Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes

1 Introduction . The idea of embedding probability distributions into a reproducing kernel Hilbert space ( RKHS ) via kernel mean embeddings ( KMEs ) has become ubiquitous in many areas of statistics and data science such as hypothesis testing [ 1 , 2 ] , non-linear regression [ 3 , 4 ] , distribution regression [ 5 , 6 ] etc . Despite strong progress in the study of KMEs , most of the examples considered in the literature tend to focus on random variables supported on some finite ( possibly high ) dimensional euclidean spaces like Rd . The study of KMEs for function-valued random variables has been largely ignored . Stochastic processes are random variables with values in some space of paths . However , reducing a stochastic process to a path-valued random variable ignores its filtration , which can be informally thought of as the flow of information carried by the process through time . A question that naturally 1University of Oxford & The Alan Turing Institute 2University of Warwick & The Alan Turing Institute 3University of Oxford 4King ’ s College London , The Alan Turing Institute & Technical University of Munich 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . emerges from the study of many random , time-evolving systems like financial markets is how does the information available up to present time affect the future evolution of the system ? Formally , this question can be addressed by conditioning a process on its filtration ( Sec . 3.1 and 3.2 ) . In this paper we introduce a family of higher order KMEs that generalizes the notion of KME to capture additional , filtration-related information ( Sec . 3.3 and 3.5 ) . In view of concrete applications , we derive empirical estimators for the associated higher order MMDs and use one of them to construct a filtration-sensitive kernel two-sample test ( Sec . 3.4 ) demonstrating with simulated data its ability to capture information that otherwise gets missed by the standard MMD test ( Sec . 4.1 ) . Furthermore , we construct a family of universal kernels on stochastic processes ( Sec . 3.6 ) that allows to solve challenging , real-world optimisation problems in quantitative finance such as the pricing of American options via classical kernel-based regression methods ( Sec . 4.2 ) . Finally , we adapt existing tests for conditional independence to the case of stochastic processes in order to design a causal-discovery algorithm able to recover the causal graph of structural dependencies among interacting bodies solely from observations of their multidimensional trajectories ( Sec . 4.3 ) . 1.1 Related work . The notion of conditioning is a powerful probabilistic tool allowing to understand possibly complex , non-linear interactions between random variables . As their unconditional counterparts , conditional distributions can also be embedded into RKHSs [ 7 ] . Recently , conditional KMEs have received increased attention , especially in the context of graphical models [ 8 ] , state-space models [ 9 ] , dynamical systems [ 10 ] , causal inference [ 11 , 12 , 13 ] , two-sample and conditional independence hypothesis testing [ 14 , 13 , 15 ] and others . Embeddings of distributions via KMEs have also shown their success in the context of distribution regression ( DR ) , which is the task of learning a function mapping a collection of samples from a probability distribution to scalar targets [ 16 , 17 , 18 ] . More recently , a framework for DR that addresses the setting where inputs are sample paths from an underlying stochastic process is proposed in [ 6 ] . The authors make extensive use of the signature transform [ 19 , 20 ] and of the signature kernel [ 21 , 22 ] , two well established tools in stochastic analysis . When it comes to stochastic processes , it was first shown in [ 23 ] that weak convergence of random variables does not always account for the information contained in the filtration , as highlighted by means of numerous numerical examples in [ 24 , 25 ] . This limitation is addressed in [ 23 , 26 ] through the construction of a sequence of so–called adapted topologies5 ( τn ) n≥1 that become progressively finer6 as n increases ( with τ1 coinciding with the weak topology ) . In particular , higher order adapted topologies are shown to capture more filtration-related information than their weak counterpart . This characteristic becomes relevant for example in some optimal stopping problems such as the pricing of American options , where the pricing function can be shown to be discontinuous with respect to the weak topology , but is continuous with respect to the second order adapted topology7 [ 24 , 28 , 27 ] . Leveraging properties of the signature transform , it has been shown that adapted topologies are intimately related to a family of higher order MMDs [ 29 ] . However , providing empirical estimators for these discrepancies that can be deployed on real-world tasks remains a challenge . In this paper we propose to address this challenge by presenting an alternative construction to this higher order MMDs using the language of kernels and KMEs . The results in [ 29 ] serve as a strong theoretical background for the present paper . 2 Preliminaries . We begin with a brief summary of tools from stochastic analysis needed to define higher order KMEs . Let X ( Rd ) = { x : [ 0 , T ] → Rd } a compact set of continuous , piecewise linear , Rd-valued paths defined over a common time interval [ 0 , T ] , obtained for example by linearly interpolating a multivariate time series . For any path x ∈ X ( Rd ) we denote its kth coordinate by x ( k ) : [ 0 , T ] → R , for k ∈ { 1 , . . . , d } . More generally we denote by X ( V ) = { x : [ 0 , T ] → V } a compact set of continuous , piecewise linear paths with values in a Hilbert space V with a countable basis . 2.1 The signature transform . The signature transform S : X ( V ) → HS ( V ) is a feature map defined for any path x ∈ X ( V ) as the following infinite collection of statistics S ( x ) = ( 1 , { S ( x ) ( k1 ) } d k1=1 , { S ( x ) ( k1 , k2 ) } d k1 , k2=1 , . . . ) ( 1 ) where each term is a real number equal to the iterated integral S ( x ) ( k1 , ... , kj ) = ∫ . . . ∫ 0 < s1 < ... < sj < T dx ( k1 ) s1 . . . dx ( kj ) sj ( 2 ) The signature feature space HS ( V ) is defined as the following direct sum of tensor powers of V HS ( V ) = ∞⊕ k=0 V ⊗k = R⊕ V ⊕ ( V ) ⊗2 ⊕ . . . ( 3 ) where ⊗ denotes the standard tensor product of vector spaces [ 30 , 19 ] . 2.2 The signature kernel . Because V is Hilbert HS ( V ) is also Hilbert [ 21 ] . The signature kernel kS : X ( V ) ×X ( V ) → R is a characteristic kernel defined for any pair of paths x , y ∈ X ( V ) as the following inner product kS ( x , y ) = 〈S ( x ) , S ( y ) 〉HS ( V ) ( 4 ) The recent article [ 22 ] establishes a surprising connection between the signature kernel and a certain class of partial differential equations ( PDEs ) , culminating in the following kernel trick for kS . Theorem 1 . [ 22 , Thm . 2.5 ] For any x , y ∈ X ( V ) the signature kernel satisfies the equation kS ( x , y ) = ux , y ( T , T ) , where ux , y : [ 0 , T ] × [ 0 , T ] → R is the solution of the hyperbolic PDE ∂2ux , y ∂s∂t = 〈ẋs , ẏt〉V ux , y ( 5 ) with boundary conditions ux , y ( 0 , · ) = ux , y ( · , 0 ) = 1 and where żs = dzrdr ∣∣ r=s . 5We say that a sequence of random variables { Xn } n∈N converges to a random variable X in the topology τ if and only if for every τ -open neighbourhood U of X there exists N ∈ N such that Xn ∈ U as soon as n ≥ N . 6A topology τ1 is said to be finer than a topology τ2 if every τ2-open set is also τ1-open . 7The second order adapted topology τ2 is equivalent to the adapted Wasserstein distance [ 27 ] . Hence , evaluating kS at a pair of paths ( x , y ) is equivalent to solving the PDE ( 5 ) ; in this paper we solve PDEs numerically via a finite difference scheme ( see Appendix B for additional details ) . In what follows , we denote byHS ( V ) the RKHS associated to kS . 2.3 Stochastic processes and filtrations . We take ( Ω , F , P ) as the underlying probability space . A ( discrete time ) stochastic process X is a random variable with values on X ( V ) . We denote by PX = P ◦X−1 the law of X . Assuming the integrability condition EX [ kS ( X , X ) ] < ∞ , the 1st order kernel mean embedding ( KME ) of X is defined as8 the following point inHS ( V ) µ1X = EX [ kS ( X , · ) ] = ∫ x∈X ( V ) kS ( x , · ) PX ( dx ) ( 6 ) Accordingly , given two stochastic processesX , Y , their 1st order maximum mean discrepancy ( MMD ) is the standard MMD distance with kernel kS given by the following expression D1S ( X , Y ) = ∥∥µ1X − µ1Y ∥∥ HS ( V ) ( 7 ) Because the signature kernel kS is characteristic , it is a classical result [ 1 , 31 ] that the 1st order MMD is a sufficient statistics to distinguish between the laws of X and Y , in other words D1S ( X , Y ) = 0 ⇐⇒ PX = PY ( 8 ) Despite the fact that stochastic processes are path-valued random variables , they encode a much richer structure compared to standard Rd-valued random variables , that goes well beyond their laws . This additional structure is described mathematically by the concept of filtration of a process X , defined as the following family of σ-algebras FX = ( FXt ) t∈ [ 0 , T ] , ( 9 ) where for any t ∈ [ 0 , T ] , FXt is the σ-algebra generated by the variables { Xs } s∈ [ 0 , t ] . Note that FX is totally ordered in the sense that FXs ⊂ FXt for all s < t , which naturally explains why filtrations are good mathematical descriptions to model the flow information carried by the process X . In the next section , we will present our main findings and introduce a family of higher order KMEs and corresponding higher order MMDs as generalizations of the standard KME and MMD respectively . We will do so by conditioning stochastic processes on elements of their filtrations .

Summary: This article study stochastic processes through the lens of kernel methods. The authors propose to use a generalization of kernel mean embeddings to define metrics between stochastic processes. In particular, they used these metrics to define a filtration-sensitive kernel two-sample test. Moreover, they showed how to use this new class of kernels in some learning tasks.