The paper concerns classical solution of path-dependent partial differential equations (PPDEs) with coefficients depending on both variables of path and path-valued measure, which are crucial to understanding large-scale mean-field interacting systems in a non-Markovian setting. We construct classical solutions of the PPDEs via solution of the forward and backward stochastic differential equations. To accommodate the intricacies introduced by the appearance of the path in the coefficients, we develop a novel technique known as the ``parameter frozen'' approach to the PPDEs.
The aim of the present work is the introduction of a viscosity type solution, called strong-viscosity solution to distinguish it from the classical one, with the following peculiarities: it is a purely analytic object; it can be easily adapted to more general equations than classical partial differential equations. First, we introduce the notion of strong-viscosity solution for semilinear parabolic partial differential equations, defining it, in a few words, as the pointwise limit of classical solutions to perturbed semilinear parabolic partial differential equations; we compare it with the st…
In insurance mathematics optimal control problems over an infinite time horizon arise when computing risk measures. Their solutions correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In this paper we propose a deep neural network algorithm for solving such partial differential equations in high dimensions. The algorithm is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with random terminal time.
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 and compare two concepts for weak solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). The first is that of mild solutions as it appears, e.g., in the log-Laplace functionals of historical superprocesses. The aim of this paper is to show that mild solutions are also solutions in a viscosity sense. This result is motivated by the fact that mild solutions can provide value functions and optimal strategies for problems of stochastic optimal control. Since unique mild solutions exist under weak conditions, we obtain as a corollary a general existence re…
We propose a fully backward representation of semilinear PDEs with application to stochastic control. Based on this, we develop a fully backward Monte-Carlo scheme allowing to generate the regression grid, backwardly in time, as the value function is computed. This offers two key advantages in terms of computational efficiency and memory. First, the grid is generated adaptively in the areas of interest and second, there is no need to store the entire grid. The performances of this technique are compared in simulations to the traditional Monte-Carlo forward-backward approach on a control proble…
In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic partial differential equations. Our approach is probabilistic. The theory of Dirichlet processes and backward stochastic differential equations play a crucial role.
In this paper, we prove that there exists a unique weak solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms and backward stochastic differential equations with singular coefficients and infinite horizon plays a crucial role.
This paper proves the existence of viscosity solutions of path dependent semilinear PDEs via Perron's method, i.e. via showing that the supremum of viscosity subsolutions is a viscosity solution. We use the notion of viscosity solutions introduced by Ekren, Keller, Touzi and Zhang, in whose work all smooth processes which are tangent in mean are considered as test functions. We also provide a comparison result for semicontinuous viscosity solutions, by using a regularization technique. As an interesting byproduct, we give a new short proof for the optimal stopping problem with semicontinuous o…
The classical Feynman-Kac identity builds a bridge between stochastic analysis and partial differential equations (PDEs) by providing stochastic representations for classical solutions of linear Kolmogorov PDEs. This opens the door for the derivation of sampling based Monte Carlo approximation methods, which can be meshfree and thereby stand a chance to approximate solutions of PDEs without suffering from the curse of dimensionality. In this article we extend the classical Feynman-Kac formula to certain semilinear Kolmogorov PDEs. More specifically, we identify suitable solutions of stochastic…
We consider the Cauchy problem for semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the eqution and mild conditions on the obstacle a unique continuous solution of the problem admits a stochastic representation in terms of reflected backward stochastic differential equations. We derive also some regularity properties of so- lutions and prove useful approximation results.
We obtain existence results for the solution u of nonlocal semilinear parabolic PDEs on $\mathbb{R}^d$ with polynomial nonlinearities in $(u, \nabla u)$, using a tree-based probabilistic representation. This probabilistic representation applies to the solution of the equation itself, as well as to its partial derivatives by associating one of d marks to the initial tree branch. Partial derivatives are dealt with by integration by parts and subordination of Brownian motion. Numerical illustrations are provided in examples for the fractional Laplacian in dimension up to 10, and for the fractiona…
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.
Explicit stabilized integrators are an efficient alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit integrators applied to stiff diffusion problems. In this paper, we provide a fully discrete strong convergence analysis of a family of explicit stabilized methods coupled with finite element methods for a class of parabolic semilinear deterministic and stochastic partial differential equations. Numerical experiments including the semilinear stochastic heat equation with space-time white noise confirm the theoretical findings.
Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear elliptic PDEs with Lipschitz nonlinearities and to prove that the proposed approximation schemes overcome the curse of dimensionality in the numerical approximation of such semilinear e…
This paper presents existence and uniqueness results for reflected backward doubly stochastic differential equations (in short RBDDSEs) in a convex domain D. Moreover, using a stochastic flow approach a probabilistic interpretation for a class of reflected SPDE's in a domain is given via such RBDSDEs. The solution is expressed as a pair (u,ν) where u is a predictable continuous process which takes values in a Sobolev space and m is a random regular measure. The bounded variation process K, component of the solution of the reflected BDSDE, controls the set when u reaches the boundary of D. This…
We consider a system of semilinear partial differential equations (PDEs) with a nonlinearity depending on both the solution and its gradient. The Neumann boundary condition depends on the solution in a nonlinear manner. The uniform ellipticity is not required to the diffusion coefficient. We show that this problem admits a viscosity solution which can be approximated by a penalization. The Lipschitz condition is required to the coefficients of the diffusion part. The nonlinear part as well as the Neumann condition are Lipschitz. Moreover, the nonlinear part is assumed monotone in the solution …
The recently introduced full-history recursive multilevel Picard (MLP) approximation methods have turned out to be quite successful in the numerical approximation of solutions of high-dimensional nonlinear PDEs. In particular, there are mathematical convergence results in the literature which prove that MLP approximation methods do overcome the curse of dimensionality in the numerical approximation of nonlinear second-order PDEs in the sense that the number of computational operations of the proposed MLP approximation method grows at most polynomially in both the reciprocal $1/ε$ of the prescr…
The full history recursive multilevel Picard approximation method for semilinear parabolic partial differential equations (PDEs) is the only method which provably overcomes the curse of dimensionality for general time horizons if the coefficient functions and the nonlinearity are globally Lipschitz continuous and the nonlinearity is gradient-independent. In this article we extend this result to locally monotone coefficient functions. Our results cover a range of semilinear PDEs with polynomial coefficient functions.
For a long time it is well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semi…
Multilevel Picard approximations overcome the curse of dimensionality in the numerical approximation of general semilinear PDEs with gradient-dependent nonlinearities
We investigate existence, uniqueness and regularity for local solutions of rough parabolic equations with subcritical noise of the form $du_t- L_tu_tdt= N(u_t)dt + \sum_{i = 1}^dF_i(u_t)d\mathbf X^i_t$ where $(L_t)_{t\in[0,T]}$ is a time-dependent family of unbounded operators acting on some scale of Banach spaces, while $\mathbf X\equiv(X,\mathbb X)$ is a two-step (non-necessarily geometric) rough path of Hölder regularity $γ>1/3.$ Besides dealing with non-autonomous evolution equations, our results also allow for unbounded operations in the noise term (up to some critical loss of regularity …
In this paper a semilinear elliptic PDE with rapidly oscillating coefficients is homogenized. The novetly of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some portion of $\R^d$.\\ Our fully probabilistic method is based on the connection between PDEs and BSDEs with random terminal time and the weak convergence of a class of diffusion processes.
We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hormander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operator $M_t$ can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection $Π$ on a subspace of sufficiently regular functions. Then the eigenfunctions of $M_t$ with small eigenvalues have only a very sma…
We study the homogenization problem of semi linear reflected partial differential equations (reflected PDEs for short) with nonlinear Neumann conditions. The non-linear term is a function of the solution but not of its gradient. The proof are fully probabilistic and uses weak convergence of associated reflected generalized backward differential stochastic equations (reflected GBSDEs in short).
We deal with a class of semilinear parabolic PDEs on the space of continuous functions that arise, for example, as Kolmogorov equations associated to the infinite-dimensional lifting of path-dependent SDEs. We investigate existence of smooth solutions through their representation via forward-backward stochastic systems, for which we provide the necessary regularity theory. Because of the lack of smoothing properties of the parabolic operators at hand, solutions in general will only share the same regularity as the coefficients of the equation. To conclude we exhibit an application to Hamilton-…
We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator $L$ has a generalized drift. We investigate existence and uniqueness of generalized solutions of class $C^1$. The generator $L$ is associated with a Markov process $X$ which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward pr…
Relying on the classical connection between Backward Stochastic Differential Equations (BSDEs) and non-linear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional semi-linear parabolic PDEs. This scheme is inspired by the approach coming from machine learning and developed using deep neural networks in Han and al. [32]. Our algorithm is based on a Picard iteration scheme in which a sequence of linear-quadratic optimisation problem is solved by means of stochastic gradient descent (SGD) algorithm. In the framework of a line…
We propose a modification of the standard linear implicit Euler integrator for the weak approximation of parabolic semilinear stochastic PDEs driven by additive space-time white noise. The new method can easily be combined with a finite difference method for the spatial discretization. The proposed method is shown to have improved qualitative properties compared with the standard method. First, for any time-step size, the spatial regularity of the solution is preserved, at all times. Second, the proposed method preserves the Gaussian invariant distribution of the infinite dimensional Ornstein-…
First, we revisit functional Itô/path-dependent calculus started by B. Dupire, R. Cont and D.-A. Fournié, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus introduced by C. Di Girolami and the second named author are explored. The second part of the paper is devoted to the study of the Kolmogorov type equation associated with the so called window Brownian motion, called path-dependent heat equation, for which well-posedness at the level of classical solutions is established. Then, a notion of strong approximating solution, calle…
We are interested in path-dependent semilinear PDEs, where the derivatives are of G{â}teaux type in specific directions k and b, being the kernel functions of a Volterra Gaussian process X. Under some conditions on k, b and the coefficients of the PDE, we prove existence and uniqueness of a decoupled mild solution, a notion introduced in a previous paper by the authors. We also show that the solution of the PDE can be represented through BSDEs where the forward (underlying) process is X.
Branching diffusion representation of semilinear PDEs and Monte Carlo approximation
Société Générale, Global Market Quantitative Research（法国兴业银行，全球市场量化研究）；EDF R&D & FiME, Laboratoire de Finance des Marchés de l’Energie（法国电力集团研发部及能源金融市场金融研究所，能源市场金融实验室）；CEREMADE, University of Paris-Dauphine, PSL Research University（巴黎第九大学数学与决策研究中心，PSL研究大学）；Ecole Polytechnique Paris, Centre de Mathématiques Appliquées（巴黎综合理工学院，应用数学中心）
Full-history recursive multilevel Picard (MLP) approximation schemes have been shown to overcome the curse of dimensionality in the numerical approximation of high-dimensional semilinear partial differential equations (PDEs) with general time horizons and Lipschitz continuous nonlinearities. However, each of the error analyses for MLP approximation schemes in the existing literature studies the $L^2$-root-mean-square distance between the exact solution of the PDE under consideration and the considered MLP approximation and none of the error analyses in the existing literature provides an upper…
We obtain estimates on the first-order Malliavin derivative of mild solutions, evaluated at fixed points in time and space, to a class of parabolic dissipative stochastic PDEs on bounded domain of $\mathbb{R}^d$. In particular, such equations are driven by multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a locally Lipschitz continuous function satisfying suitable polynomial growth bounds. The main arguments rely on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces, monotonicity, and a comparison p…
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 $α>0$.
The main objective of this paper and the accompanying one \cite{ETZ2} is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work \cite{EKTZ}, focused on the semilinear case, and is crucially based on the nonlinear optimal stopping problem analyzed in \cite{ETZ0}. We prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property and a partial comparison result. The latter is a key step for the wellposedness results established in \ci…
Using purely probabilistic methods, we prove the existence and the uniqueness of solutions fora system of coupled forward-backward stochastic differential equations (FBSDEs) with measurable, possibly discontinuous coefficients. As a corollary, we obtain the well-posedness of semilinear parabolic partial differential equations (PDEs) $$ \begin{aligned} &\mathcal{L} u(t,x)+F(t,x,u,\partial_x u)=0;\qquad u(T,x)=h(x)\\ &\mathcal{L}:=\partial_t+\frac{1}{2}\sum_{i,j=1}^m(σσ^\intercal)_{ij}(t,x)\partial^2_{x_ix_j} \end{aligned} $$ in the natural domain of the second-order linear parabolic operator $\…
Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for space-time solutions of semilinear partial differential equations
University of Wuppertal（伍珀塔尔大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；University of Münster（明斯特大学）；Toyota Financial Services（丰田金融服务公司）
In our previous paper [Ekren, Touzi and Zhang (2015)], we introduced a notion of viscosity solutions for fully nonlinear path-dependent PDEs, extending the semilinear case of Ekren et al. [Ann. Probab. 42 (2014) 204-236], which satisfies a partial comparison result under standard Lipshitz-type assumptions. The main result of this paper provides a full, well-posedness result under an additional assumption, formulated on some partial differential equation, defined locally by freezing the path. Namely, assuming further that such path-frozen standard PDEs satisfy the comparison principle and the P…
Shift Harnack and integration by part formula are establish for semilinear spde with delay and a class of stochastic semilinear evolution equation which cover the hyperdissipative Naiver-Stokes/Burges equation. For the case of stochastic equation with delay, an extension to path space is given.
This paper introduces the path derivatives, in the spirit of Dupire's functional Itô calculus, for the controlled paths in the rough path theory with possibly non-geometric rough paths. The theory allows us to deal with rough integration and rough PDEs in the same manner as standard stochastic calculus. We next study rough PDEs with coefficients depending on the rough path itself, which corresponds to stochastic PDEs with random coefficients. Such coefficients is less regular in the time variable and is not covered in the existing literature. The results are useful for studying viscosity solut…
The Feynman-Kac formula implies that every suitable classical solution of a semilinear Kolmogorov partial differential equation (PDE) is also a solution of a certain stochastic fixed point equation (SFPE). In this article we study such and related SFPEs. In particular, the main result of this work proves existence of unique solutions of certain SFPEs in a general setting. As an application of this main result we establish the existence of unique solutions of SFPEs associated with semilinear Kolmogorov PDEs with Lipschitz continuous nonlinearities even in the case where the associated semilinea…
Let P2(Rd) be the space of probability measures on Rd with finite second moment. The path independence of additive functionals of McKean-Vlasov SDEs is characterized by PDEs on the product space Rd*P2(Rd) equipped with the usual derivative in space variable and Lions derivative in distribution. These PDEs are solved by using probabilis- tic arguments developed from [2]. In particular, the path independence of the Girsanov transformation killing the drift term is identified with a nonlinear PDE on Rd*P2(Rd), which includes corresponding results derived earlier for the classical SDEs as special …
Functional Itô calculus was introduced in order to expand a functional $F(t, X\_{\cdot+t}, X\_t)$ depending on time $t$, past and present values of the process $X$. Another possibility to expand $F(t, X\_{\cdot+t}, X\_t)$ consists in considering the path $X\_{\cdot+t}=\{X\_{x+t},\,x\in[-T,0]\}$ as an element of the Banach space of continuous functions on $C([-T,0])$ and to use Banach space stochastic calculus. The aim of this paper is threefold. 1) To reformulate functional Itô calculus, separating time and past, making use of the regularization procedures which matches more naturally the noti…
This paper introduces a convenient solution space for the uniformly elliptic fully nonlinear path dependent PDEs. It provides a wellposedness result under standard Lipschitz-type assumptions on the nonlinearity and an additional assumption formulated on some partial differential equation defined locally by freezing the path.
We introduce a new definition of viscosity solution to path-dependent partial differential equations, which is a slight modification of the definition introduced in [8]. With the new definition, we prove the two important results till now missing in the literature, namely, a general stability result and a comparison result for semicontinuous sub-/super-solutions. As an application, we prove the existence of viscosity solutions using the Perron method. Moreover, we connect viscosity solutions of path-dependent PDEs with viscosity solutions of partial differential equations on Hilbert spaces.
Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore（数学科学系，物理与数学科学学院，南洋理工大学，新加坡）；Faculty of Mathematics, Bielefeld University, Germany（数学学院，比勒菲尔德大学，德国）
In this article, we adapt the definition of viscosity solutions to the obstacle problem for fully nonlinear path-dependent PDEs with data uniformly continuous in $(t,ω)$, and generator Lipschitz continuous in $(y,z,γ)$. We prove that our definition of viscosity solutions is consistent with the classical solutions, and satisfy a stability result. We show that the value functional defined via the second order reflected backward stochastic differential equation is the unique viscosity solution of the variational inequalities.
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.
