Mixed Stochastic Differential Equations: Existence and Uniqueness Result

Abstract

In this paper we establish an existence and uniqueness result for solutions of multidimensional, time-dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter $H>1/2$ and a multidimensional standard Brownian motion under a weaker condition than the Lipschitz one.

Appendix

In this appendix, we recall some results which play a great role in this work. We also show a technical lemma that has been used in the proof of pathwise uniqueness. We begin with some a priori estimates from the paper of Nualart and Rascanu [21].

Proposition 11

We have

• (i) ${\displaystyle \left\| \int _{0}^{.}f(s)\,\mathrm{d}s\right\| _{\alpha ,t}\le C\int _{0}^{t}\frac{\left| f(s)\right| }{(t-s)^{\alpha }}\,\mathrm{d}s.}$

• (ii) ${\displaystyle \left\| \int _{0}^{.}f(s)\,\mathrm{d}B^{H}(s)\right\| _{\alpha ,t} \le C\left\| B^{H}\right\| _{1-\alpha ,\infty ,t}\int _{0}^{t}\left( (t-s)^{-2\alpha }+s^{-\alpha }\right) \Vert f\Vert _{\alpha ,s}\,\mathrm{d}s.}$

Moreover, under the linear growth assumption, we have from Nualart and Rascanu [21], the following

Proposition 12

Assume (H.1) and (H.2). The following estimates hold

• (j) ${\displaystyle \left\| \int _{0}^{.}b(s,f(s))\, \mathrm{d}s\right\| _{\alpha ,t} \le C\left( \int _{0}^{t}\dfrac{\left| f(s)\right| }{(t-s)^{\alpha }}\, \mathrm{d}s+1\right) .}$

• (jj) ${\displaystyle \left\| \int _{0}^{.}\sigma _{H}(s,f(s))\, \mathrm{d}B^{H}(s)\right\| _{\alpha ,t} \le C\left\| B^{H}\right\| _{1-\alpha ,\infty ,t}\int _{0}^{t}\left( (t-s)^{-2\alpha }+s^{-\alpha }\right) }$ $\big (1+\Vert f\Vert _{\alpha ,s}\big ) \mathrm{d}s.$

We recall the following characterization of the convergence in probability in terms of weak convergence, see Gyöngy and Krylov [10].

Lemma 13

Let $(Z_{n})_{n\in \mathbb {N}}$ be a sequence of random elements in a Polish space $(\mathcal {E},d)$ equipped with the Borel $\sigma$-algebra. Then $(Z_{n})_{n\in \mathbb {N}}$ converges in probability to an $\mathcal {E}$-valued random element if and only if for every pair of subsequences $(Z_{m})_{m\in \mathbb {N}}$ and $(Z_{k})_{k\in \mathbb {N}}$ there exists a subsequence $(Z_{m(p)},Z_{k(p)})_{p\in \mathbb {N}}$ converging weakly to a random element v supported on the diagonal $\left\{ (x,y)\in \mathcal {E}\times \mathcal {E}:x=y\right\}$.

Finally, let us give a version of Bihari’s lemma.

Lemma 14

2 Fix $1/2<\alpha <1$, $a, b\ge 0$. Let $f:\left[ 0,\infty \right) \longrightarrow \left[ 0,\infty \right)$ be a continuous function such that

$$\begin{aligned} f(t)\le a+bt^{\alpha }\int _{0}^{t}(t-s)^{-\alpha }s^{-\alpha }\varrho \left( f(s)\right) \, \mathrm{d}s, \end{aligned}$$

where $\varrho$ is a concave increasing function from $\mathbb {R}_{+}$ to $\mathbb {R}_{+}$ such that $\varrho (0)=0$, $\varrho (u)>0$ for $u>0$ and satisfying (2.1) for some $q>1$. Then for any $1<p<2$ such that $\alpha <1/p$ and $q>1$ with $1/p+1/q=1$ we have

$$\begin{aligned} f(t)\le \left[ F^{-1}\left( F(2^{q-1}a^{q})+2^{q-1}b^{q}\, C_{\alpha ,p}^{q/p}t^{q\left( (1/p)-\alpha \right) +1}\right) \right] ^{1/q}, \end{aligned}$$

for all $t\in \left[ 0,T\right]$ such that

$$\begin{aligned} F(2^{q-1}a^{q})+2^{q-1}b^{q}\, C_{\alpha ,p}^{q/p}t^{q\left( (1/p)-\alpha \right) +1}\in Dom(F^{-1}), \end{aligned}$$

where

$$\begin{aligned} F(x)=\int _{1}^{x}\dfrac{\mathrm{d}u}{\varrho ^{q}(u^{1/q})},\quad for\, x \ge 0, \end{aligned}$$

and $F^{-1}$ is the inverse function of F. In particular, if moreover, $a=0$ then $f(t)=0$ for all $0<t<T$.

Proof

Let $1<p<2$ be such that $\alpha <1/p$. Using the Hölder inequality we obtain

$$\begin{aligned} f(t)\le a+bt^{\alpha }\left( \int _{0}^{t}(t-s)^{-p\alpha }s^{-p\alpha }\, \mathrm{d}s\right) ^{1/p}\left( \int _{0}^{t}\varrho ^{q}\left( f(s)\right) \, \mathrm{d}s\right) ^{1/q} \end{aligned}$$

For the first integral, using $s=tu$, we have

$$\begin{aligned} \int _{0}^{t}(t-s)^{-p\alpha }s^{-p\alpha }\, \mathrm{d}s=t^{1-2p\alpha }\int _{0}^{1}(1-u)^{-p\alpha }u^{-p\alpha }\, \mathrm{d}u=C_{\alpha ,p}t^{1-2p\alpha } \end{aligned}$$

where $C_{\alpha ,p}=B\left( 1-p\alpha ,1-p\alpha \right)$ is the beta function. It follows that

$$\begin{aligned} f(t)\le a+b\, C_{\alpha ,p}^{1/p}t^{(1/p)-\alpha }\left( \int _{0}^{t}\varrho ^{q}\left( f(s)\right) \, \mathrm{d}s\right) ^{1/q}. \end{aligned}$$

This yields

$$\begin{aligned} f^{q}(t)\le 2^{q-1}a^{q}+2^{q-1}b^{q}\, C_{\alpha ,p}^{q/p}t^{q\left( (1/p)-\alpha \right) }\int _{0}^{t}\varrho ^{q}\left( f(s)\right) \, \mathrm{d}s. \end{aligned}$$

Then it follows from Bihari’s Lemma, see [4], that

$$\begin{aligned} f(t)\le \left[ F^{-1}\left( F(2^{q-1}a^{q})+2^{q-1}b^{q}\, C_{\alpha ,p}^{q/p}t^{q\left( (1/p)-\alpha \right) +1}\right) \right] ^{1/q}, \end{aligned}$$

for all $t\in \left[ 0,T\right]$ such that

$$\begin{aligned} F(2^{q-1}a^{q})+2^{q-1}b^{q}\, C_{\alpha ,p}^{q/p}t^{q\left( (1/p)-\alpha \right) +1}\in \mathrm {Dom}(F^{-1}). \end{aligned}$$

Now, it is simple to see from Eq. (2.1) that $\lim _{x\rightarrow 0}F(x)=\infty$; therefore $\lim _{x\rightarrow \infty }F^{-1}(x)=0$ which implies that $f(t)=0$ when $a=0$ for $t\in [0,T]$.

$\square$