Error distribution of the Euler approximation scheme for stochastic Volterra equations

Abstract

The purpose of this paper is to establish the convergence in distribution of the normalized error in the Euler approximation scheme for stochastic Volterra equations driven by a standard Brownian motion, with a kernel of the form $(t-s)^\alpha$, where $\alpha \in \left( -\frac{1}{2}, \frac{1}{2}\right)$.

Appendix

We first recall the following extended version of Gronwall’s inequality (see, for instance, [15, Corollary 2]).

Lemma 6.1

Fix $\alpha >-1$. Suppose that $\{u(t), t\in [0,T]\}$ is a nonnegative integrable function such that

$$\begin{aligned} u(t) \le a + b \int _0^t (t-s)^\alpha u(s)\text {d}s, \end{aligned}$$

for all $t\in [0,T]$ and for some constants $a,b \ge 0$. Then,

$$\begin{aligned} u(t) \le a E_{\alpha +1} (b \Gamma (\alpha +1) t^{\alpha +1}), \end{aligned}$$

where $E_{\alpha +1}$ is the Mittag–Leffler function defined by $E_{\alpha +1} (z)=\sum _{k=0} ^\infty \frac{z^k}{ \Gamma ( n(\alpha +1)+1)}$.

In the next lemma, we establish the Hölder continuity in $L^2$ of the processes ${\widetilde{A}}^{n,1}_t$ and ${\widetilde{C}}_t$ defined in (4.69) and (4.8), respectively.

Lemma 6.2

Consider the processes ${\widetilde{A}}^{n,1}_t$ and ${\widetilde{C}}_t$ defined in (4.69) and (4.8), respectively. There exists a constant $C>0$, such that for all $t_1, t_2\in [0,T]$,

$$\begin{aligned} E[| {\widetilde{A}}^{n,1}_{t_1} -{\widetilde{A}}^{n,1}_{t_2}|^2] \le C |t_1-t_2|^{2\alpha +1}, \end{aligned}$$

(6.1)

and

$$\begin{aligned} E[| {\widetilde{C}}_{t_1} -{\widetilde{C}}_{t_2}|^2] \le C |t_1-t_2|^{2\alpha +1}. \end{aligned}$$

(6.2)

Proof

Let us first show (6.1). Recall that, for $0\le u \le s \le T$, $\psi _{n,1}(u,s)=(s-\eta _n(u))^\alpha -(s-u)^\alpha$ and $\eta _n(u)= \frac{\lfloor nu \rfloor }{n}$. For any $0\le t_1<t_2\le T$, we have

$$\begin{aligned} E[| {\widetilde{A}}^{n,1}_{t_1} -{\widetilde{A}}^{n,1}_{t_2}|^2]&= n^{2\alpha +1} \int _0^{t_1} |(t_2-s)^{ \alpha } -(t_1-s)^\alpha |^2 E\left[ (\sigma '\sigma )^2(X_s) \left( \int _0^s \psi _{n,1}(u,s)\text {d}W_u \right) ^2 \right] \text {d}s \\&\qquad + n^{2\alpha +1} \int _{t_1} ^{t_2} (t_2-s)^{2 \alpha } E\left[ (\sigma '\sigma )^2(X_s) \left( \int _0^s \psi _{n,1}(u,s)\text {d}W_u \right) ^2 \right] \text {d}s \\&\le C n^{2\alpha +1} \int _0^{t_1} |(t_2-s)^{ \alpha } -(t_1-s)^\alpha |^2 \int _0^s \psi ^2_{n,1}(u,s) \text {d}u \text {d}s \\&\qquad + C n^{2\alpha +1} \int _{t_1} ^{t_2} (t_2-s)^{2 \alpha } \int _0^s \psi ^2_{n,1}(u,s) \text {d}u \text {d}s . \end{aligned}$$

Applying Lemma 6.5 yields,

$$\begin{aligned} E[| {\widetilde{A}}^{n,1}_{t_1} -{\widetilde{A}}^{n,1}_{t_2}|^2]&\le C \int _0^{t_1} |(t_2-s)^{ \alpha } -(t_1-s)^\alpha |^2 \text {d}s + C \int _{t_1} ^{t_2} (t_2-s)^{2 \alpha }\text {d}s. \end{aligned}$$

Making the change of variable $s= t_1 - (t_2-t_1)y$ in the first integral of the above display, yields

$$\begin{aligned} E[| {\widetilde{A}}^{n,1}_{t_1} -{\widetilde{A}}^{n,1}_{t_2}|^2]&\le C (t_2-t_1)^{2\alpha +1} \int _0^{\frac{t_1}{t_2-t_1}} |(y+1)^\alpha -y^\alpha |^2 \text {d}y+ C \int _{t_1} ^{t_2} (t_2-s)^{2 \alpha }\text {d}s \\&\le C |t_1-t_2|^{2\alpha +1}. \end{aligned}$$

The proof of the inequality (6.2) is analogous, but instead of Lemma 6.5 we make use of inequality (3.2).

The next results are technical lemmas that have been used several time in the proofs.

Lemma 6.3

Recall that $\eta _n(s)= \frac{\lfloor ns \rfloor }{n}$. For any $\alpha \in \left( -\frac{1}{2},\frac{1}{2}\right)$ and for any $\delta >0$, there are positive constants $C_1$, $C_2$ and $C_3$, such that for all $t\in [0,1]$,

$$\begin{aligned} \int ^{t}_0\left[ (t-\eta _n(s))^{\alpha }-(t-s)^{\alpha }\right] ^2\, \text {d}s \le C_1\, n^{-2\alpha -1} \end{aligned}$$

(6.3)

and

$$\begin{aligned} \int ^{(t-\delta )_+}_0\left[ (t-\eta _n(s))^{\alpha }-(t-s)^{\alpha }\right] ^2\, \text {d}s \le C_2 n^{-2} \delta ^{2\alpha -1}. \end{aligned}$$

(6.4)

Proof

By the mean value theorem, for any $0 \le s<t \le 1$, we have

$$\begin{aligned} \left[ (t-\eta _n(s))^{\alpha }-(t-s)^{\alpha }\right] ^2\le (t-s)^{2\alpha -2}(s-\eta _n(s))^2. \end{aligned}$$

(6.5)

Let us first show the inequality (6.3). We can assume that $t>\frac{1}{n}$, because in the case $t\le \frac{1}{n}$, the inequality can be easily proved. We decompose the interval [0, t] into the union of the intervals $\left[ 0, \frac{\lfloor nt\rfloor -1}{n}\right]$ and $\left[ \frac{\lfloor nt\rfloor -1}{n},t\right]$. For the first interval, we can write

$$\begin{aligned} \int ^{\frac{\lfloor nt\rfloor -1}{n}}_0\left[ (t-\eta _n(s))^{\alpha }-(t-s)^{\alpha }\right] ^2\, \text {d}s \nonumber \\&\le \sum _{i=0}^{\lfloor nt\rfloor -2}\int _{\frac{i}{n}}^{\frac{i+1}{n}}(t-s)^{2\alpha -2}(s-\eta _n(s))^2\text {d}s \nonumber \\&\le \sum _{i=0}^{\lfloor nt\rfloor -2}\int _{\frac{i}{n}}^{\frac{i+1}{n}}\left( t-\frac{i+1}{n}\right) ^{2\alpha -2} n^{-2}\text {d}s \nonumber \\&\le \sum _{i=0}^{\lfloor nt\rfloor -2}\frac{(nt-i-1)^{2\alpha -2}}{n^{2\alpha -2}} n^{-3} \nonumber \\&=\frac{1}{n^{2\alpha +1}}\sum _{i=0}^{\lfloor nt\rfloor -2}(nt-i-1)^{2\alpha -2}\nonumber \\&\le \frac{1}{n^{2\alpha +1}} \sum _{j=1} ^\infty j^{2\alpha -2}. \end{aligned}$$

(6.6)

Next for any $s\in \left[ \frac{\lfloor nt\rfloor -1}{n},t\right]$, we have

$$\begin{aligned} \left[ (t-\eta _n(s))^{\alpha }-(t-s)^{\alpha }\right] ^2&\le C\left[ (t-\eta _n(s))^{2\alpha }+(t-s)^{2\alpha }\right] \\&\le {\left\{ \begin{array}{ll} C n^{-2\alpha } &{} \alpha \ge 0 \\ C(t-s)^{2\alpha }&{} \alpha < 0 \end{array}\right. } . \end{aligned}$$

Therefore,

$$\begin{aligned} \int _{\frac{\lfloor nt\rfloor -1}{n}}^t \left[ (t-\eta _n(s))^{\alpha }-(t-s)^{\alpha }\right] ^2\text {d}s \le Cn^{-2\alpha -1}. \end{aligned}$$

(6.7)

Thus, combining (6.6) and (6.7) we obtain (6.3).

The inequality (6.4) follows easily from (6.5). Indeed, we can assume that $t>\delta$, and in this case

$$\begin{aligned} \int _{0}^{t-\delta }\left[ (t-\eta _n(s))^{\alpha }-(t-s)^{\alpha }\right] ^2\, \text {d}s&\le \int _{0}^{t-\delta }(t-s)^{2\alpha -2}(s-\eta _n(s))^2\text {d}s\\&\le \frac{1}{n^2}\int _{0}^{t-\delta }(t-s)^{2\alpha -2}\text {d}s\\&= \frac{1}{n^2} \int _\delta ^t s^{2\alpha -2} \text {d}s \\&\le \frac{2\delta ^{2\alpha -1}}{n^2(1-2\alpha )} = C_2{n^{-2}} \delta ^{2\alpha -1}. \end{aligned}$$

Lemma 6.4

Let $\psi _{n,2}(u,s)= (s- \eta _n(u))^\alpha - (\eta _n(s) -\eta _n(u))^\alpha$ for $0\le u \le s \le T$ (see (4.28)) and recall that $\eta _n(u)= \frac{\lfloor nu \rfloor }{n}$. Then, there exists a constant $C>0$ such that for any $s\ge 0$, we have

$$\begin{aligned} n^{2\alpha +1} \int _0^{\eta _n(s)} \psi ^2_{n,2}(u,s) \text {d}u \le C. \end{aligned}$$

Proof

We have, using the mean value theorem,

$$\begin{aligned} \int _0^{\eta _n(s)} \psi ^2_{n,2}(u,s) \text {d}u&= \int _0^{\eta _n(s)} [(s-\eta _n(u))^{\alpha }-(\eta _n(s)-\eta _n(u))^{\alpha }]^2 \text {d}u\\&\le \alpha ^2 (s- \eta _n(s))^2 \int _0^{\eta _n(s)} (\eta _n(s)-\eta _n(u))^{2\alpha -2} \text {d}u \\&\le C n^{-3} \sum _{i=0} ^{\lfloor ns \rfloor -1} \left( \eta _n(s) -\frac{i}{n} \right) ^{2\alpha -2} \\&= Cn^{-2\alpha -1} \sum _{i=0} ^{\lfloor ns \rfloor -1} (\eta _n-i)^{2\alpha -2}. \end{aligned}$$

Since $2\alpha -2 <-1$ the series $\sum _{j=1} ^\infty j^{2\alpha -2}$ is convergent, and this completes the proof of the lemma.

Lemma 6.5

Recall that (see (4.10)), for $0\le u \le s \le T$, $\psi _{n,1}(u,s)=(s-\eta _n(u))^\alpha -(s-u)^\alpha$ and $\eta _n(u)= \frac{\lfloor nu \rfloor }{n}$. Then, there exists a constant $C>0$ such that for any $s\ge 0$, we have

$$\begin{aligned} n^{2\alpha +1} \int _0^{s} \psi ^2_{n,1}(u,s) \text {d}u \le C. \end{aligned}$$

Proof

We have, using the mean value theorem,

$$\begin{aligned} \int _0^{s} \psi ^2_{n,1}(u,s) \text {d}u&= \int _0^{s} [(s-\eta _n(u))^{\alpha }-(s-u)^{\alpha }]^2 \text {d}u\\&= \sum _{i=0} ^{\lfloor ns \rfloor } \int _{\frac{i}{n}} ^{\frac{i+1}{n} \wedge s} \left[ \left( s-\frac{i}{n}\right) ^\alpha - (s-u)^\alpha \right] ^2 \text {d}u\\&= n^{-2\alpha -1}\sum _{i=0} ^{\lfloor ns \rfloor } \int _{i} ^{ (i+1) \wedge ns } [ (ns-i)^\alpha - (ns-x)^\alpha ]^2 \text {d}x \\&\le n^{-2\alpha -1}\sum _{k=0} ^{\infty } \int _{0} ^{ 1 \wedge (ns -\lfloor ns \rfloor +k)} [ (ns-\lfloor ns \rfloor +k)^\alpha - (ns-\lfloor ns \rfloor +k-x)^\alpha ]^2 \text {d}x, \end{aligned}$$

which is uniformly bounded by a constant C that does not depend on n and s.