1. Introduction and the main theorem
In analysis and probability theory, we often consider the convergence of sums
Here, $\pi $ is a partition of an interval $[0, T]$ , and we consider the limit of
For instance, if , then we consider a Riemann sum approximation of $\int _0^T f(s) \, \mathrm {d} s$ , and if , where W is a Brownian motion and X is an adapted process, then we consider the Itô approximation of the stochastic integral $\int _0^T X_r \, \mathrm {d} W_r$ .
Gubinelli [Reference Gubinelli17], inspired by Lyons’ results on almost multiplicative functionals in the theory of rough paths [Reference Lyons30], showed that if
satisfies $ \lvert {\delta A_{s, u, t}}\rvert \lesssim \lvert {t - s} \rvert ^{1 + \epsilon }$ for some $\epsilon> 0$ , then the sums (1.1) converge. This result is now called the sewing lemma, named so in the work of Feyel and de La Pradelle [Reference Feyel and de La Pradelle13]. This lemma is so powerful that many applications and many extensions are known. For instance, it can be used to define rough integrals (see [Reference Gubinelli17] and the monograph [Reference Friz and Hairer14] of Friz and Hairer).
When $(A_{s, t})_{s \leq t}$ is random, and when we want to prove the convergence of the sums (1.1), the above sewing lemma is often not sufficient. For instance, if , the sums converge to the quadratic variation of the Brownian motion. However, we only have
almost surely for every $\epsilon> 0$ , and hence, we cannot apply the sewing lemma.
Lê [Reference Lê24] proved a stochastic version of the sewing lemma (stochastic sewing lemma): if a filtration $(\mathcal {F}_t)_{t \in [0, T]}$ is given, such that
-
• $A_{s, t}$ is $\mathcal {F}_t$ -measurable and
-
• for some $\epsilon _1, \epsilon _2> 0$ and $m \in [2, \infty )$ , we have for every $s < u < t$ ,
(1.3) $$ \begin{align} \lVert {\mathbb{E}[\delta A_{s, u, t} \vert \mathcal{F}_s]} \rVert _{L_m(\mathbb{P})} \lesssim \lvert {t - s} \rvert ^{1 + \epsilon_2}, \end{align} $$(1.4) $$ \begin{align} \lVert {\delta A_{s, u, t}} \rVert _{L_m(\mathbb{P})} \lesssim \lvert {t - s} \rvert ^{\frac{1}{2} + \epsilon_1}, \end{align} $$
then the sums (1.1) converge in $L_m(\mathbb {P})$ . As usual, the Banach space $L_m(\mathbb {P})$ is equipped with the norm
If , then we have $\mathbb {E}[\delta A_{s, u, t} \vert \mathcal {F}_s] = 0$ and (1.4) is satisfied with $\epsilon _1 = \frac {1}{2}$ . Therefore, we can prove the convergence of (1.1) in $L_m(\mathbb {P})$ . The stochastic sewing lemma has been already shown to be very powerful in the original work [Reference Lê24] of Lê, and an increasing number of papers are appearing that take advantage of the lemma.
However, there are situations where Lê’s stochastic sewing lemma seems insufficient. For instance, consider
where B is a fractional Brownian motion with Hurst parameter $H \in (0, 1)$ . It is well-known that the sums (1.1) converge to $c_H T$ in $L_m(\mathbb {P})$ . Although we have the estimate (1.4), we fail to obtain the estimate (1.3) unless $H = \frac {1}{2}$ .
To get an idea on how Lê’s stochastic sewing lemma should be modified for this problem, observe the following trivial fact:
This suggests that we consider estimates that interpolate $\mathbb {E}[\delta A_{s, u, t}]$ and $\mathbb {E}[\delta A_{s, u, t} \vert \mathcal {F}_s]$ . In fact, we can obtain the following estimates:
We can prove (1.6), for instance, by applying Picard’s result [Reference Picard37, Lemma A.1] on the asymptotic independence of fractional Brownian increments, or more directly by doing a similar calculation as in Section 4. This discussion motivates the following main theorem of our paper.
Theorem 1.1. Suppose that we have a filtration $(\mathcal {F}_t)_{t \in [0, T]}$ and a family of $\mathbb {R}^d$ -valued random variables $(A_{s, t})_{0 \leq s \leq t \leq T}$ , such that $A_{s, s} = 0$ for every $s \in [0, T]$ and such that $A_{s, t}$ is $\mathcal {F}_t$ -measurable. We define $\delta A_{s, u, t}$ by (1.2). Furthermore, suppose that there exist constants
such that the following conditions are satisfied.
-
• For every $0 \leq t_0 < t_1 < t_2 < t_3 \leq T$ , we have
(1.7) $$ \begin{align} \lVert {\mathbb{E}[\delta A_{t_1, t_2, t_3} \vert \mathcal{F}_{t_0}]} \rVert _{L_m(\mathbb{P})} &\leq \Gamma_1 (t_1 - t_0)^{-\alpha} (t_3 - t_1)^{\beta_1}, \quad \text{if } M(t_3 - t_1) \leq t_1 - t_0, \end{align} $$(1.8) $$ \begin{align} \lVert {\delta A_{t_0, t_1, t_2}} \rVert _{L_m(\mathbb{P})} &\leq \Gamma_2 (t_2 - t_0)^{\beta_2}. \end{align} $$ -
• We have
(1.9) $$ \begin{align} \beta_1> 1, \quad \beta_2 > \frac{1}{2}, \quad \beta_1 - \alpha > \frac{1}{2}. \end{align} $$
Then, there exists a unique, up to modifications, $\mathbb {R}^d$ -valued stochastic process $(\mathcal {A}_t)_{t \in [0, T]}$ with the following properties.
-
• $\mathcal {A}_0 = 0$ , $\mathcal {A}_t$ is $\mathcal {F}_t$ -measurable, and $\mathcal {A}_t$ belongs to $L_m(\mathbb {P})$ .
-
• There exist nonnegative constants $C_1$ , $C_2$ , and $C_3$ , such that
(1.10) $$ \begin{align} \lVert {\mathbb{E}[\mathcal{A}_{t_2} - \mathcal{A}_{t_1} - A_{t_1, t_2} \vert \mathcal{F}_{t_0}]} \rVert _{L_m(\mathbb{P})} \leq C_1 \lvert {t_1 - t_0} \rvert ^{-\alpha} \lvert {t_2 - t_1} \rvert ^{\beta_1} , \end{align} $$(1.11) $$ \begin{align} \lVert {\mathcal{A}_{t_2} - \mathcal{A}_{t_1} - A_{t_1, t_2} } \rVert _{L_m(\mathbb{P})} \leq C_2 \lvert {t_2 - t_1} \rvert ^{\beta_1 - \alpha} + C_3 \lvert {t_2 - t_1} \rvert ^{\beta_2}, \end{align} $$where $t_2 - t_1 \leq M^{-1}(t_1 - t_0)$ is assumed for the inequality (1.10).
In fact, we can choose $C_1$ , $C_2$ , and $C_3$ so that
where $\kappa _{m,d}$ is the constant of the Burkholder-Davis-Gundy inequality (see (1.14)). Furthermore, for $\tau \in [0, T]$ , if we set
then the family $(A^{\pi }_{\tau })_{\pi }$ converges to $\mathcal {A}_{\tau }$ in $L_m(\mathbb {P})$ as $ \lvert {\pi } \rvert $ tends to $0$ .
Remark 1.2. We discuss the optimality of the condition (1.9). By considering a deterministic $(A_{s,t})$ , we see that the condition $\beta _1> 1$ is necessary. To see that the conditions $\beta _2> \frac {1}{2}$ and $\beta _1 - \alpha> \frac {1}{2}$ are necessary, let $B^1$ and $B^2$ be two independent one-dimensional fractional Brownian motions with Hurst parameter $\frac {1}{4}$ (see Definition 3.1), and we set . It is well-known since the work [Reference Coutin and Qian10] of Coutin and Qian that the iterated integral $\int B^1 \mathrm {d} B^2$ does not exist, and, therefore, the Riemann sum with respect to $(A_{s, t})$ should not converge. In fact, the family $(A_{s, t})$ , with filtration $(\mathcal {F}_t)$ generated by $(B^1, B^2)$ , satisfies (1.7) and (1.8) with
To see this, we observe $\delta A_{t_1, t_2, t_3} = - B^1_{t_1, t_2} B^2_{t_2, t_3}$ , and
To compute the conditional expectation, we observe
and by the estimate (3.4), we have
Remark 1.3. The proof shows that if
then we have $C_2 \lesssim _{\alpha , \beta _1, \beta _2, M} \Gamma _1$ , and we can omit the factor $\kappa _{m,d}$ . This is similar to [Reference Lê24], where $C_2$ also does not depend on $\kappa _{m,d}$ . If $\alpha = 0$ and $M = 0$ , Theorem 1.1 recovers Lê’s stochastic sewing lemma [Reference Lê24, Theorem 2.1]. If $\alpha = 0$ and $M>0 $ , it recovers a lemma [Reference Gerencsér16, Lemma 2.2] by Gerencsér.
Recently, Gerencsér’s stochastic sewing lemma is called shifted stochastic sewing lemma. In the follow-up works, we continue to refer to Theorem 1.1 by the same name.
Remark 1.4. The proof shows that there exists $\epsilon = \epsilon (\alpha , \beta _1, \beta _2)> 0$ , such that
for every $\tau \in [0, T]$ and every partition $\pi $ of $[0, \tau ]$ . A similar remark holds in the setting of Corollary 2.7.
Remark 1.5. As in another work [Reference Lê26] of Lê, it should be possible to extend Theorem 1.1 so that the stochastic process $(A_{s,t})_{s,t \in [0, T]}$ takes values in a certain Banach space.
Remark 1.6. A multidimensional version of the sewing lemma is the reconstruction theorem [Reference Hairer18, Theorem 3.10] of Hairer. A stochastic version of the reconstruction theorem was obtained by Kern [Reference Kern21]. It could be possible to extend Theorem 1.1 in the multidimensional setting, but we will not pursue it in this paper.
The proof of Theorem 1.1 is given in Section 2. If $A_{s, t}$ is given by (1.5), then we can apply Theorem 1.1 with
However, the application of Theorem 1.1 goes beyond this simple problem of $\frac {1}{H}$ -variation of the fractional Brownian motion. Indeed, in Section 3, we prove the convergence of Itô and Stratonovich approximations to the stochastic integrals
with $H> \frac {1}{2}$ in Itô’s case and with $H> \frac {1}{6}$ in Stratonovich’s case, under rather general assumptions on the regularity of f, in fact, $f \in C^2_b(\mathbb {R}^d, \mathbb {R}^d)$ works for all $H> \frac 16$ . In Section 4, we obtain new representations of local times of fractional Brownian motions via discretization.
Finally, we remark that one of the most interesting applications of Lê’s stochastic sewing lemma lies in the phenomenon of regularization by noise (see, e.g. [Reference Lê24], Athreya et al. [Reference Athreya, Butkovsky, Lê and Mytnik2], [Reference Gerencsér16], and Anzeletti et al. [Reference Anzeletti, Richard and Tanré1]). In these works, they consider the stochastic differential equation (SDE)
with an additive noise Y, which is often a fractional Brownian motion. It is interesting that, although in absence of noise the coefficient b needs to belong to $C^1$ for well-posedness, the presence of noise enables us to prove the certain well-posedness of (1.13) under much weaker assumption, in fact, b can be even a distribution; hence the name regularization by noise. In Section 5 of our paper, we are interested in a related but different problem. Indeed, we are interested in improving the regularity of the diffusion coefficient rather than the drift coefficient. We consider the Young SDE
driven by a fractional Brownian motion B with Hurst parameter $H \in (\frac {1}{2}, 1)$ . The pathwise theory of Young’s differential equation requires that the regularity of $\sigma $ is better than $1/H$ for uniqueness, and this condition is sharp for general drivers B of the same regularity as the fractional Brownian motion. We will improve this regularity assumption for pathwise uniqueness and strong existence. Again, a stochastic sewing lemma (Lemma 5.5), which is a variant of Theorem 1.1, will play a key role.
Notation
We write and . Given a function $f:[S, T] \to \mathbb {R}^d$ , we write . We denote by $\kappa _{m, d}$ the best constant of the discrete Burkholder-Davis-Gundy (BDG) inequality for $\mathbb {R}^d$ -valued martingale differences [Reference Burkholder, Davis and Gundy6]. Namely, if we are given a filtration $(\mathcal {F}_n)_{n=1}^{\infty }$ and a sequence $(X_n)_{n=1}^{\infty }$ of $\mathbb {R}^d$ -valued random variables, such that $X_n$ is $\mathcal {F}_{n}$ -measurable for every $n \geq 1$ and $\mathbb {E}[X_n \vert \mathcal {F}_{n-1}] = 0$ for every $n \geq 2$ , then
Rather than (1.14), we mostly use the inequality
for $m \geq 2$ , which follows from (1.14) by Minkowski’s inequality. We write $A \lesssim B$ or $A = O(B)$ if there exists a nonnegative constant C, such that $A \leq C B$ . To emphasize the dependence of C on some parameters $a, b, \ldots $ , we write $A \lesssim _{a, b, \ldots } B$ .
2. Proof of the main theorem
The overall strategy of the proof is the same as that of the original work [Reference Lê24] of Lê. Namely, we combine the argument of the deterministic sewing lemma ([Reference Gubinelli17], [Reference Feyel and de La Pradelle13], and Yaskov [Reference Yaskov40]) with the discrete BDG inequality [Reference Burkholder, Davis and Gundy6]. However, the proof of Theorem 1.1 requires more labor at a technical level. Some proofs will be postponed to Appendix A.
As in [Reference Lê24], the following lemma, which originates from [Reference Yaskov40], will be needed. It allows us to replace general partitions by dyadic partitions.
Lemma 2.1 [Reference Lê24, Lemma 2.14].
Under the setting of Theorem 1.1, let
Then, we have
where
and
and where $R^n_i = 0$ for all sufficiently large n.
The next two lemmas (Lemmas 2.2 and 2.3) correspond to the estimates [Reference Lê24, (2.50) and (2.51)], respectively.
Lemma 2.2. Under the setting of Theorem 1.1, let
Then,
Proof. In view of the decomposition (2.1), the triangle inequality gives
Therefore, recalling $\beta _1> 1$ from (1.9), the claim follows.
The following lemma is the most important technical ingredient for the proof of Theorem 1.1.
Lemma 2.3. Under the setting of Theorem 1.1, let
Then,
Under (1.12), we can replace $\kappa _{m, d} \Gamma _1$ by $\Gamma _1$ .
Proof under (1.12).
To simplify the proof, here, we assume (1.12), that is, that the additional technical condition $1+\alpha -\beta _1 < 2\alpha \beta _2 - \alpha $ holds. The proof in the general setting will be given in Appendix A.
We, again, use the representation (2.1). We fix a large $n \in \mathbb {N}$ and set . Fix an integer $L = L_n \in [M + 1, 2^n]$ , which will be chosen later. We have
We estimate the first term of (2.3). By the BDG inequality together with Minkowski’s inequality (see (1.15)), we have
Using (1.8) and (2.2) and noting that we include more terms in the sum by requiring $j \le 2^n/L$ only instead of $Lj+l \le 2^n - 1$ , we get
Therefore,
We next estimate the second term of (2.3). The triangle inequality yields
By (1.7),
Therefore,
In conclusion,
We wish to choose $L = L_n$ so that (2.4) is summable with respect to n. We therefore set , where
Such a $\delta $ exists exactly under the additional technical assumption (1.12), namely, if $1 + \alpha - \beta _1 < 2 \alpha \beta _2 - \alpha $ . Then, (2.4) yields
To estimate the contribution coming from the small n with $2^{n\delta } < M+2$ , we apply (1.8) which yields
Thus, we conclude
where the fact $\kappa _{m, d} \geq 1$ is used.
Lemma 2.4. Under the setting of Theorem 1.1, let $\pi , \pi '$ be partitions of $[0, T]$ , such that $\pi $ refines $\pi '$ . Suppose that we have
Then, there exists $\epsilon \in (0, 1)$ , such that
Sketch of the proof.
Here, we give a sketch of the proof under (1.12). The complete proof is given in Appendix A. The argument is similar to Lemma 2.3.
Write
and
We set , where $\delta $ satisfies (2.5). We set
As in Lemma 2.3, we consider the decomposition $A^{\pi '}_T - A^{\pi }_T = A + B$ , where
We estimate A by using the BDG inequality, Lemma 2.3, and (2.6), to obtain
We estimate B by using the triangle inequality, Lemma 2.2, and (2.6), to obtain
As in Lemma 2.3, we choose with $\delta $ satisfying (2.5). We then obtain the claimed estimate.
Remark 2.5. In the setting of Lemma 2.4, assume that the adapted process $(\mathcal A_t)_{t \in [0,T]}$ satisfies (1.10) and (1.11). Then we obtain for some $\varepsilon> 0$ :
Indeed, it suffices to replace $A_{t_{jL + l}, t_{jL + l +1}}$ by $\mathcal A_{t_{jL + l}, t_{jL + l +1}}$ in the previous proof.
Lemma 2.6. Let $\pi $ be a partition of $[0, T]$ . Then, there exists a partition $\pi '$ of $[0, T]$ , such that $\pi $ refines $\pi '$ , $ \lvert {\pi '} \rvert \leq 3 \lvert {\pi } \rvert $ and
Proof. We write $\pi = \{0=t_0 < t_1 < \cdots <t_{N-1} < t_N = T\}$ . We set , and for $l\in \mathbb {N}$ , we inductively set
Set . Then, we define
By construction, $\pi '= \{s_j\}_{j=1}^L$ satisfies the claimed properties: $s_{j+1} - s_j \le 2|\pi |$ if $j<L-2$ , and $s_L - s_{L-1} \le 3 |\pi |$ , so $|\pi '| \le 3 |\pi |$ ; moreover, $\min _{[s,t] \in \pi '}|t-s| \ge |\pi | \ge 3^{-1} |\pi '|$ .
Proof of Theorem 1.1.
We will not write down dependence on $\alpha , \beta _1, \beta _2, M, m, d, T$ . We first prove the convergence of $(A^{\pi }_{\tau })_{\pi }$ . Without loss of generality, we assume $\tau = T$ . Let $\pi _1, \pi _2$ be partitions of $[0, T]$ . By Lemma 2.6, there exist partitions $\pi _1'$ , $\pi _2'$ , such that for $j \in \{1, 2\}$ , the partition $\pi _j$ refines $\pi _j'$ , $ \lvert {\pi ^{\prime }_j} \rvert \leq 3 \lvert {\pi _j} \rvert $ and
Lemma 2.4 shows that for some $\epsilon> 0$ , we have
Therefore, by the triangle inequality,
Let $\pi $ refine both $\pi _1'$ and $\pi _2'$ . Lemma 2.4 implies that
The estimates (2.7) and (2.8) show
Thus, $\{A^{\pi }_T\}_{\pi }$ forms a Cauchy net in $L_m(\mathbb {P})$ . We denote the limit by $\mathscr {S}_T$ . We next prove that $(\mathscr {S}_t)_{t \in [0, T]}$ satisfies (1.10) and (1.11). Let $t_0 < t_1 < t_2$ be such that $M(t_2 - t_1) \leq t_1 - t_0$ . Let $\pi _n = \{ t_1 + k 2^{-n}(t_2-t_1): k=0,\dots , 2^n\}$ be the nth dyadic partition of $[t_1, t_2]$ , and we write
We have
By Lemma 2.2,
In this estimate, we can replace $A^n_{t_1, t_2}$ by $\mathscr {S}_{t_1, t_2}$ in view of (2.9). Similarly, by Lemma 2.3, we obtain
Under (1.12), we can replace $ \kappa _{m, d} \Gamma _1$ by $\Gamma _1$ .
Finally, let us prove the uniqueness of $\mathcal {A}$ . Let $(\tilde {\mathcal {A}}_t)_{t \in [0, T]}$ be another adapted process satisfying $\tilde {\mathcal {A}}_0 = 0$ , (1.10) and (1.11). It suffices to show $\mathcal {A}_T = \tilde {\mathcal {A}}_T$ almost surely. Let $\pi _n$ be the nth dyadic partition of $[0, T]$ . By Remark 2.5, we have
Since $n \in \mathbb {N}$ is arbitrary, we must have $\mathcal {A}_T = \tilde {\mathcal {A}}_T$ almost surely.
As in [Reference Athreya, Butkovsky, Lê and Mytnik2, Theorem 4.1] of Athreya et al., we will give an extension of Theorem 1.1 that allows singularity at $t = 0$ , which will be needed in Section 4.
Corollary 2.7. Suppose that we have a filtration $(\mathcal {F}_t)_{t \in [0, T]}$ and a family of $\mathbb {R}^d$ -valued random variables $(A_{s, t})_{0 \leq s \leq t \leq T}$ , such that $A_{s, s} = 0$ for every $s \in [0, T]$ and such that $A_{s, t}$ is $\mathcal {F}_t$ -measurable. Furthermore, suppose that there exist constants
such that the following conditions are satisfied.
-
• For every $0 \leq t_0 < t_1 < t_2 < t_3 \leq T$ , we have
(2.10) $$ \begin{align} \lVert {\mathbb{E}[\delta A_{t_1, t_2, t_3} \vert \mathcal{F}_{t_0}]} \rVert _{L_m(\mathbb{P})} &\leq \Gamma_1 t_1^{-\gamma_1} (t_1 - t_0)^{-\alpha} (t_3 - t_1)^{\beta_1}, \end{align} $$(2.11) $$ \begin{align} \lVert {\delta A_{t_0, t_1, t_2}} \rVert _{L_m(\mathbb{P})} &\leq \Gamma_2 t_0^{-\gamma_2} (t_2 - t_0)^{\beta_2}, \end{align} $$(2.12) $$ \begin{align} \lVert {\delta A_{t_0, t_1, t_2}} \rVert _{L_m(\mathbb{P})} &\leq \Gamma_3 (t_2 - t_0)^{\beta_3}, \end{align} $$where $M(t_3 - t_1) \leq t_1 - t_0$ is assumed for (2.10) and $t_0> 0$ is assumed for (2.11). -
• We have
(2.13) $$ \begin{align} \beta_1> 1, \quad \beta_2 > \frac{1}{2}, \quad \beta_1 - \alpha > \frac{1}{2}, \quad \gamma_1, \gamma_2 < \frac{1}{2}, \quad \beta_3> 0. \end{align} $$
Then, there exists a unique, up to modifications, $\mathbb {R}^d$ -valued stochastic process $(\mathcal {A}_t)_{t \in [0, T]}$ with the following properties.
-
• $\mathcal {A}_0 = 0$ , $\mathcal {A}_t$ is $\mathcal {F}_t$ -measurable and $\mathcal {A}_t$ belongs to $L_m(\mathbb {P})$ .
-
• There exist nonnegative constants $C_1, \ldots , C_6$ , such that
(2.14) $$ \begin{align} & \lVert {\mathbb{E}[\mathcal{A}_{t_2} - \mathcal{A}_{t_1} - A_{t_1, t_2} \vert \mathcal{F}_{t_0}]} \rVert _{L_m(\mathbb{P})} \leq C_1 t_1^{-\gamma_1} \lvert {t_1 - t_0} \rvert ^{-\alpha} \lvert {t_2 - t_1} \rvert ^{\beta_1},\end{align} $$(2.15) $$ \begin{align} & \lVert {\mathcal{A}_{t_2} - \mathcal{A}_{t_1} - A_{t_1, t_2} } \rVert _{L_m(\mathbb{P})} \leq C_2 t_1^{-\gamma_1} \lvert {t_2 - t_1} \rvert ^{\beta_1 - \alpha} + C_3 t_1^{-\gamma_2} \lvert {t_2 - t_1} \rvert ^{\beta_2}, \end{align} $$(2.16) $$ \begin{align} & \lVert {\mathcal{A}_{t_2} - \mathcal{A}_{t_1} - A_{t_1, t_2} } \rVert _{L_m(\mathbb{P})} \leq C_4 \lvert {t_2 - t_1} \rvert ^{\beta_1 - \alpha_1 -\gamma_1} + C_5 \lvert {t_2 - t_1} \rvert ^{\beta_2 - \gamma_2} + C_6 \lvert {t_2 - t_1} \rvert ^{\beta_3}, \end{align} $$where $t_2 - t_1 \leq M^{-1}(t_1 - t_0)$ is assumed for the inequality (2.14) and $t_1> 0$ is assumed for the inequality (2.15).
In fact, we can choose $C_1, \ldots , C_6$ so that
Furthermore, for $\tau \in [0, T]$ , if we set
then the family $(A^{\pi }_{\tau })_{\pi }$ converges to $\mathcal {A}_{\tau }$ in $L_m(\mathbb {P})$ as $ \lvert {\pi } \rvert \to 0$ .
The proof is given in Appendix A.
3. Integration along fractional Brownian motions
The goal of this section is to prove the convergence of Itô and Stratonovich approximations of
along a multidimensional fractional Brownian motion B with Hurst parameter H, using Theorem 1.1. For Itô’s case, we let $H \in (\frac {1}{2}, 1)$ , and for Stratonovich’s case, we let $H \in (\frac {1}{6}, \frac {1}{2})$ .
Definition 3.1. Let $(\mathcal {F})_{t \in \mathbb {R}}$ be a filtration. We say that a process B is an $(\mathcal {F}_t)$ -fractional Brownian motion with Hurst parameter $H \in (0, 1)$ if
-
• a two-sided d-dimensional $(\mathcal {F}_t)$ -Brownian motion $(W_t)_{t \in \mathbb {R}}$ is given;
-
• a random variable $B(0)$ is a (not necessarily centered) $\mathcal {F}_0$ -measurable $\mathbb {R}^d$ -valued Gaussian random variable and is independent of $(W_t)_{t \in \mathbb {R}}$ ;
-
• we set
(3.1) $$ \begin{align} B_t = B(0) + \int_{\mathbb{R}} K_H(t, s) \, \mathrm{d} W_s. \end{align} $$
If B has the representation (3.1), then
where we write $B = (B^i)_{i=1}^d$ in components and $\mathcal {B}$ is the Beta function
Regarding the expression of the constant $c_H$ , see [Reference Picard, Donati-Martin, Lejay and Rouault38, Appendix B]. In particular, we have
In this section, we always write B for an $(\mathcal {F}_t)$ -fractional Brownian motion. An advantage of the representation (3.1) is that given $v < s$ , we have the decomposition
where the second term $\int _v^s K(s, r) \, \mathrm {d} W_r$ is independent of $\mathcal {F}_v$ . Later, we will need to estimate the correlation of
We note that for $s \leq t$
Lemma 3.2. Let $H \neq \frac {1}{2}$ . Let $0 \leq v < s \leq t$ be such that $t-s \leq s - v$ . Then,
where we have
uniformly over such $v, s, t$ .
Proof. See Appendix A.
We apply Theorem 1.1 to construct a stochastic integral
as the limit of Riemann type approximations. An advantage of the stochastic sewing lemma is that we do not need any regularity of f. We denote by $L_{\infty }(\mathbb {R}^d,\mathbb {R}^d)$ the space of bounded measurable maps from $\mathbb {R}^d$ to $\mathbb {R}^d$ . We write
for the inner product of $\mathbb {R}^d$ .
Proposition 3.3. Let $H \in (1/2, 1)$ and $f \in L_{\infty }(\mathbb {R}^d, \mathbb {R}^d)$ . Then, for any $\tau \in [0, T]$ and $m \in [2, \infty )$ , the sequence
converges in $L_m(\mathbb {P})$ for every $m < \infty $ as $ \lvert {\pi } \rvert \to 0$ . Furthermore, if we denote the limit by $\int _0^{\tau } f(B_r) \, \mathrm {d} B_r$ and if we write
then for every $0 \leq s < t \leq T$ ,
Remark 3.4. We can replace $f(B_s)$ by $f(B_u)$ for any $u \in [s, t]$ . It is well-known that the sums converge to the Young integral if $f \in C^{\gamma } (\mathbb {R})$ with $\gamma> H^{-1} (1 - H)$ . Yaskov [Reference Yaskov40, Theorem 3.7] proves that the sums converge in some $L_p(\mathbb {P})$ -space if f is of bounded variation.
Proof. We will not write down dependence on d, H, and m. The filtration $(\mathcal {F}_t)_{t \in \mathbb {R}}$ is generated by the Brownian motion W appearing in the Mandelbrot–Van Ness representation (3.1). We will apply Theorem 1.1 with . Let $m \geq 2$ . We have
To estimate conditional expectations, let $0 \leq v < s < t$ be such that $t-s \leq s-v$ and set
We write , if conditioned under $\mathcal {F}_v$ . Namely, we write, for instance
We are going to compute $\mathbb {E}[A_{s, t} \vert \mathcal {F}_v]$ . Conditionally on $\mathcal {F}_v$ , we have the Wiener chaos expansion [Reference Nualart34, Theorem 1.1.1]
where $\tilde {B}^{\perp }_s$ is orthogonal in $L_2(\mathbb {P})$ to the subspace spanned by the constant $1$ and
Note that
Then, by the orthogonality of the Wiener chaos decomposition,
Hence, for $u \in (s, t)$ ,
where
Here, $\boldsymbol {e}_i$ is the ith unit vector of $\mathbb {R}^d$ . We first estimate $A^0_{s, u, t}$ , for which we begin with estimating $a_0(s) - a_0(u)$ . We set
where X has the standard normal distribution in $\mathbb {R}^d$ . Note that
and similarly for $a_0(u)$ , we have
Therefore,
This yields
Therefore,
The random variable $Y_{s, u}$ is Gaussian and
We have a similar estimate for $Y_{u, t}$ . Therefore,
Now we move to estimate $A^i_{s, u, t}$ . By Lemma 3.2, we have
Therefore, if we write ,
If we set
then $a_i^i(s) = G_i(Y_s, (2H)^{-\frac {1}{2}} (s - v)^H)$ and similarly for $a_i^i(u)$ . Since
we have
Therefore,
and thus
This yields
and
if $t-s \leq s - v$ .
Therefore, by (3.3) and (3.5),
if $t-s \leq s - v$ . Hence, $(A_{s, t})$ satisfies the assumption of Theorem 1.1 with
Next, we consider the case $H \in (\frac {1}{6}, \frac {1}{2})$ . The following result reproduces [Reference Nourdin35, Theorem 3.5], with a more elementary proof and with improvement of the regularity of f. More precisely, the cited result requires $f \in C^6$ while here $f \in C^\gamma $ with $\gamma> \frac {1}{2H}-1$ is sufficient and thus, in particular, $f \in C^2$ works for all $H \in (\frac {1}{6}, \frac {1}{2})$ . We denote by $C^{\gamma } (\mathbb {R}^d,\mathbb {R}^d)$ the space of $\gamma $ -Hölder maps from $\mathbb {R}^d$ to $\mathbb {R}^d$ , with the norm
if $\gamma \in (0, 1)$ and
if $\gamma \in (1, 2)$ .
Proposition 3.5. Let $H \in (\frac {1}{6}, \frac {1}{2})$ , $\gamma> \frac {1}{2H} - 1$ and $f \in C^{\gamma }(\mathbb {R}^d, \mathbb {R}^d)$ . If $H \leq \frac {1}{4}$ and $d> 1$ , assume furthermore that
Then, for every $m \in [2, \infty )$ and $\tau \in [0, T]$ , the family of Stratonovich approximations
converges in $L_m(\mathbb {P})$ as $ \lvert {\pi } \rvert \to 0$ . Moreover, if we denote the limit by $\int _0^{\tau } f(B_r) \circ \, \mathrm {d} B_r$ and if we write
then for every $0 \leq s < t \leq T$ , we have
Proof. We will not write down dependence on d, H, m, and $\gamma $ . The filtration $(\mathcal {F}_t)_{t \in \mathbb {R}}$ is generated by the Brownian motion W appearing in the Mandelbrot–Van Ness representation (3.1). We can assume
We will apply Theorem 1.1 with
We first claim
Observe
If $H> \frac {1}{4}$ , the claim (3.7) follows from the estimates
If $H \leq \frac {1}{4}$ , then $\gamma> 1$ , and we have
where (3.6) is used. Then, the claim (3.7) follows, again, from the Hölder estimate of f. Note that the condition $\gamma> \frac {1}{2H} - 1$ is equivalent to $(\gamma + 1) H> \frac {1}{2}$ .
The rest of the proof consists of estimating the conditional expectation $\mathbb {E}[\delta A_{s, u, t} \vert \mathcal {F}_v]$ . Let $t -s \leq s - v$ . We will use the same notation as in the proof of Proposition 3.3. We have
where
and
We first estimate $D^0_{s, u, t}$ . Suppose that $H> \frac {1}{4}$ . Recall
Therefore,
This yields
Therefore, by (3.4),
Now suppose that $H \leq \frac {1}{4}$ . To simplify notation, we write . Since (3.6) gives $\partial _{m^i} I^j = \partial _{m^j} I^i$ for every $i, j$ , we have
Since
we have
Notice
Therefore,
This yields
Hence, we obtain the estimate (3.10) when $H \leq \frac {1}{4}$ .
We move to estimate $D^i_{s, u, t}$ . By using the identity,
we obtain
Since the other terms can be estimated similarly, we only estimate $(a_i^i(t) - a_i^i(u)) \mathbb {E}[\tilde {B}_t^i \tilde {B}_{s, t}^i]$ . By Lemma 3.2,
Now we estimate $ \lvert {a_i^i(t) - a_i^i(u)} \rvert $ . Recall $a_i^i(s) = G_i(Y_s, (2H)^{-\frac {1}{2}} (s - v)^H)$ ,
If $H \leq \frac {1}{4}$ , we can replace $f^i(m + \sigma x) - f^i(m)$ by
Therefore,
This yields
and hence
Therefore, we obtain
By (3.10) and (3.12), we conclude
if $t-s \leq s- v$ . Therefore, we can apply Theorem 1.1 with
4. Local times of fractional Brownian motions
In this section, we set $d = 1$ , and we are interested in local times of fractional Brownian motions. In case of a Brownian motion W, or, more generally, semimartingales as discussed in Łochowski et al. [Reference Łochowski, Obłój, Prömel and Siorpaes28], there are three major methods to construct its local time.
-
1. Via occupation measure. The local time $L_T^W(\cdot )$ of W is defined as the density with respect to the Lebesgue measure of
$$ \begin{align*} L_T^W(a) = \int_0^T \delta(W_s - a) \, \mathrm{d} s, \end{align*} $$where $\delta $ is Dirac’s delta function concentrated at $0$ . -
2. Via discretization. The local time $L_T^W(a)$ is defined by
-
3. Via numbers of interval crossing. For $n \in \mathbb {N}$ , we set and inductively
In case of a fractional Brownian motion, the construction of its local time via the method 1 is well-known, see the survey [Reference Geman and Horowitz15] and the monograph [Reference Biagini, Hu, Øksendal and Zhang5]. In contrast, there are few results in the literature in which the local time of a fractional Brownian motion is constructed via the method 2 or 3. Because of this, the construction of the local time via the method 3 was stated as a conjecture in [Reference Cont and Perkowski9]. We are aware of only two results in this direction. One is the work [Reference Azaïs4] of Azaïs, who proves Corollary 4.8 below. The other is the work [Reference Mukeru33] of Mukeru, who proves that the local time $L_T(a)$ of a fractional Brownian motion with Hurst parameter less than $\frac {1}{2}$ is represented as
Our goal in this section is to give new representations of the local times of fractional Brownian motions in the spirit of the method (b) along deterministic partitions. The representation in Corollary 4.9 is compatible with [Reference Cont and Perkowski9, Definition 3.1].
Theorem 4.1. Let B be an $(\mathcal {F}_t)$ -fractional Brownian motion with Hurst parameter $H \neq \frac {1}{2}$ , in the sense of Definition 3.1. Let $m \in [2, \infty )$ , $\gamma \in [0, \infty )$ , and $a \in \mathbb {R}$ . If $H> \frac {1}{2}$ , assume that m satisfies
Then, as $ \lvert {\pi } \rvert \to 0$ , where $\pi $ is a partition of $[0, T]$ , the family of
converges in $L_m(\mathbb {P})$ to $\mathfrak {c}_{H, \gamma } L_T(a)$ , where $L_T(a)$ is the local time of B at level a and
Furthermore, we have
Remark 4.2. A similar result holds for a Brownian motion ( $H = \frac {1}{2}$ ). However, we omit a proof since it is easier but requires a special treatment.
Remark 4.3. We can similarly prove
Consequently,
where the convergence is in $L_m(\mathbb {P})$ .
Proof. We will not write down dependence on H, $\gamma $ , and m. Without loss of generality, we can assume $\mathbb {E}[B(0)] = 0$ . To apply Theorem 1.1 (for $H < \frac {1}{2}$ ) or Corollary 2.7 (for $H> \frac {1}{2}$ ), respectively, we set
If we set , it suffices to show that the estimates (1.10) and (1.11) are satisfied for $H < \frac {1}{2}$ , and that the estimates (2.14), (2.15), and (2.16) are satisfied for $H> \frac {1}{2}$ . Since the proof is rather long, we split the main arguments into three lemmas.
Lemma 4.4. We have
where in either case there exists a constant $c = c(H)> 0$ , such that
Remark 4.5. Due to (4.1), the exponent $1 + \frac {H}{m} - H$ is greater than $\frac {1}{2}$ .
Proof. We have
Now we consider the case $H> \frac {1}{2}$ .
Set
Since $H> \frac {1}{2}$ , $\chi _1 \geq 0$ , and
Then, if X and Y are two independent standard normal distributions on $\mathbb {R}$ , we have
Therefore, if we set
We first estimate . Using the estimate
we have
Then,
where in the third line we applied
And,
Therefore,
We now estimate . Similarly to (4.5), we have
and similarly
Therefore, we conclude
which completes the proof of the lemma.
Recall the Mandelbrot–Van Ness representation (3.1), and recall that W is an $(\mathcal {F}_t)_{t \in \mathbb {R}}$ -Brownian motion.
Lemma 4.6. Let $v < s < t$ , and set
If $\frac {t-s}{s-v}$ is sufficiently small, then
where for some $c = c(H, m)> 0$ ,
Proof. As in the proof of Proposition 3.3, for $s> v$ , we set
and we write under the conditioning of $\mathcal {F}_v$ . Then, recalling $\hat {A}_{s,t}$ from (4.4), we have
To compute, we set
By Lemma 3.2,
and
We have the decomposition
where the second term is independent of $\tilde {B}_{s}$ . If we set
and if we write X and Y for two independent standard normal distributions, then the quantity (4.6) equals to
where
For a while, assume $\gamma> 0$ . Using the estimate
we have
We set
We have for $y> 0$
and if $q \geq 2 \lvert {p} \rvert $ and $y \in [q - \lvert {p} \rvert , q + \lvert {p} \rvert ]$ , then
Therefore, if $q \geq 2 \lvert {p} \rvert $ , we have
Therefore,
When $\gamma = 0$ , we have
and thus (4.9) holds for $\gamma = 0$ . We estimate the expectation (with respect to Y) of each term.
We have
By using the estimate
we obtain
Next, we estimate the second term of (4.9). We have for $n \in \{0, 1\}$ ,
where we applied (4.10) to get the last inequality. Therefore, we obtain
Finally, we estimate the third term of (4.9). Suppose that $\frac {t-s}{s-v}$ is so small that $ \lvert {\sigma _s^{-2} \rho _{s, t}} \rvert \leq \frac {1}{24}$ , and then we have
Hence,
and
This gives the estimate of the third term.
In summary, recalling that $\mathbb {E}[\hat {A}_{s,t} \vert \mathcal {F}_v]$ equals to (4.8), we obtain
where
Let us estimate $ \lVert {R_1} \rVert _{L_m(\mathbb {P})}$ . Recall that
and
We have the estimate (3.4) of $Y_{s, t}$ . Since
there is a constant $c = c(H)> 0$ , such that
Therefore, for some constant $c_1 = c(H, m)>0$ ,
and finally
After this long calculation, we conclude
if $\frac {t-s}{s-v}$ is sufficiently small.
By (4.7), we have
where
Therefore,
This completes the proof of the lemma.
Lemma 4.7. We have
where $f_H(a)$ satisfies the estimate (4.3). Moreover, if $\frac {t-s}{s-v}$ is sufficiently small, then
where for some $c = c(H, m)> 0$ ,
Proof. The estimate in (4.15) follows from [Reference Ayache, Wu and Xiao3, (3.38)]. However, since this is not entirely obvious, we sketch here an alternative derivation, which is motivated by [Reference Butkovsky, Lê and Mytnik7]. In view of the formal expression $L_t(a) = \int _0^t \delta _a(B_r) \, \mathrm {d} r$ , we set
We note $\mathbb {E}[\delta \bar {A}_{s,u,t}(a) \vert \mathcal {F}_s] =0$ . By Lê’s stochastic sewing lemma [Reference Lê24], to prove (4.15), it suffices to show
-
• the estimate
$$ \begin{align*} \lVert {e^{-\frac{ H(Y_s - a)^2}{c_H (r-s)^{2H}}}} \rVert _{L_m(\mathbb{P})} \lesssim_{T} \begin{cases} f_H(a), & H < \frac{1}{2}, \\ (\mathbb{E}[B(0)^2] + s^{2H})^{-\frac{1}{2m}} \lvert {r-s} \rvert ^{\frac{H}{m}} f_H(a), & H> \frac{1}{2}, \end{cases} \end{align*} $$ -
• and the identity $L_t(a) = \lim _{ \lvert {\pi } \rvert \to 0} \sum _{[u,v] \in \pi } \bar {A}_{u, v}(a)$ , where $\pi $ is a partition of $[0, t]$ .
The first point is essentially given in Lemma 4.4. The second point follows from the identity
Thus, we now focus on the estimate (4.16). We have the identity
Indeed, we can convince ourselves of the validity of the identity from the formal expression
We have the decomposition
By (4.11), we obtain
We have
Hence, by (4.11),
To estimate $R_5$ , observe
By (4.12),
To estimate $\mathbb {P}( \lvert {Y_s - a} \rvert \leq 2 \lvert {Y_{s,r}} \rvert )$ , consider the decomposition
If $\frac {t-s}{s-v}$ is sufficiently small, then $ \lvert {\mathbb {E}[Y_s^2]^{-1} \mathbb {E}[Y_s Y_{s,r}]} \rvert \leq \frac {1}{4}$ . Therefore,
This gives an estimate of $R_5$ . Thus, we conclude
where
This completes the proof of the lemma.
Now we can complete the proof of Theorem 4.1. The above lemmas show
and if $\frac {t-s}{s-v}$ is sufficiently small, then
Noting that the exponents satisfy the assumption of Theorem 1.1 or Corollary 2.7, Remark 1.4 implies
Hence, we complete the proof of Theorem 4.1.
Corollary 4.8. We have
where the convergence is in $L_m(\mathbb {P})$ with m satisfying (4.1).
Proof. The claim is a special case of Theorem 4.1 with $\gamma = 0$ . When $m = 2$ , it is proved in [Reference Azaïs4, Theorem 5].
For applications to pathwise stochastic calculus, a representation of the local time as in (b) above is more useful. In [Reference Cont and Perkowski9, Theorem 3.2], a pathwise Itô-Tanaka formula is derived under the assumption that
converges weakly in $L_m(\mathbb {R})$ for some $m> 1$ . But as already suggested by [Reference Cont and Perkowski9, Lemma 3.5], this weak convergence in $L_m(\mathbb {R})$ follows from our convergence result in Theorem 4.1:
Corollary 4.9. Let $B \in C([0,T], \mathbb {R})$ , and for any partition $\pi $ of $[0,T]$ , let $\tilde {L}^{\pi }_T(a)$ be defined as in (4.18). Assume that $m> 1$ and that $(\pi _n)$ is a sequence of partitions of $[0,T]$ , such that $\lim _{n \to \infty } \sup _{[s,t] \in \pi _n} |B_t - B_s| = 0$ and for a limit $L_T \in L_m(\mathbb {R})$ :
Then
Remark 4.10. If B is a sample path of the fractional Brownian motion with Hurst index $H \in (0,1)$ , then by Theorem 4.1, the convergence (4.19) holds in probability for any sequence of partitions $(\pi _n)_{n \in \mathbb {N}}$ , provided that m satisfies (4.1). Therefore, we can find a subsequence so that the convergence along the subsequence holds almost surely. In fact, by (4.17), we even control the convergence rate in terms of the mesh size of the partition, and this easily gives us specific sequences of partitions along which the convergence holds almost surely and not only in probability. For example, if $\pi _n$ is the nth dyadic partition of $[0, T]$ , the estimate (4.17) gives
Since the right-hand side is summable with respect to n, the Borel–Cantelli lemma implies the almost sure convergence. Along any such sequence of partitions, we therefore obtain the almost sure weak convergence of $\tilde L^{\pi _n}_T$ in $L_m(\mathbb {R})$ .
Proof. Set
It suffices to show that $\sum _{[s,t] \in \pi _n} \tilde {A}_{s, t}(\cdot )$ converge weakly to $0$ in $L_m(\mathbb {R})$ . Since
and since is bounded in $L_m(\mathbb {P})$ by assumption, it suffices to show that
for every compactly supported continuous function g. Since for $B_s < B_t$ we have
we obtain
Therefore,
which converges to $0$ .
Remark 4.11. As noted in [Reference Mukeru33], we can use Theorem 4.1 to simulate the local time of a fractional Brownian motion (see Figures 1 ( $H = 0.1$ ) and 2 ( $H=0.6$ )).Footnote 1
5. Regularization by noise for diffusion coefficients
Let $y \in C^{\alpha } ([0, T] , \mathbb {R}^d)$ with $\alpha \in ( \frac {1}{2}, 1 )$ . We consider a Young differential equation
We suppose that the drift coefficient b belongs to $C^1_b (\mathbb {R}^d , \mathbb {R}^d)$ , where $C^1_b(\mathbb {R}^d , \mathbb {R}^d)$ is the space of continuously differentiable bounded functions between $\mathbb {R}^d$ with bounded derivatives. If the diffusion coefficient $\sigma $ belongs to $C^1_b (\mathbb {R}^d , \mathcal {M}_d)$ , where $\mathcal {M}_d$ is the space of $d\times d$ matrices, then we can prove the existence of a solution to (5.1). However, to prove the uniqueness of solutions, the coefficient $\sigma $ needs to be more regular. The following result is well-known (e.g. [Reference Lyons29]), but we give a proof for the sake of later discussion.
Proposition 5.1. Let $b \in C^1_b(\mathbb {R}^d, \mathbb {R}^d)$ and $\sigma \in C^{1 + \delta }(\mathbb {R}^d, \mathcal {M}_d)$ with $\delta> \frac {1-\alpha }{\alpha }$ . Then the Young differential equation (5.1) has a unique solution.
Proof. The argument is very similar to that of [Reference Lê24, Theorem 6.2]. Let $x^{(i)}$ ( $i = 1, 2$ ) be two solutions to (5.1). Then,
where
Note that the second term is well-defined as a Young integral since
is $\delta \alpha $ -Hölder continuous and $\delta \alpha + \alpha> 1$ by our assumption of $\delta $ . Therefore, $x^{(1)} - x^{(2)}$ is a solution of the Young differential equation
The uniqueness of this linear Young differential equation is known. Hence, $x^{(1)} - x^{(2)} = 0.$
Proposition 5.1 is sharp in the sense that for any $\alpha \in (1,2)$ and any $\delta \in (0, \frac {1-\alpha }{\alpha })$ , we can find $\sigma \in C^{\gamma }(\mathbb {R}^2,\mathcal {M}_2)$ and $y \in C^{\alpha }([0,T],\mathbb {R}^2)$ , such that the Young differential equation
has more than one solution (see Davie [Reference Davie11, Section 5]). However, if the driver y is random, we could hope to obtain the uniqueness of solutions in a probabilistic sense even when the regularity of $\sigma $ does not satisfy the assumption of Proposition 5.1. For instance, if the driver y is a Brownian motion and the integral is understood in Itô’s sense, the condition $\sigma \in C^1_b$ is sufficient to prove pathwise uniqueness.
The goal of this section is to prove the following.
Theorem 5.2. Suppose that B is an $(\mathcal {F}_t)$ -fractional Brownian motion with Hurst parameter $H \in (\frac {1}{2}, 1)$ in the sense of Definition 3.1. Let $b \in C^1_b (\mathbb {R}^d , \mathbb {R}^d)$ and $\sigma \in C^{1}_b (\mathbb {R}^d , \mathcal {M}_d)$ . Assume one of the following.
-
1. We have $b \equiv 0$ and $\sigma \in C^{1+\delta } (\mathbb {R}^d , \mathcal {M}_d)$ with
(5.2) $$ \begin{align} \delta> \frac{(1 - H) (2 - H)}{H (3 - H)}. \end{align} $$ -
2. For all $x \in \mathbb {R}^d$ , the matrix $\sigma (x)$ is symmetric and satisfies
$$ \begin{align*} y \cdot \sigma(x) y> 0, \quad \forall y \in \mathbb{R}^d, \end{align*} $$and $\sigma \in C^{1+\delta }(\mathbb {R}^d , \mathcal {M}_d)$ with $\delta $ satisfying (5.2). -
3. We have $\sigma \in C^{1+\delta }(\mathbb {R}^d; \mathcal {M}_d)$ with
(5.3) $$ \begin{align} \delta> \frac{(1-H)(2-H)}{1+H-H^2}. \end{align} $$
The graphs of (5.2) and (5.3) can be found in Figure 3.
Then, for every $x \in \mathbb {R}^d$ , there exists a unique, up to modifications, process $(X_t)_{t \in [0, \infty )}$ with the following properties.
-
• The process $(X_t)$ is $(\mathcal {F}_t)$ -adapted and is $\alpha $ -Hölder continuous for every $\alpha < H$ .
-
• The process $(X_t)$ solves the Young differential equation
(5.4) $$ \begin{align} \, \mathrm{d} X_t = b (X_t) \, \mathrm{d} t + \sigma (X_t) \, \mathrm{d} B_t, \quad X_0 = x. \end{align} $$
Furthermore, in that case the process $(X_t)_{t \in [0, \infty )}$ is a strong solution, that is, it is adapted to the natural filtration generated by the Brownian motion W appearing in the Mandelbrot–Van Ness representation (3.1).
Remark 5.3. In the case 2, we assume the positive-definiteness of $\sigma $ to ensure that for every $x, y \in \mathbb {R}^d$ and $\theta \in [0,1]$ , the matrix $\theta \sigma (x) + (1 - \theta ) \sigma (y)$ is invertible.
Remark 5.4. Since the seminal work [Reference Catellier and Gubinelli8] of Catellier and Gubinelli, many works have appeared to establish weak or strong existence or uniqueness to the SDE
for an irregular drift b and a fractional Brownian motion B. In contrast, there are much fewer works that attempt to optimize the regularity of the diffusion coefficient $\sigma $ . The work [Reference Hinz, Tölle and Viitasaari20] by Hinz et al., where $b \equiv 0$ , considers certain existence and uniqueness for $\sigma $ that is merely of bounded variation, at the cost of additional restrictive assumptions (variability and [Reference Hinz, Tölle and Viitasaari20, Assumption 3.15]). It seems that Theorem 5.2 is the first result to improve the regularity of $\sigma $ without any additional assumption (except $\sigma $ being invertible for the case 2). However, we believe that our assumption of $\delta $ is not optimal (see Remark 5.9).
The proof of Proposition 5.1 suggests that the pathwise uniqueness holds if, for any two $(\mathcal {F}_t)$ -adapted solutions $X^{(1)}$ and $X^{(2)}$ to (5.4), we can construct the integral
If $\theta X^{(1)}_s + (1 - \theta ) X^{(2)}_s$ is replaced by $B_s$ , then the integral is constructed in Proposition 3.3. The difficulty here is that $X^{(i)}$ is not Gaussian and the Wiener chaos decomposition crucially used in the proof of Proposition 3.3 cannot be applied. Yet, the process $X^{(i)}$ is locally controlled by the Gaussian process B (whose precise meaning will be clarified later), and by taking advantage of this fact, we can still make sense of the integral (5.5).
As a technical ingredient, we need a variant of Theorem 1.1.
Lemma 5.5. Let $(A_{s, t})_{0 \leq s < t \leq T}$ be a family of two-parameter random variables, and let $(\mathcal {F}_t)_{t \in [0, T]}$ be a filtration, such that $A_{s, t}$ is $\mathcal {F}_t$ -measurable for every $0 \leq s \leq t \leq T$ . Suppose that for some $m \geq 2$ , $\Gamma _1, \Gamma _{2,} \Gamma _3 \in [0, \infty )$ and $\alpha , \gamma , \beta _1, \beta _2, \beta _3 \in [0, \infty )$ , we have for every $0 \leq v < s < u < t \leq T$
Suppose that
Finally, suppose that there exists a stochastic process $(\mathcal {A}_t)_{t \in [0,T]}$ , such that
where the convergence is in $L_m(\mathbb {P})$ . Then, we have
Remark 5.6. It should be possible to formulate Lemma 5.5 at the generality of Theorem 1.1. However, such generality is irrelevant to prove Theorem 5.2, and we do not pursue the generality to simplify the presentation.
Proof. Here, we consider dyadic partitions. Fix $s < t$ , and set
Since $\mathcal {A}_{s,t} = \lim _{n \to \infty } A^n_{s, t}$ , it suffices to show
for some $\delta> 0$ and all sufficiently large n. As in the proof of Theorem 1.1, we decompose
By the BDG inequality,
Thus,
Furthermore,
Therefore,
We choose $L = \lfloor 2^{n \varepsilon } \rfloor $ with $\varepsilon \in (0, 1)$ so that
Namely,
Such an $\epsilon $ exists exactly under our assumption (5.6).
We mentioned that a solution to (5.4) is controlled by B. Here comes a more precise statement. We fix $\alpha \in ( \frac {1}{2}, H )$ and let X be a solution to (5.4). We have the estimates
Furthermore, the a priori estimate of the Young differential equation ([Reference Friz and Hairer14, Proposition 8.1]) gives
Therefore, we have
where
This motivates the following definition. Recall that B is an $(\mathcal {F}_t)$ -fractional Brownian motion in the sense of Definition 3.1.
Definition 5.7. Let Z be a random path in $C^{\alpha } ([0, T] , \mathbb {R}^d)$ . For $\beta \in (\alpha , \infty )$ , we write $Z \in \mathcal {D} (\alpha , \beta )$ , if for every $s < t$ , we have
where
-
• the random variables $z^{(1)}(s), z^{(3)}(s) \in \mathbb {R}^d$ , and $z^{(2)}(s) \in \mathcal {M}_d$ are $\mathcal {F}_s$ -measurable and
-
• there exists a (random) constant $C \in [0, \infty )$ , such that for all $s < t$
$$\begin{align*}| R_{s, t} | \leq C | t - s |^{\beta}. \end{align*}$$
We set
Furthermore, we set
and .
Proposition 5.8. Let $f \in C^1_b (\mathbb {R}^d; \mathbb {R}^d)$ and $Z \in \mathcal {D}_1(\alpha , \beta )$ . If
and if $\alpha $ is sufficiently close to H, then for every $m \in [2, \infty )$ ,
If $Z \in \mathcal {D}_0(\alpha , \beta )$ , a similar estimate holds with $ \lVert {Z} \rVert _{\mathcal {D}_1(\alpha ,\beta )}$ replaced by $ \lVert {Z} \rVert _{\mathcal {D}(\alpha ,\beta )}$ .
Proof. Our tool is Lemma 5.5. Since arguments are similar, we only prove the claim for $Z \in \mathcal {D}_1(\alpha , \beta )$ . We set
We have
Hence
Let $v < s$ with $t-s \leq s- v$ . As $Z \in \mathcal {D}_1 (\alpha , \beta )$ , we write
Then, if we write ,
Hence, if we write , then
Next, we will estimate $ \lVert {\mathbb {E} [\delta \hat {A}_{s, u, t} \vert \mathcal {F}_v]} \rVert _{L_m(\mathbb {P})}$ . The rest of the calculation resembles the proof of Proposition 3.5. We write and as before. We can decompose
where $\hat {Y}_r$ is $\mathcal {F}_v$ -measurable and $\tilde {B}_r$ is independent of $\mathcal {F}_v$ . We set
Then, as in the proof of Proposition 3.5, we have the decomposition
where, as in (3.8) and (3.11),
The map $\mathbb {R}^d \ni x \mapsto f (z^{(2)} (v)^{} x) \in \mathbb {R}^d$ belongs to $C^{\delta } (\mathbb {R}^d)$ with its norm bounded by
Therefore, by repeating the argument used to obtain (3.9), we obtain
Referring to (3.4), we have
Therefore,
Similarly as before, we have
As $H> \frac {1}{2}$ , by Lemma 3.2, we have
Therefore,
Combining our estimates, we have
Hence, combining (5.11), (5.12), and (5.13)
and
To apply Lemma 5.5, we need
which, if $\alpha $ is sufficiently close to H, are fulfilled under (5.10).
Proof of Theorem 5.2.
We first prove the pathwise uniqueness. We suppose that the assumption in the case 2 mentioned in Theorem 5.2 holds, and the other cases will be discussed later. Let $X^{(1)}$ and $X^{(2)}$ be $(\mathcal {F}_t)$ -adapted solutions. Our strategy is similar to Proposition 5.1, but, here, we must construct the integral (5.5) stochastically. For each $k \in \mathbb {N}$ , we set
we let $\sigma ^{k} \in C^{1 + \delta }(\mathbb {R}^d; \mathcal {M}_d)$ be such that $\sigma ^k = \sigma $ in $\{x \mid \lvert {x} \rvert \leq k\}$ and
and we set
If we write
then in the event $\Omega _k$ , we have $X^{(i)}_t = X^{(i), k}_t$ , $t \in [0, T]$ .
Let $\{\sigma ^{k, n}\}_{n=1}^{\infty }$ be a smooth approximation of $\sigma ^k$ . In general, we can only guarantee the convergence in $C^{1+\delta '}(\mathbb {R}^d, \mathcal {M}_d)$ for any $\delta '<\delta $ , which is still sufficient to make the following argument work. To simplify the notation, we assume that we can take $\delta '=\delta $ .
We have
and in $\Omega _k$
where
For a fixed $\theta \in (0, 1)$ , we set
By the a priori estimate (5.9), for $\alpha \in (\frac {1}{2}, H)$ , we have
with
Note that we have
and hence
Therefore, we have $Z^{\theta , k} \in \mathcal {D}_1(\alpha , 2 \alpha )$ with
Since
if $\alpha $ is sufficiently close to H, by Proposition 5.8,
By Kolmogorov’s continuity theorem, the sequence $(V^{k, n})_{n \in \mathbb {N}}$ converges to some $V^k$ in $C^{\alpha }([0, T], \mathbb {R}^d)$ .
Therefore, we conclude that almost surely in $\Omega _k$ , the path $z = X^{(1)} - X^{(2)}$ solves the linear Young equation
and hence $X^{(1)} = X^{(2)}$ . Since $\mathbb {P}(\Omega _k) \to 1$ , we conclude $X^{(1)} = X^{(2)}$ almost surely. Thus, we completed the proof of the uniqueness under the case 2. The other cases can be handled similarly. Indeed, under the case 1, we have $X^{(i)} \in \mathcal {D}_0(\alpha , 2 \alpha )$ , and under the case 3, we have $X^{(i)} \in \mathcal {D}_0(\alpha , 1)$ .
Now, it remains to prove the existence of a strong solution. However, in view of the Yamada-Watanabe theorem (Proposition B.2), it suffices to show the existence of a weak solution, which will be proved in Lemma B.3 based on a standard compactness argument.
Remark 5.9. We believe that our assumption in Theorem 5.2 is not optimal. One possible approach to relax the assumption is to consider a higher order approximation in (5.7). Yet, we believe that this will not lead to an optimal assumption, as long as we apply Lemma 5.5. Thus, finding an optimal regularity of $\sigma $ for the pathwise uniqueness and the strong existence remains an interesting open question that is likely to require a new idea.
A. Proofs of technical results
Proofs of Lemmas 2.3 and 2.4
Proof of Lemma 2.3 without (1.12).
Let us first recall our previous strategy under (1.12). We used Lemma 2.1 to write
Then, we decomposed
where . We estimated the first term of (A.2) by the BDG inequality and (1.8):
In the proof under (1.12), we estimated the second term of (A.2) by the triangle inequality and (1.7):
Then, we chose L so that both (A.3) and (A.4) are summable with respect to n, for which to be possible, we had to assume (1.12).
In order to remove the assumption (1.12), let us think again why we did the decomposition (A.2). This is because we do not want to apply the simplest estimate, namely, the triangle inequality, since the condition (1.7) implies that $(A_{s,t})_{[s,t] \in \pi }$ are not so correlated. This point of view teaches us that, to estimate
we should not simply apply the triangle inequality. That is, we should again apply the decomposition as in (A.2).
To carry out our new strategy, set
For this new strategy, we can set . In particular, L does not depend on n. We use the convention $\mathbb {E}[X \vert \mathcal {G}^{(1), l}_j] = 0$ for $j \leq 0$ . Then,
By setting
the quantity (A.5) equals to
The $L_m(\mathbb {P})$ -norm of the first term can be estimated by the BDG inequality: it is bounded by
By (1.7), we have
Therefore, the quantity (A.6) is bounded by
As the reader may realize, we will repeat the same argument for
and continue. To state more precisely, set inductively,
We claim that, if $L^k \leq 2^n$ , we have
The proof of the claim is based on induction. The case $k=1$ and $k=2$ is obtained. Suppose that the claim is correct for $k \geq 2$ , and consider the case $k+1$ . Again, decompose
To prove the claim, it suffices to estimate the first sum in the right-hand side. By the BDG inequality, its $L_m(\mathbb {P})$ -norm is bounded by
By (1.7),
Therefore, the quantity (A.8) is bounded by
and the claim follows.
Now let us estimate
by the triangle inequality:
By (1.7) (or essentially the estimate (A.9)),
and hence, the quantity (A.10) is bounded by
In conclusion, we obtained for $L^k \leq 2^n$ ,
where
We recall that $L = \max \{2, \lceil M \rceil \}$ . To respect $L^k \leq 2^n$ , we set . We then have
and
Therefore, we note that the right-hand side of (A.11) is summable with respect to n and
Proof of Lemma 2.4.
The proof is similar to Lemma 2.3. Write
and
By (2.6), we have $N \leq 3 \lvert {\pi '} \rvert ^{-1} T$ . We fix a parameter L, which will be chosen later, and set
Inductively, we set
As in Lemma 2.3, for each $k \in \mathbb {N}$ , we consider the decomposition
where
For this decomposition, we must have $L^k \leq N$ . By the BDG inequality and the Cauchy-Schwarz inequality,
By Lemma 2.3,
For $p \geq 2$ , by Lemma 2.2 and (2.6),
Therefore, we obtain
where $f_k$ is defined by (A.12).
We move to estimate B. By Lemma 2.2 and (2.6),
Therefore,
Combining (A.13) and (A.14), we obtain
As in the proof of Lemma 2.3, we set and . We then obtain the claimed estimate.
Proof of Corollary 2.7
The argument is similar to that of Theorem 1.1. Therefore, we only prove an analogue of Lemma 2.3.
Analogue of Lemma 2.3.
Given a partition
we have
where $t_0> 0$ is assumed for (A.15). In fact, the proof of (A.15) is the same as that of Lemma 2.3, since we can simply replace $\Gamma _1$ and $\Gamma _2$ by $t_0^{-\gamma _1} \Gamma _1$ and $t_0^{-\gamma _2} \Gamma _2$ . Therefore, we focus on proving (A.16).
Yet, the proof of (A.16) is similar to that of Lemma 2.3. Recalling the notation therein, namely, (A.1) and (A.7), we have
We fix a large n. To estimate the first term of (A.17), we apply the BDG inequality to obtain
For $p=1$ , since $S^{(1), l_1}_j = R^n_{j L + l_1}$ , by the Cauchy-Schwarz inequality,
For $i \geq 1$ , by (2.11), we have
and by (2.12)
Therefore,
We observe
where the condition $\gamma _2 < \frac {1}{2}$ is used. We conclude
For $2 \leq p \leq k$ , the argument is similar but now we use (2.10). By the Cauchy-Schwarz inequality,
We note that for each index $l_1, \ldots , l_p$ and j in the sum, there exists a unique $i = i(l_1, \ldots , l_p; j)$ , such that
As $p \geq 2$ , we know $i \geq L$ . By (2.10) (as in the estimate (A.9)),
Therefore,
where to obtain the second inequality, we applied the estimate (A.18).
Now we consider the estimate of the second term of (A.17). By the triangle inequality,
But the estimate of the right-hand side was just discussed. In fact, we have
Hence, we obtain the estimate
By choosing L and k exactly as in the proof of Lemma 2.3, we conclude that there exists an $\epsilon = \epsilon (\alpha , \beta _1, \beta _2, \beta _3)> 0$ , such that for all large n
from which we obtain (A.16).
Proof of Lemma 3.2
Let $d = 1$ . For $u> v$ , we set
so that $B_u - B(0) = B^{(1)}_u + B^{(2)}_u$ and $B^{(1)}$ and $B^{(2)}$ are independent. Then, we have
and by (3.2), we have
and thus, we will estimate $\mathbb {E}[B^{(1)}_s B^{(1)}_t]$ . We have
By [Reference Picard, Donati-Martin, Lejay and Rouault38, Theorem 33], the first term of (A.19) equals to
Since
the second term of (A.20) equals to
By [Reference Picard, Donati-Martin, Lejay and Rouault38, Theorem 33],
Similarly, the second term of (A.19) equals to
Therefore, $\mathbb {E}[B^{(1)}_s B^{(1)}_t]$ equals to
Since
the proof is complete.
B. Yamada-Watanabe theorem for fractional SDEs
We consider a Young differential equation
where $b \in L_{\infty }(\mathbb {R}^d,\mathbb {R}^d)$ and B is an $(\mathcal {F}_t)_{t \in \mathbb {R}}$ fractional Brownian motion with Hurst parameter $H \in (\frac {1}{2}, 1)$ . We fix $\alpha \in (\frac {1}{2}, H)$ , and we assume that $\sigma \in C^{\frac {1-\alpha }{\alpha }}(\mathbb {R}^d; \mathcal {M}_d)$ so that the integral
is interpreted as a Young integral.
Definition B.1. We say that a quintuple $(\Omega , (\mathcal {F}_t)_{t \in \mathbb {R}}, \mathbb {P}, B, X)$ is a weak solution to (B.1) if $(B, X)$ are random variables defined on the filtered probability space $(\Omega , (\mathcal {F}_t), \mathbb {R})$ , if B is an $(\mathcal {F}_t)$ -fractional Brownian motion, if $X \in C^{\alpha }([0, T])$ is adapted to $(\mathcal {F}_t)$ , and if X solves the Young differential equation (B.1). Given a filtered probability space $(\Omega , (\mathcal {F}_t)_{t \in \mathbb {R}}, \mathbb {P})$ and an $(\mathcal {F}_t)$ -fractional Brownian motion B, we say that a $C^{\alpha }([0,T])$ -valued random variable X defined on $(\Omega , (\mathcal {F}_t)_{t \in \mathbb {R}}, \mathbb {P})$ is a strong solution if it solves (B.1) and if it is adapted to the natural filtration generated by B. We say that the pathwise uniqueness holds for (B.1) if, for any two adapted $C^{\alpha }([0, T])$ -valued random process X and Y defined on a common filtered probability space that solve (B.1) driven by a common $(\mathcal {F}_t)$ -Brownian motion, we have $X = Y$ almost surely.
We will prove a Yamada-Watanabe type theorem for (B.1) based on Kurtz [Reference Kurtz25]. To this end, we recall that an $(\mathcal {F}_t)$ -fractional Brownian motion has the representation (3.1), and we view (B.1) as an equation of X and the Brownian motion W.
Proposition B.2. Suppose that a weak solution to (B.1) exists and that the pathwise uniqueness holds for (B.1). Then, there exists a strong solution to (B.1).
Proof. We would like to apply [Reference Kurtz25, Theorem 3.14]. For this purpose, we need a setup. We follow the notation in [Reference Kurtz25]. We fix $\beta> 0$ that is less than but sufficiently close to $\frac {1}{2}$ . As before, we set , and we set and define $S_2$ as a subspace of
that is Polish and the Brownian motion lives in $S_2$ . We note that for $w \in S_2$ , the improper integral
is well-defined. For $t \in [0, T]$ , we denote by $(\mathcal {B}^{S_1}_t)_{t \in [0, T]}$ and $(\mathcal {B}^{S_2}_t)_{t \in [0, T]}$ the filtration generated by the coordinate maps in $S_1$ and $S_2$ , respectively. We set
as our compatibility structure in the sense of [Reference Kurtz25, Definition 3.4]. We denote by $\mathcal {S}_{\Gamma , \mathcal {C}, W}$ the set of probability measures $\mu $ on $S_1 \times S_2$ , such that
-
• we have
$$ \begin{align*} \mu(\{(x,y) \in S_1 \times S_2 \mid x_t = x + \int_0^t b(x_r) \, \mathrm{d} r + \int_0^t \sigma(x_r) \, \mathrm{d} Iy_r\,\, \text{for all }t \in [0,T]\} ) = 1, \end{align*} $$where ; -
• $\mu $ is $\mathcal {C}$ -compatible in the sense of [Reference Kurtz25, Definition 3.6];
-
• $\mu (S_1 \times \cdot )$ has the law of the Brownian motion.
By [Reference Kurtz25, Lemma 3.8], $\mathcal {S}_{\Gamma , \mathcal {C}, W}$ is convex. In view of [Reference Kurtz25, Lemma 3.2], the existence of weak solutions implies $\mathcal {S}_{\Gamma , \mathcal {C}, W} \neq \varnothing $ .
Therefore, to apply [Reference Kurtz25, Theorem 3.14], it remains to prove the pointwise uniqueness in the sense of [Reference Kurtz25, Definition 3.12]. Suppose that $(X_1, X_2, W)$ are defined on a common probability space, that the laws of $(X_1, Y)$ and $(X_2, Y)$ belong to $\mathcal {S}_{\Gamma , \mathcal {C}, W}$ , and that $(X_1, X_2)$ are jointly compatible with W in the sense of [Reference Kurtz25, Definition 3.12]. But then, if we denote by $(\mathcal {F}_t)$ the filtration generated by $(X_1, X_2, W)$ , by [Reference Kurtz25, Lemma 3.2], the joint compatibility implies that W is an $(\mathcal {F}_t)$ -Brownian motion, and therefore the pathwise uniqueness implies $X_1 = X_2$ almost surely.
Hence, by [Reference Kurtz25, Theorem 3.14], there exists a measurable map $F: S_2 \to S_1$ , such that for a Brownian motion W, the law of $(F(W), W)$ belongs to $\mathcal {S}_{\Gamma , \mathcal {C}, W}$ . Then, [Reference Kurtz25, Lemma 3.11] implies that $F(W)$ is a strong solution.
Lemma B.3. Let $b \in C_b^1(\mathbb {R}^d)$ and $\sigma \in C^1_b(\mathbb {R}^d)$ . Then, there exists a weak solution to (B.1).
Proof. Let $(\sigma ^n)_{n \in \mathbb {N}}$ be a smooth approximation to $\sigma $ , and let $X^n$ be the solution to
Let W be the Brownian motion, such that $B_t = \int _{-\infty }^t K_H(t, r) \, \mathrm {d} W_r$ . Let $\epsilon $ be greater than but sufficiently close to $0$ , and let S be a subspace of
that is Polish and where the Brownian motion lives. By the a priori estimate (5.8), we see that a sequence of the laws of $(X^n, W)$ is tight in $C^{H - \epsilon }([0, T]) \times S$ . Thus, replacing it with a subsequence, we suppose that the sequence $(X^n, W)$ converges to some limit $(\tilde {X}, \tilde {W})$ in law.
To see that $(\tilde {X}, \tilde {W})$ solves (5.4), we write , and for $\delta>0$ , we set
Then, we have
However, by the estimate of Young’s integral,
Thus, combined with the a priori estimate (5.8), we observe
and hence, $\mathbb {P}((\tilde {X}, \tilde {W}) \in A_{\delta }) = 0$ . Since $\delta $ is arbitrary, this implies
Finally, since $W_t - W_s$ is independent of the $\sigma $ -algebra generated by $(X^n_r)_{r \leq s}$ and $(W_r)_{r \leq s}$ , we know that $\tilde {W}_t - \tilde {W}_s$ is independent of
or equivalently, $\tilde {W}$ is an $(\tilde {\mathcal {F}}_t)$ -Brownian motion.
Acknowledgements
We thank Khoa Lê for suggesting a simpler proof of Lemma 2.3 in Appendix A than the original one. TM thanks Henri Altman, Hannes Kern, and Helena Kremp for the discussions related to this paper. The main part of the work was done while TM was a Ph.D. student at Freie Universität Berlin under the financial support of the German Science Foundation (DFG) via the IRTG 2544. NP gratefully acknowledges funding by DFG through the Heinz Maier-Leibnitz Prize.
Competing interest
The author has no competing interest to declare.