arXiv subjects
Publications and source records attributed to Peter K. Friz.
We establish a simultaneous generalization of Itô's theory of stochastic and Lyons' theory of rough differential equations. The interest in such a unification comes from a variety of applications, including pathwise stochastic filtering, - control and the conditional analysis of stochastic systems with common noise.
We develop an intrinsic well-posedness theory for nonlinear backward stochastic differential equations (BSDEs) driven simultaneously by Brownian motion and a (level-$2$) rough path of finite $p$-variation. Unlike earlier approaches based on smooth approximation or transformation methods, we formulate the equation directly by viewing its solution as a rough semimartingale (in the sense of \cite{friz2023rough}). This framework is particularly suited to BSDEs, whose Brownian martingale component is only implicitly defined and lacks the \emph{a priori} time regularity required by stochastic-sewing-based controlled rough path methods \cite{fhl21,allan2024rough}. We establish comparison, existence, uniqueness, and stability under the natural regularity condition $H\in C_b^γ$, $γ>p$. The main analytical difficulty, namely, the loss of integrability arising from nonlinear composition, is overcome through conditional $p$-variation norms with BMO-type properties. Finally, by randomizing the rough driver as a Brownian rough path, we establish a direct correspondence between rough BSDEs and backward doubly stochastic differential equations (BDSDEs).
We develop novel duality methods for stochastic optimal control problems under two sources of randomness and non-Markovian random coefficients adapted to just one of them. The Hamilton-Jacobi-Bellman (HJB) equation is a second-order backward stochastic partial differential equation. Our duality theory provides an alternative description of the random value function in terms of a suitable pathwise optimal control problem parameterized by the realizations of one of the Brownian motions. This allows us to regain Markovianity using the theory of rough optimal control problems, leading to rough second order HJB equations that we solve in a suitable viscosity sense.
Motivated by mean-field games (MFG) with common noise on the one hand and pathwise stochastic control theory on the other, we formulate here a linear-quadratic (LQ) MFG with rough common noise, along with a satisfactory well-posedness theory for the linear-quadratic case. A novel Volterra-type (or mild) formulation allows to keep technical (rough-stochastic) consideration to a minimum. We derive a characterization of the optimal state and optimal control through a rough forward-backward SDE (rough FBSDE), and provide an existence and uniqueness result under the usual assumptions. Our theory is accompanied by stability estimates with respect to initial data and common noise while we also establish continuity of what we call the Itô-Lions-Lyons map for rough mean-field games. Finally, we discuss randomization of the rough common noise under appropriate conditions on the coefficients. When the latter is given by the Stratonovich lift of a Brownian motion independent of the idiosyncratic noise, we show that solutions of the rough LQ MFG coincide with those obtained by conditioning on the common noise.
We break the mold in flow-based generative modeling by proposing a new model based on the damped wave equation, also known as telegrapher's equation. Similar to the diffusion equation and Brownian motion, there is a Feynman-Kac type relation between the telegrapher's equation and the stochastic Kac process in 1D. The Kac flow evolves stepwise linearly in time, so that the probability flow is Lipschitz continuous in the Wasserstein distance and, in contrast to diffusion flows, the norm of the velocity remains globally bounded. Furthermore, the Kac model has the diffusion model as its asymptotic limit. We extend these considerations to a multi-dimensional stochastic process which consists of independent 1D Kac processes in each spatial component. We show that this process gives rise to an absolutely continuous curve in the Wasserstein space and analytically compute the conditional velocity field when starting in a Dirac point. Using the framework of flow matching, we train a neural network to approximate the velocity field and use it for sample generation. Our numerical experiments demonstrate the scalability of our approach, and show its advantages over diffusion models.
We develop a calculus of space-time controlled fields for rough stochastic systems. This approach provides a unified composition rule for evaluating random fields along rough semimartingales and yields a rough stochastic Itô-Wentzell formula under natural and verifiable regularity assumptions. Our motivation comes from works of Hudde et al. (2024) and, independently, Del Moral and Singh (2022) where the authors established, respectively, Itô-Alekseev-Gröbner, backward Itô-Wentzell, and diffusion interpolation formulas.
We consider Talagrand-type transportation inequalities for the law of Brownian motion on Carnot groups. An important example is the lift of standard Brownian motion to the Brownian rough path. We present a direct proof on enhanced path space, which also yields equality when restricting to adapted couplings in the transport problem. Moreover, we prove a Talagrand inequality for the heat kernel measure on Carnot groups and deduce the inequality for the law of Brownian motion on Carnot groups via a bottom-up argument. Our study of this enhanced Wiener measure contributes to a longstanding programme to extend key properties of Wiener measure to the non-commutative setting of the enhanced Wiener measure, which is of central importance in Lyons' rough path theory. With a non-commutative sub-Riemannian state space, we observe phenomena that differ from the Euclidean case. In particular, while a top-down projection argument recovers Talagrand's inequality on Euclidean space from the corresponding inequality on the path space, such a projection argument breaks down in the Carnot group setting. We further study a Riemannian approximation of the Heisenberg group, in which case the failure of the top-down projection can be partially overcome. Finally, we show that the cost function used in the Talagrand inequality is a natural choice, in that it arises as a limit of discretised costs in the sense of $Γ$-convergence.
This article is concerned with the well-posedness of the "filtering equations", due to Zakai and Kushner-Stratonovich, arising in nonlinear stochastic filtering. In general situations, notably in correlated diffusion models and when signal coefficients depend on the observation process, the well-posedness is a difficult problem, mainly due to conflicting martingale structures of the involved forward and backward equations. Crisan-Pardoux (2024) address this classical problem with BSPDE techniques, Du et al. (2013), a Sobolev-based approach that however requires increasingly strong regularity assumptions in high dimensions. In this work, we take a new mixed rough stochastic perspective which allows us to derive well-posed rough counterparts of the filtering equations. Importantly, the rough filtering equations are seen, upon randomization, to coincide with the classical filtering equations. Our framework yields well-posedness (existence, uniqueness, stability) under dimension-independent regularity assumptions, providing a robust and conceptually unified solution to a longstanding problem in stochastic filtering theory. To illustrate the flexibility of the method, we also treat rough versions of the classical Kalman-Bucy filter, with characteristics described by a new class of RDEs of rough Riccati type.
We study stochastic optimal control of rough stochastic differential equations (RSDEs). This is in the spirit of the pathwise control problem (Lions--Souganidis 1998, Buckdahn--Ma 2007; also Davis--Burstein 1992), with renewed interest and recent works drawing motivation from filtering, SPDEs, and reinforcement learning. Results include regularity of rough value functions, validity of a rough dynamic programming principles and new rough stability results for HJB equations, removing excessive regularity demands previously imposed by flow transformation methods. Measurable selection is used to relate RSDEs to "doubly stochastic" SDEs under conditioning. In contrast to previous works, Brownian statistics for the to-be-conditioned-on noise are not required, aligned with the "pathwise" intuition that these should not matter upon conditioning. Depending on the chosen class of admissible controls, the involved processes may also be anticipating. The resulting stochastic value functions coincide in great generality for different classes of controls. RSDE theory offers a powerful and unified perspective on this problem class.
Wiener spaces are in many ways the decisive setting for fundamental results on Gaussian measures: large deviations (Schilder), quasi-invariance (Cameron--Martin), differential calculus (Malliavin), support description (Stroock--Varadhan), concentration of measure (Fernique), etc. Analogues of these classical results have been derived in the "enhanced" context of Gaussian rough paths and, more recently, regularity structures equipped with Gaussian models. The aim of this article is to propose a similar notion directly on this enhanced level - an abstract Wiener model space - that encompasses the aforementioned. More specifically, we focus here on enhanced Schilder type results, Cameron--Martin shifts and Fernique estimates, offering a somewhat unified view on results of Friz--Victoir and Hairer--Weber.
The expected signature kernel arises in statistical learning tasks as a similarity measure of probability measures on path space. Computing this kernel for known classes of stochastic processes is an important problem that, in particular, can help reduce computational costs. Building on the representation of the expected signature of (inhomogeneous) Lévy processes with absolutely continuous characteristics as the development of an absolutely continuous path in the extended tensor algebra [F.-H.-Tapia, Forum of Mathematics: Sigma (2022), "Unified signature cumulants and generalized Magnus expansions"], we extend the arguments developed for smooth rough paths in [Lemercier-Lyons-Salvi, "Log-PDE Methods for Rough Signature Kernels"] to derive a PDE system for the expected signature of inhomogeneous Lévy processes. As a specific example, we see that the expected signature kernel of Gaussian martingales satisfies a Goursat PDE.
Rough stochastic differential equations (rough SDEs), recently introduced by Friz, Hocquet and Lê in arXiv:2106.10340, have emerged as a versatile tool to study "doubly" SDEs under partial conditioning (with motivation from pathwise filtering and control, volatility modelling in finance and mean-field stochastic dynamics with common noise ...). While the full dynamics may be highly non-Markovian, the conditional dynamics often are. In natural (and even linear) situations, the resulting stochastic PDEs can be beyond existing technology. The present work then tackles a key problem in this context, which is the well-posedness of regular solution to the rough Kolmogorov backward equation. To this end, we study parameter dependent rough SDEs in sense of $\mathscr{L}$-differentiability (as in Krylov, 2008). In companion works, we will show how this removes dimension-dependent regularity assumptions for well-posedness of the Zakai, Kushner-Stratonovich and nonlinear Fokker-Planck stochastic equations.
McKean-Vlasov SDEs describe systems where the dynamics depend on the law of the process. The corresponding Fokker-Planck equation is a nonlinear, nonlocal PDE for the corresponding measure flow. In the presence of common noise and conditional law dependence, the evolution becomes random and is governed by a stochastic Fokker-Planck equation; that is, a nonlinear, nonlocal SPDE in the space of measures. (Such equations constitute an important ingredient in the theory of mean-field games with common noise.) Well-posedness of such SPDEs is a difficult problem, the best result to date due to Coghi-Gess (2019), which however comes with dimension-dependent regularity assumptions. In the present work, we show how rough path techniques can circumvent these entirely. Hence, and somewhat contrarily to common believe, the use of rough paths leads to substantially less regularity demands on the coefficients than methods rooted in classical stochastic analysis methods.
We show well-posedness for McKean--Vlasov equations with rough common noise and progressively measurable coefficients. Our results are valid under natural regularity assumptions on the coefficients, in agreement with the respective requirements of Ito and rough path theory. To achieve these goals, we work in the framework of rough stochastic differential equations recently developed by the authors of this article.
Rough stochastic differential equations (RSDEs) are common generalisations of Ito SDEs and Lyons RDEs and have emerged as new tool in several areas of applied probability, including non-linear stochastic filtering, pathwise stochastic optimal control, volatility modelling in finance and mean-fields analysis of common noise system. We here take a unified perspective on rough Ito processes and discuss in particular when and how they become, upon randomisation, "doubly stochastic" Ito processes, and what can be said about their conditional laws.
Local stochastic volatility refers to a popular model class in applied mathematical finance that allows for "calibration-on-the-fly", typically via a particle method, derived from a formal McKean-Vlasov equation. Well-posedness of this limit is a well-known problem in the field; the general case is largely open, despite recent progress in Markovian situations. Our take is to start with a well-defined Euler approximation to the formal McKean-Vlasov equation, followed by a newly established half-step-scheme, allowing for good approximations of conditional expectations. In a sense, we do Euler first, particle second in contrast to previous works that start with the particle approximation. We show weak order one for the Euler discretization, plus error terms that account for the said approximation. The case of particle approximation is discussed in detail and the error rate is given in dependence of all parameters used.
The concept of signatures and expected signatures is vital in data science, especially for sequential data analysis. The signature transform, a Cartan type development, translates paths into high-dimensional feature vectors, capturing their intrinsic characteristics. Under natural conditions, the expectation of the signature determines the law of the signature, providing a statistical summary of the data distribution. This property facilitates robust modeling and inference in machine learning and stochastic processes. Building on previous work by the present authors [Unified signature cumulants and generalized Magnus expansions, FoM Sigma '22] we here revisit the actual computation of expected signatures, in a general semimartingale setting. Several new formulae are given. A log-transform of (expected) signatures leads to log-signatures (signature cumulants), offering a significant reduction in complexity.
In this work, we introduce a novel pricing methodology in general, possibly non-Markovian local stochastic volatility (LSV) models. We observe that by conditioning the LSV dynamics on the Brownian motion that drives the volatility, one obtains a time-inhomogeneous Markov process. Using tools from rough path theory, we describe how to precisely understand the conditional LSV dynamics and reveal their Markovian nature. The latter allows us to connect the conditional dynamics to so-called rough partial differential equations (RPDEs), through a Feynman-Kac type of formula. In terms of European pricing, conditional on realizations of one Brownian motion, we can compute conditional option prices by solving the corresponding linear RPDEs, and then average over all samples to find unconditional prices. Our approach depends only minimally on the specification of the volatility, making it applicable for a wide range of classical and rough LSV models, and it establishes a PDE pricing method for non-Markovian models. Finally, we present a first glimpse at numerical methods for RPDEs and apply them to price European options in several rough LSV models.