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A Stratonovich-Skorohod Integral Formula for Gaussian Rough Paths

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A Stratonovich-Skorohod Integral Formula for Gaussian Rough Paths Submitted to the Annals of Probability A STRATONOVICH-SKOROHOD INTEGRAL FORMULA FOR GAUSSIAN ROUGH PATHS By Thomas Cass ∗ and Nengli Lim † Imperial College London and Singapore University of Technology and Design Given a Gaussian process X, its canonical geometric rough path lift X, and a solution Y to the rough differential equation (RDE) dYt = V (Yt)○ dXt, we present a closed-form correction formula for ∫ Y ○dX−∫ Y dX, i.e. the difference between the rough and Skorohod integrals of Y with respect to X. When X is standard Brownian mo- tion, we recover the classical Stratonovich-to-Itˆoconversion formula, which we generalize to Gaussian rough paths with finite p-variation, p < 3, and satisfying an additional natural condition. This encom- passes many familiar examples, including fractional Brownian motion 1 with H > 3 . To prove the formula, we first show that the Riemann- 2 sum approximants of the Skorohod integral converge in L (Ω) by using a novel characterization of the Cameron-Martin norm in terms of higher-dimensional Young-Stieltjes integrals. Next, we append the approximants of the Skorohod integral with a suitable compensation term without altering the limit, and the formula is finally obtained after a re-balancing of terms. 1. Introduction. Gaussian processes are used in modeling natural phe- nomena, from rough stochastic volatility models in high-frequency trading [2], to models of vortex filaments based on fractional Brownian motion [30]. To analyze stochastic processes with regularity lower than standard Brow- nian motion, one can employ the theory of rough paths [25]. In particular, given a Gaussian process X, one can lift it canonically to a geometric rough path X [17], and this allows one to study the properties of rough differential equations (RDEs) arXiv:1604.06846v3 [math.PR] 30 Jan 2018 e (1) dYt V Yt dXt, Y0 y0 R , = ( ) ○ = ∈ ∗Supported by EPSRC grant EP/M00516X/1. †Part of the work was carried out while the second author was a PhD candidate at ICL, supported by the Roth scholarship and the Imperial College-NUS joint PhD scholarship. MSC 2010 subject classifications: Primary 60H05; secondary 60H07, 60H10 Keywords and phrases: Rough paths theory, Malliavin calculus, generalized Itˆo- Stratonovich correction formulas 1 2 T. CASS AND N. LIM and of rough integrals of the form T (2) Yt dXt. S0 ○ Furthermore, this geometric calculus generalizes Stratonovich’s stochastic calculus, and as such, it finds natural applications, e.g. in stochastic geom- etry where the change-of-variable formula allows one to give an intrinsic and coordinate-invariant definition of a rough path on a general smooth manifold, cf. [9], [5]. Itˆointegrals, by contrast, preserve the local martingale property, which is a useful feature when computing probabilistic quantities such as exit dis- tributions and conditional expectations. One can often gain insight into a problem by transforming Stratonovich integrals to Itˆointegrals and vice versa, depending on the requirement at hand. Now if Y and X are two continuous semi-martingales, both Rd-valued, it is well-known that the difference between the two integrals is given in terms of the quadratic covariation through the formula, cf. [24], [33], T T 1 Yt, dXt Yt, dXt Y, X T . S0 ⟨ ○ ⟩= S0 ⟨ ⟩+ 2 [ ] In the case where Yt solves RDE (1) and Xt is taken to be standard Brownian motion Bt, this becomes T T 1 T (3) Yt, dBt Yt, dBt tr V Yt dt, S0 ○ = S0 + 2 S0 ⟨ ⟩ ⟨ ⟩ [ ( )] where in the second term on the right side we have the usual trace of V Yt Rd Rd considered as a d-by-d matrix. ∈ ⊗ On the other hand, if Yt f Xt , where f is sufficiently smooth,( then) we get Itˆo’s formula, ≡ ∇ ( ) T (4) f XT f X0 f Xt , dXt − = S0 ∇ ○ T 1 T (5) ( ) ( ) ⟨ f(Xt), dXt ⟩ ∆f Xt dR t , = S0 ∇ + 2 S0 where the first term on the right⟨ ( side) is the⟩ Skorohod( integral) (of) f X 2 ∇ with respect to X, and R t is the variance E X(1) . This has been t ( ) well-studied for general Gaussian processes, particularly fractional Brownian ( ) motion, over the past two decades; see [32], [1], [4], [29], and in particular [21], which uses rough path techniques to prove the formula. STRATONOVICH-SKOROHOD FORMULA FOR GAUSSIAN RPS 3 Our main result is the following theorem, where the driving signal X is constructed from the limit of the piecewise-linear approximations of a Gaussian process with i.i.d. components. Theorem. For 2 p 3, let Y p−var 0, T ; Rd denote the path-level solution to ≤ < ∈ C [ ] dYt V Yt dXt, Y0 y0, = ○ = 6 Rd Rd Rd ( ) 0,p−var ⌊p⌋ Rd where V b ; and X 0, T ; G . We assume ∈ ⊗ ∈ 3 the covariance function R of X is of finite 2D ρ-variation, 1 ρ 2 , and satisfies C C [ ] ≤ < 1 (6) R t, ⋅ − R s, ⋅ C t − s ρ , ρ−var;[0,T ] ≤ for all s,t 0, T .Y Then( ) almost( surely,)Y we have S S ∈ (7) [ ] T T 1 T Yt, ○dXt Yt, dXt + tr V Yt dR t S0 = S0 2 S0 X X 1 ⟨ ⟩ + ⟨ 1⟩ s tr J[ (J )]− V (Y) − V Y dR s,t . 2 [0,t) t s s t S[0,T ] X ( ) ( ) ( ) ( ) Here, Jt denotes the Jacobian of the flow map y0 → Yt. The last term on the right side of (7) is a proper 2D Young-Stieltjes integral with respect to the covariance function. When X is standard Brownian motion, it vanishes since the integrand is zero on the diagonal and dR s,t δ = ds dt. This, {s t} together with the fact that R t t, allows us to recover= the usual Itˆo- Stratonovich conversion formula (3=). ( ) Hence, an immediate contribution( ) of the theorem is the generalization of formula (3) to the setting where the integrands are solutions to Gaussian RDEs. Here, we are able to give a formula for integrators other than stan- dard Brownian motion without restriction on the regularity of the integrand; compare this to [12] e.g., where essentially Young complementary regularity is required. Furthermore, the novel 2D Young-Stieltjes integral can be un- derstood as measuring the failure of the commutativity of V with respect to the covariance of the Gaussian process. For studying the dynamics of Gaus- sian processes in cases where the correction formula is indispensable, e.g. Gaussian processes evolving on manifolds, this could lead to new insights. We now provide the main idea behind the proof of the theorem. Denoting X 1, X, X2 , the solution Y to RDE (1) can be integrated against this = ( ) 4 T. CASS AND N. LIM rough path and T 2 (8) Yt, ○dXt lim Yt , Xt ,t 1 + V Yt X → i i i+ i ti,ti+1 S0 = YπY 0 Qi almost surely.⟨ Continuing,⟩ we devote⟨ Section 4⟩ to proving( ) two claims. The first is that Y lies in the domain of the Skorohod integral operator w.r.t. X, and the second is that, as a limit in L2 Ω we have T Yt, dXt ( ) S0 (9) t i X X −1 ⟨ lim ⟩ Yt , Xt ,t 1 − tr J J V Ys R ∆i, ds . → i i i+ ti s = YπY 0 Qi S0 ⟨ ⟩ ( ) ( ) Proving these facts is less obvious than in the case where Yt ∇f Xt , and we need to use the tail estimates of [10] in a fundamental way.= In Section 5, we use condition (6) to prove that ( ) 1 2 (10) V Y X2 E X(1) ti ti,ti+1 − ti,ti+1 d Qi 2 ( ) I has a vanishing L2 Ω limit as π → 0. The proof of this relies on estimates coming from a delicate interplay between the theory of Malliavin calculus and Gaussian rough( ) paths; seeY PropositionY 5.1. After augmenting (10) to (9) and extracting an almost sure subsequence, we can take the difference between this subsequence and (8). A careful rearrangement of the terms in this difference will then yield the correction term. We now outline the structure of the paper, as well as highlight other contributions that are of independent interest. We begin in Section 2 with a summary of Gaussian rough path concepts and a primer on the Malliavin calculus as applied to RDE solutions. In Section 3.1, we provide a general closed-form expression and a novel bound for the higher-order Malliavin derivatives of RDE solutions relative to the driving rough path (cf. [22], [23], [8]). This will be then used in Section 5 to show that (10) has vanishing L2 Ω limit. In Section 4, we give a new characterization of the Cameron-Martin norm in( terms) of multi-dimensional Young-Stieltjes integrals. We show that one p var can identify − 0, T with a dense subspace of 1, the Hilbert space generated by the indicator functions which is isomorphic to the Cameron- Martin space.C In particular,([ ]) for f p−var 0, T , weH have ∈ f C f ([f R])s,t . (11) H1 2 t s d = ¾S[0,T ] Y Y ( ) STRATONOVICH-SKOROHOD FORMULA FOR GAUSSIAN RPS 5 We also give a corresponding characterization with regards to the tensor product of the Cameron-Martin space in Section 4.2, and revisit the clas- sical Ito-Skorohod isometry in Section 4.3 by giving it a new formulation in terms of multi-dimensional Young-Stieltjes integrals. Finally, Section 4 is concluded with a proof of (9). The main theorem and its proof is given in Section 6. 2. Preliminaries. The following is a summary of basic notation that will be used throughout the paper.
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