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 paper aims to construct a family of kernel mean embeddings that capture filtration information from stochastic processes. This is achieved through the use of the signature kernel, a characteristic kernel defined on pairs of paths that can be evaluated by solving a hyperbolic PDE, and by forming conditional kernel mean embeddings of the stochastic process conditioned on its own filtration up to some time in the past.  The resulting conditional kernel mean operator itself is a stochastic process, and thus this construction can be repeated to produce higher ordered conditional kernel mean embeddings of the original stochastic process. The corresponding MMD defined from the conditional kernel mean embeddings at each order can then be used to measure distances between two stochastic processes at each order.  The first major contribution shows that higher ordered MMDs are stronger discrepancy measures than lower ordered MMDs (Theorem 2 and 4). They can then be used to perform a kernel two-sample test between paths from two stochastic processes. In particular, the paper focuses on using the second order MMD and derive consistent estimators for it (Theorem 3).  The second major contribution applies these constructions for distribution regression on stochastic processes, where the input is a collection of sample paths and the output is a vector of scalars. This is done by embedding the sampled paths from the stochastic process into a kernel mean embedding through the signature kernel, then learning the mapping between this kernel mean embedding to the vector of scalar outputs. The authors pay strong attention to the application in financial markets where the functions of interest are often discontinuous with respect to the first order mean embedding, and show they can still be learned by proving that the RKHS induced in higher orders is dense in the space of functions of interest (Theorem 5).