a theory of SPDEs with rather general coefficients. In particular, we can then study SPDEs driven by general Lévy processes in this context.
In this article we show that a finite dimensional stochastic differential equation driven by a Lévy process can be formulated as a stochastic partial differential equation. We prove the existence and uniqueness of strong solutions of such stochastic PDEs. The solutions that we construct have the `translation invariance' property. The special case of this correspondence for diffusion processes was proved in [Rajeev, Translation invariant diffusion in the space of tempered distributions, Indian J. Pure Appl. Math. 44 (2013), no.~2, 231--258].
In this paper we address again the problem of the connection between multitime Brownian sheet and heat type PDEs. The main results include: the volumetric character of the solutions of the forward (backward) diffusion-like PDEs; the forward mean value of a Brownian process as the solution of the forward heat PDE; the backward mean value of a Brownian process as the solution of the backward heat PDE; the multitime stochastic processes with volumetric dependence; the stochastic partial derivative of a stochastic process with respect to a multitime Wiener process; Hermite polynomials stochastic p…
In this paper, we present a sufficient condition for the large deviation criteria of Budhiraja, Dupuis and Maroulas for functionals of Brownian motions. We then establish a large deviation principle for obstacle problems of quasi-linear stochastic partial differential equations. It turns out that the backward stochastic differential equations will play an important role.
We prove the applicability of the Weighted Energy-Dissipation (WED) variational principle [50] to nonlinear parabolic stochastic partial differential equations in abstract form. The WED principle consists in the minimization of a parameter-dependent convex functional on entire trajectories. Its unique minimizers correspond to elliptic-in-time regularizations of the stochastic differential problem. As the regularization parameter tends to zero, solutions of the limiting problem are recovered. This in particular provides a direct approch via convex optimization to the approximation of nonlinear …
We prove the existence and uniqueness of solution of quasilinear stochastic partial differential equations with obstacle (OSPDEs in short) in degenerate case. Using De Giorgi's iteration, we deduce the $L^p-$estimates for the time-space uniform norm of weak solutions.
This paper explores the relationship between non-Markovian fully coupled forward-backward stochastic systems and path-dependent PDEs. The definition of classical solution for the path-dependent PDE is given within the framework of functional Itô calculus. Under mild hypotheses, we prove that the forward-backward stochastic system provides the unique classical solution to the path-dependent PDE.
We define multiple stochastic integrals with respect to càdlàg martingales and prove moment bounds and chaos expansions, which allow to work with them in a way similar to Wiener stochastic integrals. In combination with the discretization framework of Erhard and Hairer (2017), our results give a tool for proving convergence of interacting particle systems to stochastic PDEs using regularity structures. As examples, we prove convergence of martingale-driven discretizations of the $3$-dimensional stochastic quantization equation and the KPZ equation.
We prove a version of the Wong-Zakai theorem for one-dimensional parabolic nonlinear stochastic PDEs driven by space-time white noise. As a corollary, we obtain a detailed local description of solutions. Dedicated to the memory of Kiyosi Itô on the occasion of the 100th anniversary of his birth.
In this paper, we first explore certain structural properties of Lévy flows and use this information to obtain the existence of strong solutions to a class of Stochastic PDEs in the space of tempered distributions, driven by Lévy noise. The uniqueness of the solutions follows from Monotonicity inequality. These results extend an earlier work Bhar (2017) on the diffusion case.
We prove that a version of Smagorinsky Large Eddy model for a 2D fluid in vorticity form is the scaling limit of suitable stochastic models for large scales, where the influence of small turbulent eddies is modeled by a transport type noise.
Considering stochastic partial differential equations of parabolic type with random coefficients in vector-valued Hölder spaces, we obtain a sharp Schauder estimate. As an application, the existence and uniqueness of solution to the Cauchy problem is also proved.
Stochastic PDEs are ubiquitous in mathematical modeling. Yet, many such equations are too singular to admit classical treatment. In this article we review some recent progress in defining, approximating and studying the properties of a few examples of such equations. We focus mainly on the dynamical Phi4 equation, KPZ equation and Parabolic Anderson Model, as well as touch on a few other equations which arise mainly in physics.
We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $Φ^4_3$ model. As a corollary, we prove that the Brownian bridge measure is the unique invariant measure for the KPZ equation with periodic boundary conditions.
In this paper, we establish an existence and uniqueness result for system of quasilinear stochastic partial differential equations (SPDEs for short) with reflection in a convex domain in R^k by analytical approach. The method is based on the approximation of the penalized systems of SPDEs.
We consider a system of interacting diffusions on the integer lattice. By letting the mesh size go to zero and by using a suitable scaling, we show that the system converges (in a strong sense) to a solution of the stochastic heat equation on the real line. As a consequence, we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities.
The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. We review in this paper the main ideas a…
The optimal stochastic control problem with a quadratic cost functional for linear partial differential equations (PDEs) driven by a state-and control-dependent white noise is formulated and studied. Both finite-and infinite-time horizons are considered. The multi-plicative white noise dynamics of the system give rise to a new phenomenon of singularity to the associated Riccati equation and even to the Lyapunov equation. Well-posedness of both Riccati equation and Lyapunov equation are obtained for the first time. The linear feedback coefficient of the optimal control turns out to be singular …
This paper is devoted to the study of the differentiability of solutions to real-valued backward stochastic differential equations (BSDEs for short) with quadratic generators driven by a cylindrical Wiener process. The main novelty of this problem consists in the fact that the gradient equation of a quadratic BSDE has generators which satisfy stochastic Lipschitz conditions involving BMO martingales. We show some applications to the nonlinear Kolmogorov equations.
This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain Hölder-type classes in which a random field is treated as a space-time function taking values in $L^{p}$-space of random variables. A modified stochastic parabolicity condition involving $p$ is proposed to ensure the finiteness of the associated norm of the solution, which is showed to be sharp by examples. The Schauder-type estimates and the solvability theorem are proved.
These are the notes for a course at the 18th Brazilian School of Probability held from August 3rd to 9th, 2014 in Mambucaba. The aim of the course is to introduce the basic problems of non--linear PDEs with stochastic and irregular terms. We explain how it is possible to handle them using two main techniques: the notion of energy solutions and that of paracontrolled distributions. In order to maintain a link with physical intuitions, we motivate such singular SPDEs via a homogenisation result for a diffusion in a random potential.
We consider parabolic PDEs with randomly switching boundary conditions. In order to analyze these random PDEs, we consider more general stochastic hybrid systems and prove convergence to, and properties of, a stationary distribution. Applying these general results to the heat equation with randomly switching boundary conditions, we find explicit formulae for various statistics of the solution and obtain almost sure results about its regularity and structure. These results are of particular interest for biological applications as well as for their significant departure from behavior seen in PDE…
We study utility maximization problem for general utility functions using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are described by an $R^d$-valued continuous semimartingale. Under some regularity assumptions we derive backward stochastic partial differential equation (BSPDE) related directly to the primal problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As examples the cases of power, exponential and logarithmic utilities are considered.
We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with non-homogeneous second order operator. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair $(u,ν)$ where $u$ is a predictable continuous process which takes values in a proper Sobolev space and $ν$ is a random regular measure satisfying minimal Skohorod condition. Moreover, we establish a maximum principle for local solutions of such class of stochastic PDEs. The proofs are based on a version of Itô's formula and e…
In this paper, a class of non-Markovian forward-backward doubly stochastic systems is studied. By using the technique of functional Itô (or path-dependent) calculus, the relationship between the systems and related path-dependent quasi-linear stochastic partial differential equations (SPDEs in short) is established, and the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Relat. Fields 98 (1994), pp. 209--227] is developed to the non-Markovian situation. Moreover, we obtai…
We consider the problem of estimating stochastic volatility for a class of second-order parabolic stochastic PDEs. Assuming that the solution is observed at a high temporal frequency, we use limit theorems for multipower variations and related functionals to construct consistent nonparametric estimators and asymptotic confidence bounds for the integrated volatility process. As a byproduct of our analysis, we also obtain feasible estimators for the regularity of the spatial covariance function of the noise.
We consider parabolic stochastic partial differential equations driven by white noise in time. We prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions. These conditions amount essentially to the fact that the equation transmits the noise to all its determining modes. Several examples are investigated, including some where the noise does not act on every determining mode directly.
In this paper, we study intermittency properties for various stochastic PDEs with varieties of space time Gaussian noises via matching upper and lower moment bounds of the solution. Due to the absence of the powerful Feynman Kac formula, the lower moment bounds have been missing for many interesting equations except for the stochastic heat equation. This work introduces and explores the Feynman diagram formula for the moments of the solution and the small ball nondegeneracy for the Green's function to obtain the lower bounds for all moments which match the upper moment bounds. Our upper and lo…
The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly - and in a sense, arbitrarily - bad: as shown by Krylov, for any $α>0$ one can find a simple $1$-dimensional constant coefficient linear equation whose solution at the boundary is not $α$-Hölder continuous. We obtain a positive counterpart of this: under some mild regularity assumptions on the coefficients, solutions of semilinear SPDEs on $C^1$ domains are proved to be $α$-Hölder continuous up to the boundary with some $α>0$.
This paper presents existence and uniqueness results for reflected system of quasilinear stochastic partial differential equations in a convex domain D from Rk. The method is based on the probabilistic interpretation of the solution by using the backward doubly stochastic differential equation. The solution is expressed as a pair (u,ν) where u is a predictable continuous process which takes values in a proper Sobolev space and ν is a random signed regular measure satisfying the minimal Skohorod condition.
We prove an existence and uniqueness result for quasilinear Stochastic PDEs with obstacle (OSPDE in short). Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair $(u,ν)$ where $u$ is a predictable continuous process which takes values in a proper Sobolev space and $ν$ is a random regular measure satisfying the minimal Skohorod condition.
Reflected generalized BDSDEs driven by non-homogeneous Lévy processes and obstacle problems for stochastic integro-PDEs with nonlinear Neumann boundary conditions
Stemmed from the derivation of the optimal control to a stochastic linear-quadratic control problem with Markov jumps, we study one kind of backward stochastic differential equations (BSDEs) that the generator f is affected by a Markovian switching. Then, the case that the Markov chain is involved in a large state space is considered. Following the classical approach, a hierarchical approach is adopted to reduce the complexity and a singularly perturbed Markov chain is involved. We will study the asymptotic property of BSDE with the singularly perturbed Markov chain. At last, as an application…
