Limit Theorems for Additive Functionals of Path-Dependent SDEs

By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.


Introduction and Main Results
Since W. Doeblin [9] in 1938 established the law of large numbers and central limit theorem for denumerable Markov chains, limit theory for additive functionals of Markov processes has been extensively investigated. In general, for an ergodic Markov process (X t ) t≥0 on a Polish space E, as t → ∞ one describes the convergence of the empirical distribution µ t := 1 t t 0 δ Xs ds to the unique invariant probability measure µ ∞ . A standard way is to look at the convergence rate of for f in a class of reference functions.This leads to the study of limit theorems for additive functionals of ergodic Markov processes. Classical limit theorems include • Strong law of large numbers (SLLN): P-a.s. convergence of A f t to µ ∞ (f ); • Central limit theorem (CLT): The weak convergence of 1 √ t t 0 {f (X s ) − µ ∞ (f )}ds to a normal random variable; • Law of iterated logarithm (LIL): the asymptotic range of t log log t t 0 f (X s )ds. Once CLT is established, one may further investigate the large/moderate deviations principles, see for instance [12] and references within.
When the Markov processes are exponentially ergodic in L 2 (µ ∞ ) or total variational norm, limit theorems of A f t have been established for reference functions f ∈ L 2 (µ ∞ ) or B b (E), respectively; see the recent monograph [22] and earlier references [6,13,17,20,19,23,28]. However, these results do not apply to highly degenerate models which are exponentially ergodic merely under a Wasserstein distance; see for instance [15] for 2D Navier-Stokes equations with degenerate stochastic forcing, and [2,4,5,14] for stochastic differential equations (SDEs) with memory.
In this paper, we aim to establish limit theorems for path-dependent SDEs, which were initiated by Itô-Nisio [18]. Due to the path-dependence of the noise term, the corresponding segment solutions are no longer ergodic in the total variational norm (see e.g. [22,Example 5.1.3]). Moreover, the L 2 -ergodicity is also unknown because of the lack of Dirichlet form for path-dependent SDEs. So far, there are a few of papers on LLN and CLT for stochastic dynamical systems which are weakly ergodic; see e.g. [21,22,24,27].
In particular, f in [21,27] is assumed to be (bounded) Lipschitz with respect to a metric and the weak LLN is investigated; In [22], the LLN is established under some additional technical conditions (see [22,Theorem 5.1.10] for more details). In this paper, we will show that limit theorems established in [24] for uniformly mixing Markov processes apply well to the present model for f being Lipchitz continuous with respect to a quasi-metric.
For a fixed number r 0 ∈ (0, ∞), let C = C([−r 0 , 0]; R d ) be the collection of all continuous functions f : [−r 0 , 0] → R d endowed with the uniform norm For any continuous path (γ(t)) t≥−r 0 on R d , its segment (γ t ) t≥0 is a continuous path on C defined by Consider the following path-dependent SDE on R d : where (W (t)) t≥0 is a d-dimensional Brownian motion on a complete filtration probability space (Ω, F , (F t ) t≥0 , P), and are measurable maps satisfying the following assumptions.
Let P t be the associated Markov process, i.e., For a probability measure µ on C , let µP t be the law of X t with initial distribution µ.
holds for a family of random variables To state the CLT, we introduce the corrector R f for f ∈ C p,γ (C ) defined by This function is well-defined since (1.2) and µ ∞ ∈ P p,γ (C ) imply for some constants c 1 , c 2 > 0. Let For any D ∈ [0, ∞), let Φ D be the normal distribution function with zero mean and variance D, where Φ 0 (z) := 1 [0,∞) (z) for D = 0. We have the following CLT.
and the following assertion holds: (2) When D f = 0, there exists an increasing function h 0 : Finally, to investigate the LIL, we consider the unit ball in the Camron-Martin space of C([0, 1]; R): and the following discrete version of R f and ϕ f for f ∈ C p,γ (C ) with µ ∞ (f ) = 0: which are well defined due to (1.4). For any n ≥ 1, consider the following random variable on C([0, 1]; R): Then the sequence {Λ f,ξ n (·)} n≥1 is almost surely relatively compact in C([0, 1]; R), and when n → ∞ the set of limit points coincides with H. Consequently, P-a.s.
Note that the LIL has been intensively investigated for many different models, see e.g. [3,7,8,10,13,20,25] and references therein. Theorem 1.3 is a supplement in the setting of path-dependent SDEs.
The remainder of this paper is arranged as follows. In Section 2, we recall some known results on SLLN, CLT and LIL for Markov processes, which are then applied to prove the above three results in Sections 3-5 respectively.

Some known results
We first state some results presented in [24] for continuous Markov processes on separable Hilbert spaces. Since proofs of these results only use the norm rather than the inner product of the space, they apply also to a Banach space.
Let {X x t : x ∈ B, t ≥ 0} be a continuous Markov process on a separable Banach space (B, · ) with respect to a complete filtration probability space (Ω, F , (F t ) t≥0 , P) such that the associate Markov semigroup has a unique invariant probability measure µ ∞ . For a constant γ ∈ (0, 1] and an increasing function w ∈ C([0, ∞); [1, ∞)), let C w,γ (B) be the class of measurable functions on B such that Note that in [24] f w,γ is defined by using x − y γ instead of 1 ∧ x − y γ , but this does not make essential differences since these two definitions are equivalent up to a constant multiplication. We take the present formulation in order to apply the ergodicity result derived in [2]. By [24, Proposition 2.6], we have the following result.
and for some k ∈ N, then for any f ∈ C w,γ (B), where w −1 is the inverse of w. Then for any ε ∈ (0, 1 2 ), there exist a constant c ε > 0 and a family of random variables and Let f ∈ C w,γ (B) and x ∈ B, assume that is a well-defined square integrable martingale. Consider its discrete time quadratic variation process Let ⌊t⌋ = sup{k ∈ Z + : k ≤ t} be the integer part of t ≥ 0. The following CLT is due to [24,Theorem 2.8].
holds for some constant α > 0 and continuous function κ : R + → R + . Then (2) There exists an increasing function h : Finally, let (M n ) n≥0 be a square integrable martingale and let Z n = M n − M n−1 be the martingale difference. The following result is taken from [16, Theorem 1].
Lemma 2.4. Assume that S n := EM n → ∞ as n → ∞, and there exists a constant δ > 0 such that and P-a.s.
Then the sequence (Λ n ) n≥1 of random variables on C([0, 1]; R) defined by is almost surely relatively compact, and the set of its limits points coincides with H in (1.6).
Proof. (1) By Jensen's inequality, concerning (3.1) we only need to consider p ≥ 2. Since λ 1 − λ 2 e λ 1 r 0 > 0, there exists a constant ε ∈ (0, λ 1 ) such that According to (A1) and (A3), we may find a constant c 0 > 0 such that So, by Itô's formula, holds for some constant c 1 > 0 and the martingale Noting that where N ξ (t) := sup 0≤s≤t M ξ (s). By invoking Gronwall's inequality (see e.g. [11,Theorem 11]), this implies Combining this with Hölder's inequality, for fixed p ≥ 2 we may find constants c 2 , c 3 > 0 such that On the other hand, by means of (A3) and using BDG's and Hölder's inequalities, there exist constants c 4 , c 5 > 0 such that Substituting this into (3.5), and noting that due to λ 1 − ε > λ ε > 0 we have we may find a constant C > 0 such that By a truncation argument with stopping times, we may and do assume that E X ξ t p ∞ < ∞, so that by Gronwall's inequality, this implies the desired estimate (3.1) for some constants c, β > 0.

Proof of Theorem 1.3
Let us fix f ∈ C p,γ (C ) with µ ∞ (f ) = 0. To apply Lemma 2.4, for any ξ ∈ C , we consider The argument after (4.1) implies that (M ξ n ) n≥0 is a well-defined square integrable martingale. Let Proof. According to the proof of [3, Lemma 3.2], it suffices to show that the maps are continuous. For simplicity, we only prove the continuity of Λ 1 as that of the other is completely similar. By definition it is easy to see that Combining this with (1.4), we find constants c 1 , c 2 > 0 such that ∞ k=n e −β(k−n) ρ p,γ (X ξ n , X η n ) + ρ p,γ (X ξ n−1 , X η n−1 ) ≤ c 2 ρ p,γ (X ξ n , X η n ) + ρ p,γ (X ξ n−1 , X η n−1 ) .
So, (2.11) holds for (S n , Z n ) = (S ξ n , Z ξ n ) as well, and hence the assertion in (1) follows from Lemma 2.4.
(2) It remains to prove (1.8). By the first assertion, Λ f,ξ n (t) is almost surely relatively compact in C([0, 1]; R) and the set of its limits points coincides with H. Since h H ≤ 1 for any h ∈ H, this implies P-a.s. On the other hand, since the limits points of (Λ f,ξ n (t)) coincides with H and h ∈ H with h(t) = t, t ∈ [0, 1], there exists a subsequence n k ↑ ∞ as k → ∞ such that P-a.s. In particular, combining this with (1.7) for k = n − 1 and t = k n , we deduce P-a.s.