Comparison Theorem for Stochastic Functional Differential Equations and Applications

Abstract

The comparison theorem is proved for stochastic functional differential equations whose drift term satisfies the quasimonotone condition and diffusion term is independent of delay. Application is given to stochastic neutral networks with delays.

Appendix: The Proof of Theorem 1

The idea for the proof is partially borrowed from [16].

Since $\rho$ is concave with $\rho (0) = 0$, there are two positive constants $a_1, a_2$ such that

$$\begin{aligned} \rho (x) \le a_1x + a_2\ \mathrm{for}\ x \ge 0. \end{aligned}$$

(5.1)

As before, $\varrho (x) \triangleq x + \rho (x)\ \mathrm{for}\ x \ge 0.$

Lemma 12

For any $T > 1$ and $\xi \in L^2(\varOmega , C(I,\mathbb {R}^d))$, where $I$ is an interval, let

$$\begin{aligned} \tilde{C}_1(\xi )&\triangleq 3\big [(1+\gamma T^2)\mathrm{E}\parallel \xi \parallel ^2+\gamma T^2+4\gamma T\big ]e^{6\gamma T(T+2)},\\ C_1&\triangleq 3\gamma T(T+4)e^{6\gamma T(T+2)},\\ C_2&\triangleq 4d(TL+2),\\ C_3&\triangleq C_2\varrho (4\tilde{C}_1(\xi )). \end{aligned}$$

Then $T_1 \triangleq \frac{4C_1}{C_2(4C_1+4C_1a_1+a_2)}$, depending only on $T$ and $\rho$ rather than on $\xi$, satisfies that

$$\begin{aligned} C_2\varrho (C_3T_1)\le C_3. \end{aligned}$$

(5.2)

Proof

$\tilde{T}_1 \triangleq \frac{4\tilde{C}_1(\xi )}{C_2(4\tilde{C}_1(\xi )+4\tilde{C}_1(\xi )a_1+a_2)}.$ Then $T_1 < \tilde{T}_1 < 1$. Since $\varrho$ is increasing on $[0,\infty )$, it suffices to show that

$$\begin{aligned} C_2\varrho \big (C_3\tilde{T}_1\big )\le C_3=C_2\varrho \big (4\tilde{C}_1(\xi )\big ). \end{aligned}$$

Equivalently, $(C_3\tilde{T}_1)\le 4\tilde{C}_1(\xi ),$ i.e.,

$$\begin{aligned} C_2\varrho \big (4\tilde{C}_1(\xi )\big )\tilde{T}_1 \le 4\tilde{C}_1(\xi ). \end{aligned}$$

By (5.1),

$$\begin{aligned} C_2\varrho \big (4\tilde{C}_1(\xi )\big )\tilde{T}_1 \le C_2\tilde{T}_1\big (4\tilde{C}_1(\xi )+4\tilde{C}_1(\xi )a_1+a_2\big )= 4\tilde{C}_1(\xi ). \end{aligned}$$

This proves (5.2). $\square$

From now on, fix $T > 1$ and let $k$ be an integer with $\frac{T}{k} < T_1$. Denote by

$$\begin{aligned} J_i \triangleq \Big [\frac{iT}{k}-\tau ,\frac{iT}{k}\Big ]\ \mathrm{and}\ I_i \triangleq \Big [\frac{iT}{k},\frac{(i+1)T}{k}\Big ]\quad \mathrm{for}\ i = 0, 1,\ldots , k-1. \end{aligned}$$

Consider the stochastic functional differential equations

$$\begin{aligned} \left\{ \begin{array}{ll} x(t)=\xi \big (\frac{iT}{k}\big ) +\int \limits _{\frac{iT}{k}}^{t} f\big (s,x(s),x_{s}\big )ds+ \int _{\frac{iT}{k}}^{t}\sigma \big (s,x(s)\big )dW_{s},\quad t\in I_i\\ x(\theta )=\xi (\theta ), \quad \theta \in J_i. \end{array}\right. \end{aligned}$$

(5.3)

Proposition 13

Let $\xi \in L^2(\varOmega , C(J_i,\mathbb {R}^d))$ be $\{\mathfrak {F}_t\}_{t\in J_i}$ adapted. Then (5.3) has a solution $x\in L^2(\varOmega , C([\frac{iT}{k}-\tau ,\frac{(i+1)T}{k}],\mathbb {R}^d))$ adapted to $\{\mathfrak {F}_{t}\}_{t\in J_i\cup I_i}$ and with initial process $\xi$.

Proof

To prove the existence of solution to (5.3), let us construct the following sequence of successive approximations by setting

$$\begin{aligned} \left\{ \begin{array}{ll} x^{n}(t)=\xi \big (\frac{iT}{k}\big ) +\int \limits _{\frac{iT}{k}}^{t} f\big (s,x^{n-1}(s),x^{n-1}_{s}\big )ds+\int _{\frac{iT}{k}}^{t} \sigma \big (s,x^{n-1}(s)\big )dW_{s},\quad t\in I_i\qquad \qquad \\ x^{n}(\theta )=\xi (\theta ), \quad \theta \in J_i, \end{array}\right. \end{aligned}$$

(5.4)

for $n\ge 1,$ and

$$\begin{aligned} \left\{ \begin{array}{ll} x^{0}(t)=\xi \big (\frac{iT}{k}\big ), \quad t\in I_i\\ x^{0}(\theta )=\xi (\theta ), \quad \theta \in J_i. \end{array} \right. \end{aligned}$$

For every $t\in I_i$, by (5.4) and the Hölder inequality we have

$$\begin{aligned} \sup \limits _{\frac{iT}{k}\le \chi \le t}|x^{n}(\chi )|^{2}&\le \sup \limits _{\frac{iT}{k}\le \chi \le t}\Bigg (3|\xi \big (\frac{iT}{k}\big )|^{2}+3|\int \limits _{\frac{iT}{k}}^{\chi } f(s,x^{n-1}(s),x^{n-1}_{s})ds|^{2}\nonumber \\&+\, 3|\int _{\frac{iT}{k}}^{\chi } \sigma (s,x^{n-1}(s))dW_{s}|^{2}\Bigg )\\&\le 3\Big |\xi \left( \frac{iT}{k}\right) \Big |^{2}+3\Big (t-\frac{iT}{k}\Big )\int \limits _{\frac{iT}{k}}^{t} \big |f(s,x^{n-1}(s),x^{n-1}_{s})\big |^{2}ds \\&+\, 3\sup \limits _{\frac{iT}{k}\le \chi \le t}\Big |\int _{\frac{iT}{k}}^{\chi }\sigma (s,x^{n-1}(s))dW_{s}\Big |^{2}. \end{aligned}$$

By the Burkholder–Davis–Gundy inequality we have

$$\begin{aligned} E\sup \limits _{\frac{iT}{k}\le \chi \le t}|x^{n}(\chi )|^{2}&\le 3 E\Big |\xi \left( \frac{iT}{k}\right) \Big |^{2}+3\Big (t-\frac{iT}{k}\Big )E\Big (\int \limits _{\frac{iT}{k}}^{t} |f(s,x^{n-1}(s),x^{n-1}_{s})|^{2}ds\Big )\nonumber \\&+\, 12E\int _{\frac{iT}{k}}^{t} \Big |\sigma (s,x^{n-1}(s))\Big |^{2}ds. \end{aligned}$$

It follows from (H4) and $t-\frac{iT}{k}\le T$ for $t\in I_i$ that

$$\begin{aligned} E\sup \limits _{\frac{iT}{k}\le \chi \le t}|x^{n}(\chi )|^{2}\!&\le \! 3E\left| \xi (\frac{iT}{k})\right| ^{2}\!+\!3\gamma \left( t\!-\!\frac{iT}{k}\right) \left( \int \limits _{\frac{iT}{k}}^{t} (1\!+\!E|x^{n-1}(s)|^{2}\!+\!E\Vert x^{n-1}_{s}\Vert ^{2})ds\right) \\&\quad +\, 12\gamma \int _{\frac{iT}{k}}^{t}\left( 1+E|x^{n-1}(s)|^{2}\right) ds\\&\le 3\left[ (1+\gamma T^{2})E\Vert \xi \Vert ^{2}+\gamma T^{2}+4\gamma T)\right] \\&\quad +\, 6\gamma (T+2)\int _{\frac{iT}{k}}^{t}\left( E\sup \limits _{\frac{iT}{k}\le \chi \le s}|x^{n-1}(\chi )|^{2}\right) ds, \end{aligned}$$

which shows

$$\begin{aligned} \sup \limits _{0\le j\le n}E\sup \limits _{\frac{iT}{k}\le \chi \le t}|x^{j}(\chi )|^{2}&\le 3\left[ (1+\gamma T^{2})E\Vert \xi \Vert ^{2}+\gamma T^{2}+4\gamma T)\right] \\&\quad +\, 6\gamma (T+2)\int _{\frac{iT}{k}}^{t}\left( \sup \limits _{0\le j\le n}E\sup \limits _{\frac{iT}{k}\le \chi \le s}|x^{j}(\chi )|^{2}\right) ds. \end{aligned}$$

Using the classical Gronwall inequality we get that for $t\in I_i$

$$\begin{aligned} \sup \limits _{0\le j\le n}E\sup \limits _{\frac{iT}{k}\le \chi \le t}|x^{j}(\chi )|^{2}&\le 3\left[ (1+\gamma T^{2})E\Vert \xi \Vert ^{2}+\gamma T^{2}+4\gamma T)\right] e^{6\gamma (T+2)(t-\frac{iT}{k})}\\&\le 3\left[ (1+\gamma T^{2})E\Vert \xi \Vert ^{2}+\gamma T^{2}+4\gamma T)\right] e^{6\gamma T(T+2)}. \end{aligned}$$

In particular,

$$\begin{aligned} E\sup \limits _{t\in I_i}|x^{n}(t)|^{2}\le 3\left[ (1+\gamma T^{2})E\Vert \xi \Vert ^{2}+\gamma T^{2}+4\gamma T)\right] e^{6\gamma T(T+2)}=\tilde{C}_1(\xi ),\ n\ge 1.\nonumber \\ \end{aligned}$$

(5.5)

From (5.4) it follows that $\{x^{n}(t),n\ge 1,t\in I_i\}$ are adapted to $(\mathfrak {F}_{t})_{t\in I_i}$ with continuous sample paths. Again by (5.4) and the Hölder inequality we get that for any $n\ge 1,m\ge 1$ and $t\in I_i$

$$\begin{aligned} \sup \limits _{\frac{iT}{k}\le \chi \le t}|x^{n+m}(\chi )-x^{n}(\chi )|^{2}\!&\le \! \sup \limits _{\frac{iT}{k}\le \chi \le t}\Big (2|\int \limits _{\frac{iT}{k}}^{\chi } (f(s,x^{n+m-1}(s),x^{n+m-1}_{s})-f(s,x^{n-1} (s),\nonumber \\&\quad \times x^{n-1}_{s}))ds|^{2}\\&+\, 2|\int _{\frac{iT}{k}}^{\chi }(\sigma (s,x^{n+m-1}(s))\!-\!\sigma (s,x^{n-1}(s))) dW_{s}|^{2}\Big )\\ \!&\le \! 2T\int \limits _{\frac{iT}{k}}^{t}\big | f(s,x^{n+m-1}(s),x^{n+m-1}_{s})-f(s,x^{n-1}(s),x^{n-1}_{s})\big |^{2}ds\\&+\, 2\sup \limits _{\frac{iT}{k}\le \chi \le t}\big |\int _{\frac{iT}{k}}^{\chi }(\sigma (s,x^{n+m-1}(s))-\sigma (s,x^{n-1}(s))) dW_{s}\big |^{2}. \end{aligned}$$

By (H2), (H3$^*$), the concavity of $\rho$ and the Burkholder–Davis–Gundy inequality, we have

$$\begin{aligned} E\sup \limits _{\frac{iT}{k}\le \chi \le t}|x^{n+m}(\chi )-x^{n}(\chi )|^{2}\!&\le \! 2{ TdL} E\int \limits _{\frac{iT}{k}}^{t}\left( | x^{n+m-1}(s)\!-\!x^{n-1}(s)|^{2}\!+\!\Vert x^{n+m-1}_{s}\!-\!x^{n-1}_{s}\Vert ^{2}\right) ds \nonumber \\&+\, 8E\int _{\frac{iT}{k}}^{t}\left| \sigma (s,x^{n+m-1}(s))- \sigma (s,x^{n-1}(s))\right| ^{2}ds \nonumber \\ \!&\le \! 4{ TdL} \int \limits _{\frac{iT}{k}}^{t}\left( E\sup \limits _{\frac{iT}{k}\le \chi \le s}| x^{n+m-1}(\chi )-x^{n-1}(\chi )|^{2}\right) ds \nonumber \\&+\, 8d\int \limits _{\frac{iT}{k}}^{t}\rho \left( E\sup \limits _ {\frac{iT}{k}\le \chi \le s}|x^{n+m-1}(\chi )-x^{n-1}(\chi )|^{2}\right) ds\nonumber \\ \!&\le \! C_{2}\int \limits _{\frac{iT}{k}}^{t}\varrho \left( E\sup \limits _{\frac{iT}{k}\le \chi \le s}| x^{n+m-1}(\chi )-x^{n-1}(\chi )|^{2}\right) ds, \end{aligned}$$

(5.6)

where $C_{2}=4TdL+8d$. The above inequality and (5.5) show that

$$\begin{aligned} E\sup \limits _{\frac{iT}{k}\le \chi \le t}\big |x^{n+m}(\chi )-x^{n}(\chi )\big |^{2}\le C_{3}\Big (t-\frac{iT}{k}\Big ), \end{aligned}$$

(5.7)

where $C_{3}\triangleq C_{2}\varrho (4\tilde{C}_{1}(\xi )).$

Define two sequences of functions $\{\varphi _{n}(t)\}$ and $\{\widetilde{\varphi }_{n,m}(t)\}$ on $I_i$ as follows:

$$\begin{aligned} \varphi _{1}(t)&=C_{3}\left( t-\frac{iT}{k}\right) ,\\ \varphi _{n+1}(t)&=C_{2}\int _{\frac{iT}{k}}^{t}\varrho (\varphi _{n}(s))ds,\quad n\ge 1,\\ \widetilde{\varphi }_{n,m}(t)&=E\sup \limits _{\frac{iT}{k}\le \chi \le t}|x^{n+m}(\chi )-x^{n}(\chi )|^{2},\quad n\ge 1, m\ge 1. \end{aligned}$$

We claim that for every $n\ge 1$ and $m\ge 1$,

$$\begin{aligned} 0\le \widetilde{\varphi }_{n,m}(t)\le \varphi _{n}(t)\le \varphi _{n-1}(t)\le \cdots \le \varphi _{1}(t),\quad t\in I_i. \end{aligned}$$

(5.8)

First of all, By (5.2), the assumption $\frac{T}{k} < T_1$ and the monotonicity for $\varrho$, we have for any $t\in I_i,$

$$\begin{aligned} C_{2}\varrho \left( C_{3}(t-\frac{iT}{k})\right) \le C_{3}. \end{aligned}$$

(5.9)

In view of (5.7) and (5.6), we have

$$\begin{aligned} \widetilde{\varphi }_{1,m}(t)&=E\sup \limits _{\frac{iT}{k}\le \chi \le t}\left| x^{1+m}(\chi )-x^{1}(\chi )\right| ^{2}\le C_{3}\left( t-\frac{iT}{k}\right) =\varphi _{1}(t).\\ \widetilde{\varphi }_{2,m}(t)&=E\sup \limits _{\frac{iT}{k}\le \chi \le t}\left| x^{2+m}(\chi )-x^{2}(\chi )\right| ^{2}\\&\le C_{2}\int \limits _{\frac{iT}{k}}^{t}\varrho \left( E\sup \limits _{\frac{iT}{k}\le \chi \le s}|x^{1+m}(\chi )-x^{1}(\chi )|^{2}\right) ds \\&\le C_{2}\int \limits _{\frac{iT}{k}}^{t}\varrho \left( \varphi _{1}(s)\right) ds=\varphi _{2}(t). \end{aligned}$$

But by (5.9) we also have

$$\begin{aligned} \varphi _{2}(t)=C_{2}\int _{\frac{iT}{k}}^{t}\varrho (\varphi _{1}(s))ds \le \varphi _{1}(t). \end{aligned}$$

Now we have already shown that for $t\in I_i,$

$$\begin{aligned} \widetilde{\varphi }_{2,m}(t)\le \varphi _{2}(t)\le \varphi _{1}(t). \end{aligned}$$

Next we assume that (5.8) holds for some $n\ge 2$, then by (5.6)

$$\begin{aligned} \widetilde{\varphi }_{n+1,m}(t)&\le C_{2} \int \limits _{\frac{iT}{k}}^{t}\varrho \left( E\sup \limits _{\frac{iT}{k}\le \chi \le s}| x^{n+m}(\chi )-x^{n}(\chi )|^{2}\right) ds\\&\le C_{2}\int \limits _{\frac{iT}{k}}^{t}\varrho (\varphi _{n}(s))ds=\varphi _{n+1}(t)\\&\le C_{2}\int \limits _{\frac{iT}{k}}^{t}\varrho (\varphi _{n-1}(s))ds=\varphi _{n}(t), \end{aligned}$$

that is, (5.8) holds for $n+1$ as well. Consequently by induction (5.8) must hold for $n\ge 1.$

Now our purpose is to prove

$$\begin{aligned} E\sup \limits _{t\in I_i}\left| x^{l}(t)-x^{n}(t)\right| ^{2}\rightarrow 0 \end{aligned}$$

(5.10)

as $l,n\rightarrow \infty .$ Note that for every $n\ge 1, \varphi _{n}(t)$ is increasing on $I_i$ and for each $t, \varphi _{n}(t)$ is monotonically nonincreasing as $n\rightarrow \infty .$ Hence we can define the function $\varphi (t)$ by $\varphi _{n}(t)\downarrow \varphi (t).$ It is easy to see that $\varphi (t)$ is continuous and increasing on $I_i.$ By the definition of $\varphi _{n}(t)$ and $\varphi (t)$ we have

$$\begin{aligned} \varphi (t)=C_{2}\int _{\frac{iT}{k}}^{t}\varrho (\varphi (s))ds,\quad t\in I_i. \end{aligned}$$

The proof of Corollary 2.3 shows that $\varphi (t)=0,\ t\in I_i.$ Clearly $\varphi _{n}(\frac{(i+1)T}{k})\downarrow 0$ as $n\rightarrow \infty .$ Hence for any $\epsilon >0$, there exists an integer $N\ge 1$ such that $\varphi _{n}(\frac{(i+1)T}{k})<\epsilon$ whenever $n>N.$ For any $m\ge 1$ and $n>N$, the above claim deduces that

$$\begin{aligned} E\sup \limits _{t\in I_i}\left| x^{n+m}(t)-x^{n}(t)\right| ^{2}=\widetilde{\varphi }_{n,m}\left( \frac{(i+1)T}{k}\right) \le \varphi _{n}\left( \frac{(i+1)T}{k}\right) <\epsilon . \end{aligned}$$

So (5.10) holds. It follows that

$$\begin{aligned} \lim \limits _{n,l\rightarrow \infty } E\sup \limits _{t\in J_i\bigcup I_i}\left| x^{n}(t)-x^{l}(t)\right| ^{2}=0. \end{aligned}$$

(5.11)

For each $x(t)\in \mathbb {L}^{2}(\varOmega ,C(J_i\bigcup I_i,\mathbb {R}^{d}))$, define $\Vert x\Vert _{\star }\triangleq (E\sup \nolimits _{t\in J_i\bigcup I_i}|x(t)|^{2})^{\frac{1}{2}}$. Then the space $\mathbb {L}^{2}(\varOmega ,C(J_i\bigcup I_i,\mathbb {R}^{d}))$ is a Banach space. Consequently by (5.11) there exists $x(t)\in \mathbb {L}^{2}(\varOmega ,C(J_i\bigcup I_i,\mathbb {R}^{d}))$ such that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } E\sup \limits _{t\in J_i\bigcup I_i}|x^{n}(t)-x(t)|^{2}=0. \end{aligned}$$

For any $\delta >0$, by Chebyshev’s inequality we have

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } \mathbb {P}\left( \sup \limits _{t\in J_i\bigcup I_i}|x^{n}(t)-x(t)|\ge \delta \right) =0. \end{aligned}$$

By the definition of limit, there is a subsequence $\{n_{k}\}_{k=1}^{\infty }$ satisfying that

$$\begin{aligned} \mathbb {P}\left( \sup \limits _{t\in J_i\bigcup I_i}|x^{n_{k}}(t)-x(t)|\ge \frac{1}{k}\right) \le \frac{1}{2^{k}},k\ge 1. \end{aligned}$$

The Borel–Cantelli Lemma shows that $x^{n_{k}}(t)$ converges to $x(t)$ uniformly on $J_i\bigcup I_i$ almost surely. It follows that $x(t),\ t\in I_i$ has continuous sample paths and is adapted to $\{\mathfrak {F}_{t}\}_{t\in I_i}$. Moreover, simply computation shows that

as $n\rightarrow \infty .$ Letting $n\rightarrow \infty$, we conclude that $x(t), \ t\in I_i$ satisfies system (5.3). This completes the proof. $\square$

Proof of Theorem 2.1

For any $\phi \in \mathcal {C}$ and $T > 1$, applying Proposition 13 with $i=0,\ \xi =\phi$, we get the solution $x = x(t)$ for (2.1) on $I_0$, which is adapted to $\{\mathfrak {F}_{t}\}_{t\in I_0}$. Then again using Proposition 13 with $i=1,\ \xi =x_{\frac{T}{k}}$, we have a solution for (5.3) on $I_1$ adapted to $\{\mathfrak {F}_{t}\}_{t\in I_1}$. In this way, we have extended the solution for (2.1) to the interval $I_0\bigcup I_1$. Repeatedly applying Proposition 13 with $\xi =x_{\frac{iT}{k}}$ for $i=2,\ldots ,k-1$ in order, we obtain that the existence for the strong solution for (2.1) on the interval $[0,T]$, which is adapted to $\{\mathfrak {F}_{t}\}_{t\in [0,T]}$. Since $T$ is arbitrary, the global strong solution exists. Lemma II.2.1 in [17] guarantees that $x_{t}(\phi )$ is a $\mathcal {C}$-valued process adapted to $\{\mathfrak {F}_{t}\}_{t\ge 0}$ with continuous sample paths.

Finally we finish the proof of the uniqueness of solution to system (2.1). To this end, we assume that $\{x(t),t\ge 0\}$ and $\{x^{*}(t),t\ge 0\}$ are solutions to system (2.1). Using the above similar arguments, we have

$$\begin{aligned} E\sup \limits _{0\le t \le T}|x(t)-x^{*}(t)|^{2}=0. \end{aligned}$$

This shows that $\{x(t),t\ge 0\}$ and $\{x^{*}(t),t\ge 0\}$ are modifications of one another, and thus are indistinguishable. This completes the proof. $\square$

Remark 14

Using Fatou Lemma and (5.5) one has

$$\begin{aligned} E\sup \limits _{0\le t \le T}|x(t)|^{2}\le \widetilde{C}_{1}(\phi ). \end{aligned}$$

Hence by the pathwise continuity of $x(t)$ and Lebesgue’s Theorem on dominated convergence one concludes that $t\rightarrow E\sup \nolimits _{0\le \chi \le t}|x(\chi )|^{2}$ is continuous.

Note We independently obtain the comparison theorem for SFDEs. This result was first presented in the Second International Conference on Recent Advances in Random Dynamical Systems, which held in Nanjing Normal University on June 20–23, 2011, and then in several international conferences. We submitted it to Stochastic Analysis and Applications on October 24, 2011 and withdrew the submission on August 18, 2014.