In this section, we prove moment estimates and continuity of the solution of the mixed fractional CEV system (1.2)–(1.3) by extending the idea of [13] for a mixed stochastic differential equation.
Lemma 3.1
If
X
obeys the normal distribution with zero mean, then for any constant
\(p \ge 1\), there exists a constant
\({c_{1}}(p)\), depending only on
p, such that
$$ E\bigl[ \vert X \vert ^{p}\bigr] \le {c_{1}}(p)E{ \bigl[ \vert X \vert \bigr]^{p}}. $$
(3.1)
Proof
Let \(X \sim N(0,{\sigma ^{2}})\) and \(f(x) = \frac{1}{ {\sqrt{2\pi } {\sigma ^{2}}}}\exp \{ - \frac{{{x^{2}}}}{{2{\sigma ^{2}}}}\} \). If \(n \ge 2\) is an even number, then it is easy to have
$$ E\bigl[ \vert X \vert ^{n}\bigr] = 2 \int _{0}^{ + \infty } {{x^{n}}f(x)\, \mathrm{d}x} = \frac{2}{ {\sqrt{2\pi {\sigma ^{2}}} }} \int _{0}^{ + \infty } {{x^{n}}\exp \biggl\{ - \frac{x}{{2{\sigma ^{2}}}}\biggr\} \,\mathrm{d}x}. $$
Using \(t = \frac{x}{{2{\sigma ^{2}}}}\), we have
$$ E\bigl[ \vert X \vert ^{n}\bigr] = 2 \int _{0}^{ + \infty } {{x^{n}}f(x)\, \mathrm{d}x} = \frac{ {{{(\sqrt{2} )}^{n}}}}{{\sqrt{\pi }}}{\sigma ^{n}} \int _{0}^{ + \infty } {{t^{\frac{{n - 1}}{2}}}\exp \{ - t \} \,\mathrm{d}x}. $$
Recall that \(\varGamma (n) = \int _{0}^{ + \infty } {{t^{n - 1}}\exp \{ - t\} \,\mathrm{d}x} \), Thus we use \(\varGamma (n) = (n - 1)\varGamma (n - 1)\) and \(\varGamma (0.5) = \sqrt{\pi }\) to arrive at
$$ E\bigl[ \vert X \vert ^{n}\bigr] = 2 \int _{0}^{ + \infty } {{x^{n}}f(x)\, \mathrm{d}x} = \frac{ {{{(\sqrt{2} )}^{n}}}}{{\sqrt{\pi }}}{\sigma ^{n}}\varGamma \biggl( \frac{ {n + 1}}{2}\biggr) = (n - 1)!{\sigma ^{n}}. $$
(3.2)
Let \(p \ge 1\) be a positive constant. We define
$$ n = \min \{ m|p \le m, m \text{ is an even number}\} . $$
Using the Hölder inequality and combining (3.2), we obtain
$$ E\bigl[ \vert X \vert ^{p}\bigr] \le E{\bigl[ \vert X \vert ^{n}\bigr]^{\frac{p}{n}}} = (n){!^{\frac{p}{n}}} \cdot {\sigma ^{p}}. $$
(3.3)
Now we focus on \(E[|X|]\). A change of variable for integral gives
$$\begin{aligned} E\bigl[ \vert X \vert \bigr] &= 2 \int _{\mathrm{R}} {\mathit{xf}(x)\,\mathrm{d}x} = \frac{2}{{\sqrt{2\pi } \sigma }} \int _{\mathrm{R}} {x\exp \biggl\{ - \frac{{{x^{2}}}}{{2{\sigma ^{2}}}}\biggr\} \, \mathrm{d}x} \\ &= \frac{1}{{\sqrt{2\pi } \sigma }} \int _{\mathrm{R}} {\exp \biggl\{ - \frac{ {{x^{2}}}}{{2{\sigma ^{2}}}}\biggr\} \, \mathrm{d} {x^{2}}}. \end{aligned}$$
Letting \(t = \frac{{{x^{2}}}}{{2{\sigma ^{2}}}}\) and using integration by substitution, we have
$$ E\bigl[ \vert X \vert \bigr] = \sqrt{{2 / \pi }} \cdot \sigma \int _{\mathrm{R}} {\exp \{ - t \} \,\mathrm{d}t} = \sigma \sqrt{{2 / \pi }}. $$
(3.4)
Combining (3.2), (3.3), and (3.4), the lemma is proved. □
Lemma 3.2
If
\({V_{1}}(t)\)
is the solution of volatility equation (1.3), then for any
\(p \ge 2\), we have
$$ E\biggl[ \biggl\vert \int _{0}^{t} {\sqrt{ \bigl\vert {V_{1}}(s) \bigr\vert } \,\mathrm{d}B_{s}^{H}} \biggr\vert ^{p}\biggr] \le {c_{2}}(p,H){H^{p}} {2^{p - 1}} \int _{0}^{T} {E\bigl[ \bigl\vert {V_{1}}(t) \bigr\vert ^{p}\bigr] \,\mathrm{d}t} {+} {c_{3}}(p,H,T), $$
where
\({c_{2}}(p,H) = \frac{1}{2}{c_{1}}(p){H^{p}}\)
and
\({c_{3}}(p,H,T) = \frac{1}{{{2^{p}}}}c(p){H^{p}}{2^{p - 1}}{ ( {\int _{0}^{T} {{t^{2H - 2}}\,\mathrm{d}t} } )^{p}}\).
Proof
For a partition π: \(0 = {t_{0}} < {t_{1}} < \cdots < {t_{n}} = t\), let
$$ S\bigl(\sqrt{ \vert {V_{1}} \vert },\pi \bigr) = \sum _{i = 0}^{n - 1} {\sqrt{ \bigl\vert {V_{1}}(t) \bigr\vert } \diamondsuit \bigl(B_{{t_{i + 1}}}^{(H)} - B_{{t_{i}}}^{(H)}\bigr)}. $$
(3.5)
According to the WIS integral with respect to fBm, we have
$$ \int _{0}^{t} {\sqrt{ \bigl\vert {V_{1}}(s) \bigr\vert }\,\mathrm{d}B_{s}^{(H)}} = \mathop{\lim } _{ \vert \pi \vert \to 0} S\bigl(\sqrt{ \vert {V_{1}} \vert },\pi \bigr). $$
(3.6)
Note that, for fixed \({t_{i}}\), \({V_{1}}({t_{i}})\) is a random variable. Using (3.37) in [15], we have
$$\begin{aligned} &\sqrt{ \bigl\vert {V_{1}}({t_{i}}) \bigr\vert } \diamondsuit \bigl(B_{{t_{i + 1}}} ^{(H)} - B_{{t_{i}}}^{(H)}\bigr) \\ &\quad = \sqrt{ \bigl\vert {V_{1}}(t) \bigr\vert } \cdot \int _{\mathrm{{R}}} {{I_{({t_{i}},{t _{i + 1}}]}}} \,\mathrm{d}B_{s}^{(H)} - D_{\varPhi g}^{\phi }\sqrt{ \bigl\vert {V_{1}}({t_{i}}) \bigr\vert } \\ &\quad = \sqrt{ \bigl\vert {V_{1}}(t) \bigr\vert } \cdot \int _{\mathrm{{R}}} {{I_{({t_{i}},{t _{i + 1}}]}}} \,\mathrm{d}B_{s}^{(H)} = \sqrt{ \bigl\vert {V_{1}}(t ) \bigr\vert } \cdot \bigl(B_{{t_{i + 1}}}^{(H)} - B_{{t_{i}}}^{(H)} \bigr). \end{aligned}$$
(3.7)
Substituting (3.7) into (3.5) and using the triangle and Hölder inequalities, we obtain
$$\begin{aligned} E\bigl[ \bigl\vert S\bigl(\sqrt{ \vert {V_{1}} \vert },\pi \bigr) \bigr\vert \bigr] &\le \sum_{i = 0}^{n - 1} {E\bigl[ \bigl\vert {\sqrt{ \bigl\vert {V_{1}}(t) \bigr\vert } \cdot \bigl(B_{{t_{i + 1}}}^{(H)} - B _{{t_{i}}}^{(H)} \bigr)} \bigr\vert } \bigr] \\ &\le \sum_{i = 0}^{n - 1} {\sqrt{E\bigl[ \bigl\vert {{V _{1}}(t )} \bigr\vert \bigr] \cdot E\bigl[ \bigl\vert B_{{t_{i + 1}}}^{(H)} - B_{{t_{i}}}^{(H)} \bigr\vert ^{2}\bigr]} }. \end{aligned}$$
(3.8)
Letting \(|\pi | \to 0\) and combining (3.6) and (3.8), we obtain
$$ E\biggl[ \biggl\vert \int _{0}^{t} {\sqrt{ \bigl\vert {V_{1}}(s) \bigr\vert } \,\mathrm{d}B_{s}^{H}} \biggr\vert \biggr] \le \int _{0}^{T} {\sqrt{E\bigl[ \bigl\vert {V_{1}}(t ) \bigr\vert \bigr]} \,\mathrm{d} {t^{H}}}. $$
Using Lemma 3.1, we arrive at
$$\begin{aligned} &E\biggl[ \biggl\vert \int _{0}^{t} {\sqrt{ \bigl\vert {V_{1}}(s) \bigr\vert } \,\mathrm{d}B _{s}^{H}} \biggr\vert ^{p}\biggr] \\ &\quad \le c_{1}(p)E{\biggl[ \biggl\vert \int _{0}^{t} {\sqrt{ \bigl\vert {V_{1}}(s) \bigr\vert } \,\mathrm{d}B _{s}^{H}} \biggr\vert \biggr]^{p}} \le c_{1}(p){ \biggl( { \int _{0}^{T} {\sqrt{E\bigl[ \bigl\vert {V_{1}}(t ) \bigr\vert \bigr]} \,\mathrm{d} {t^{H}}} } \biggr)^{p}}. \end{aligned}$$
(3.9)
Using the Hölder inequality twice, we have
$$\begin{aligned} E\biggl[ \biggl\vert \int _{0}^{t} {\sqrt{ \bigl\vert {V_{1}}(s) \bigr\vert } \,\mathrm{d}B_{s}^{H}} \biggr\vert ^{p}\biggr] &\le c_{1}(p){H^{p}} { \biggl( { \int _{0}^{T} {\frac{1}{2}E\bigl[ \bigl\vert {V_{1}}(t ) \bigr\vert \bigr] + \frac{1}{2}{t^{2H - 2}} \,\mathrm{d}t} } \biggr)^{p}} \\ &\le \frac{1}{{{2^{\mathrm{{p}}}}}}c_{1}(p){H^{p}} { \biggl( { \int _{0}^{T} {E\bigl[ \bigl\vert {V_{1}}(t ) \bigr\vert \bigr]\,\mathrm{d}t} {+} \int _{0}^{T} {{t^{2H - 2}}\, \mathrm{d}t} } \biggr)^{p}}. \end{aligned}$$
Using the inequality \({(a + b)^{p}} \le {2^{p - 1}}({a^{p}} + {b^{p}})\), we get
$$\begin{aligned} & E\biggl[ \biggl\vert \int _{0}^{t} {\sqrt{ \bigl\vert {V_{1}}(s) \bigr\vert } \,\mathrm{d}B _{s}^{H}} \biggr\vert ^{p}\biggr] \\ &\quad\le \frac{1}{{{2^{p}}}}c_{1}(p){H^{p}} {2^{p - 1}} \biggl( { \biggl( { \int _{0}^{T} {E\bigl[ \bigl\vert {V_{1}}(t ) \bigr\vert \bigr]\,\mathrm{d}t} } \biggr)^{p}} + { \biggl( { \int _{0}^{T} {{t^{2H - 2}}\, \mathrm{d}t} } \biggr)^{p}} \biggr). \end{aligned}$$
(3.10)
Recalling that \(H \in (0.5,1)\), we have \(2H - 2 \in ( - 1, 0)\). This implies that the integral \(\int _{0}^{T} {{t^{2H - 2}}\,\mathrm{d}t} \) is convergent. Hence the lemma is proved applying the Hölder inequality to (3.10). □
Theorem 3.1
Let
\(T > 0\)
be fixed. For
\(p \ge 2\), there are constants
\({c_{4}}\)
and
\({c_{5}}\), depending on
λ, p, \(\sigma _{i}\), \({\kappa _{i}}\), \(H,{v_{i}}\), T, such that
$$ E\Bigl[\mathop{\sup } _{t \in [0,T]} { \bigl\vert {{V_{i}}(t )} \bigr\vert ^{p}}\Bigr] \le {c_{i + 3}},\quad i = 1, 2. $$
Proof
Here we only prove the case \(i = 1\). Since for any \(t \in [0, T]\),
$$ {V_{1}}(t) = {v_{1}} + \int _{0}^{t} {{\kappa _{1}}\bigl({ \theta _{1}} - {V_{1}}(s)\bigr)\,\mathrm{d}s} + {\sigma _{1}} \int _{0}^{t} {\sqrt{ \bigl\vert {V_{1}}(s) \bigr\vert }} \,\mathrm{d}M_{2,1}^{H}(s), $$
using the Young inequality, we have for any \(p \ge 2\) that
$$ { \bigl\vert {{V_{1}}(t)} \bigr\vert ^{p}} \le {3^{p - 1}}\bigl( \vert {v_{1}} \vert ^{p} + {A_{1}} + {A_{2}}\bigr), $$
(3.11)
where \({A_{1}} = \kappa _{1}^{p}{ \vert {\int _{0}^{t} {\theta - {V _{1}}(s)\,\mathrm{d}s} } \vert ^{p}}\), \({A_{2}} = \sigma _{1}^{p} { \vert {\int _{0}^{t} {\sqrt{|{V_{1}}(s)|} } \,\mathrm{d}M_{2,1} ^{H}(s)} \vert ^{p}}\).
Now, we compute \(E[{A_{1}}]\) and \(E[{A_{2}}]\). For the second term \(A_{1}\) in (3.11), using the Hölder and Young inequalities, we have
$$\begin{aligned} E[{A_{1}}] &\le {2^{p - 1}}\kappa _{1}^{p}{\theta ^{p}}T + {2^{p - 1}} \kappa _{1}^{p}{ \biggl\vert { \int _{0}^{t} {{V_{1}}(s)\, \mathrm{d}s} } \biggr\vert ^{p}} \\ &\le {2^{p - 1}}\kappa _{1}^{p}{\theta ^{p}}T + {2^{p - 1}} \kappa _{1}^{p}{T^{p - 1}} \int _{0}^{t} {E\bigl[ \bigl\vert {V_{1}}(s) \bigr\vert ^{p}\bigr]\,\mathrm{d}s}. \end{aligned}$$
(3.12)
Noting that \(M_{t}^{H} = \lambda B(t) + {B^{H}}(t)\) and applying the inequality \({(a + b)^{n}} \le {2^{n - 1}}({a^{n}} + {b^{n}})\) to \({A_{2}}\), we have
$$ {A_{2}} \le {2^{p - 1}}\sigma _{1}^{p}{ \lambda ^{p}} {A_{3}} + {2^{p - 1}} {\lambda ^{p}}\sigma _{1}^{p}{A_{4}}, $$
(3.13)
where \({A_{3}} = { \vert {\int _{0}^{t} {\sqrt{|{V_{1}}(s)|} \,\mathrm{ {d}}B(s)} } \vert ^{p}}\) and \({A_{4}} = { \vert {\int _{0}^{t} {\sqrt{|{V_{1}}(s)|} \,\mathrm{d}B^{H}(s)} } \vert ^{p}}\). Applying the B-D-G inequality [13, 15] to \(E[{A_{3}}]\) and using the inequality \(x \le 1 + {x^{2}}\), we have
$$ E[{A_{3}}] \le { \biggl\vert { \int _{0}^{t} {E\bigl[ \bigl\vert {V_{1}}(s) \bigr\vert \bigr]\,{\mathrm{{d}}}s} } \biggr\vert ^{{p / 2}}} \le { \biggl\vert { \int _{0}^{t} {E\bigl[ \bigl\vert {V _{1}}(s) \bigr\vert \bigr]\,{\mathrm{{d}}}s} } \biggr\vert ^{p}} + 1. $$
Using the Hölder inequality, we obtain
$$ E[{A_{3}}] \le \int _{0}^{t} {E\bigl[ \bigl\vert {V_{1}}(s) \bigr\vert ^{p}\bigr]\,{\mathrm{{d}}}s} + 1. $$
(3.14)
Substituting (3.14) and Lemma 3.2 into (3.13), we have
$$ E[{A_{2}}] \le {c_{6}}(\lambda,p,\sigma,T,H) \int _{0}^{t} {E\bigl[ \bigl\vert {V_{1}}(s) \bigr\vert ^{p}\bigr]\,\mathrm{d}s} + {c_{7}}(\lambda,p,{\sigma _{1}},H,T), $$
(3.15)
where
$$\begin{aligned} &{c_{6}}(\lambda,p,\sigma,T,H) = {2^{p - 1}} {\lambda ^{p}}\sigma _{1} ^{p} + {2^{p - 1}} {\lambda ^{p}}\sigma _{1}^{p}{H^{p}} {c_{2}}(p,H), \\ &{c_{7}}(\lambda,p,{\sigma _{1}},H,T) = {2^{p - 1}}\sigma _{1}^{p}{\lambda ^{p}} {+} {2^{p - 1}} {\lambda ^{p}}\sigma _{1}^{p}{c_{3}}(p,H,T). \end{aligned}$$
Substituting (3.12) and (3.15) into (3.11) and letting
$$\begin{aligned} &{c_{8}} = {3^{p - 1}} \vert {v_{1}} \vert ^{p} + {6^{p - 1}}\kappa _{1}^{p}{ \theta ^{p}}T + {3^{p - 1}} {c_{7}}(\lambda,p,{ \sigma _{1}},H,T), \\ &{c_{9}} = {6^{p - 1}}\kappa _{1}^{p}{T^{p - 1}} + {3^{p - 1}} {c_{6}}( \lambda,p,{\sigma _{1}},T,H), \end{aligned}$$
we obtain
$$ E\bigl[{ \bigl\vert {{V_{1}}(t)} \bigr\vert ^{p}} \bigr] \le {c_{8}} + {c_{9}} \int _{0} ^{t} {E\bigl[ \bigl\vert {V_{1}}(s) \bigr\vert ^{p})\bigr]\,\mathrm{d}s}. $$
(3.16)
Hence the theorem follows from the Gronwall inequality. □
Lemma 3.3
The claim of Theorem
3.1
still holds if
\(p \in (0,2)\).
Proof
If \(1 \le p < 2\), then applying the Cauchy inequality to \({V_{1}}(t )\) in Theorem 3.1, we have
$$ E\bigl[{ \bigl\vert {{V_{1}}(t )} \bigr\vert ^{p}} \bigr] \le E{\bigl[{ \bigl\vert {{V_{1}}(t )} \bigr\vert ^{2p}}\bigr]^{\frac{1}{2}}} \le { \Bigl[ {\mathop{\sup } _{t \in [0,T]} E\bigl[{{ \bigl\vert {{V_{1}}(t )} \bigr\vert }^{2p}}\bigr]} \Bigr] ^{\frac{1}{2}}}. $$
Noting that \(2p \ge 2\) and using (3.9), we obtain
$$ E\bigl[{ \bigl\vert {{V_{1}}(t )} \bigr\vert ^{p}} \bigr] \le \sqrt{{C_{4}}(\lambda,p, \sigma _{1},{\theta _{1}},{\kappa _{1}},H,{v_{1}},T)}. $$
Because \(t \in [0,T]\) is arbitrary, (3.9) is proved for \(1 \le p < 2\).
For \(0 < p < 1\), note that
$$\begin{aligned} { \bigl\vert {{V_{1}}(t )} \bigr\vert ^{p}} &= { \bigl\vert {{V_{1}}(t )} \bigr\vert ^{p}} {I_{\{ \vert {{V_{1}}(t )} \vert \ge 1\} }} + { \bigl\vert {{v_{1}}(t )} \bigr\vert ^{p}} {I_{\{ \vert {{V_{1}}(t )} \vert < 1 \} }} \\ &\le { \bigl\vert {{V_{1}}(t )} \bigr\vert ^{p + 1}} {I_{\{ \vert {{V_{1}}(t )} \vert \ge 1\} }} + { \bigl\vert {{V_{1}}(t )} \bigr\vert ^{p}} {I_{\{ \vert {{V_{1}}(t )} \vert < 1\} }}. \end{aligned}$$
Further we have
$$ { \bigl\vert {{V_{1}}(t )} \bigr\vert ^{p}} \le { \bigl\vert {{V_{1}}(t )} \bigr\vert ^{p + 1}} {I_{\{ \vert {{V_{1}}(t )} \vert \ge 1\} }} + 1 \le { \bigl\vert {{V_{1}}(t)} \bigr\vert ^{p + 1}} + 1. $$
Hence it follows that, in the case \(1 < p < 2\),
$$ \mathop{\sup } _{t \in [0,T]} E\bigl[{ \bigl\vert {{V_{1}}(t )} \bigr\vert ^{p}}\bigr] \le \sqrt{{c_{4}}(\lambda,p, \sigma _{1},{\theta _{1}},{\kappa _{1}},H,{v_{1}},T)} + 1. $$
Thus the proof of the lemma is completed. □
Lemma 3.4
The stock price equation of the CEV model has a unique solution. For any positive constant
p, we have
$$ \mathop{\sup } _{t \in [0,T]} E\bigl[{ \bigl\vert {{S_{i}}(t)} \bigr\vert ^{p}}\bigr] \le {c_{10}}(r,\lambda,p,\sigma _{i},H,{\theta _{i}},{\kappa _{i}},{v_{i}},{s_{i}},T), \quad i = 1,2. $$
(3.17)
Proof
A similar proof of the existence and uniqueness of stock price equation can be found in [14]; (3.17) can be obtained by following the proof of Lemma 3.1. □