定义1.1 ^[1]  设$W^{H}=\{W^{H}_{t}\}$为定义在概率空间($\Omega, F$, P)上的带有分形指数$H$的分数布朗运动(fractional Brownian motion), 且其为连续的高斯过程, 其性质如下:

(1) $W^{H}_{0}=0$ a.s. P;

(2) $EW^{H}_{t}=0, EW^{H}_{t}W^{H}_{s}=\frac{1}{2}(t^{2H}+s^{2H}-|s-t|^{2H}), s, t \geq0$;

(3) $W^{H}$具有平稳增量且为自相似过程, 其轨道为几乎处处连续且不可微的.

定义1.2 定义在概率空间($\Omega, F$, P)上的随机过程$Y^{H}=\{Y^{H}_{t}(a_{1}, a_{2}, \cdots, a_{m});t\geq0\} = \left\{ {Y_t^H ;t \ge 0} \right\}$称为参数为$a_{1}, a_{2}, \cdots, a_{m}$和$H=(H_{1}, H_{2}, \cdots, H_{m})$的广义混合分数布朗运动(以下简称GMFBM), 若$\forall t\in\mathbb{R}_{+}$, 定义

$\begin{eqnarray} Y^{H}_{t}=a_{1}W^{H_{1}}_{t}+a_{2}W^{H_{2}}_{t}+\cdots+a_{m}W^{H_{m}}_{t}, \end{eqnarray}$

(1.1)

其中$(W^{H_{i}}_{t})_{t\in\mathbb{R}_{+}}$是带有分形指数$H_{i}, i=1, 2, \cdots, m$的相互独立的标准分数布朗运动.

注1.3  当$m=2$时, 定义1.2中的GMFBM就为混合分数布朗运动(见文献[4]).

文献[2, 3]研究了布朗运动与分数布朗运动组合的模型的一系列性质, 文献[4]研究了两个分数布朗运动组合时的各种性质.刘韶跃、Elliot等在文献[7, 8]中分别给出了分数布朗运动环境下欧式未定权益的定价公式, 并推出了一些相关欧式期权的定价公式.文献[9]在此基础上讨论了资产受多个分数布朗运动影响的两类欧式幂期权定价问题, 其中风险证券价格$S(t)$满足下式:

$\begin{eqnarray} dS(t) = S(t)[{( {\mu (t)-\delta (t)} )dt + d\sum\limits_{i = 1}^m {a_i W_t^{H_i } } }], \end{eqnarray}$

(1.2)

其中$W_t^{H_i }$为定义在概率空间($\Omega, F$, P)上的参数为$H_{i}$的分数布朗运动, $\mu (t), \delta (t), a_i$分别为标的资产的瞬时收益率、红利率、波动率.但该文没有研究GMFBM (1.1) 的性质(其他文章也没有研究它的各种性质), 本文将讨论GMFBM的一些性质, 它们是混合分数布朗运动相应性质的推广.

本小节将讨论GMFBM的基本性质.

定理2.1  GMFBM $\left( {Y_t^H (a_1, a_2, \cdots, a_m )} \right)_{t \in \mathbb{R}_ + }$满足下列性质:

(1) $Y^H$是中心高斯过程;

(2) 对任意$t \in \mathbb{R} +, E((Y_t^H )^2 ) = \sum\limits_{i = 1}^m {a_i^2 t^{2H_i } }$;

(3) 对任意$t \in \mathbb{R} +, {\hbox{Cov}}(Y_t^H, Y_s^H ) = \frac{1}{2}\sum\limits_{i = 1}^m {a_i^2 (\left| t \right|^{2H_i } + \left| s \right|^{2H_i } - \left| {t - s} \right|^{2H_i } );}$

(4) $Y_t^H$的增量是平稳的.

该结论的证明类似于文献[4]中引理2.1的证明, 在此略去.

定理2.2  GMFBM $\left( {Y_t^H (a_1, a_2, \cdots, a_m )} \right)_{t \in \mathbb{R} + }$满足混合自相似性, 即对任意$h>0 $, 有

$\begin{equation} \left\{ {Y_{th}^H \left( {a_1, a_2, \cdots a_m } \right)} \right\} \buildrel \Delta \over = \left\{ {Y_t^H \left( {a_1 h^{H_1 }, a_2 h^{H_2 }, \cdots a_m h^{H_m } } \right)} \right\}. \end{equation}$

(2.1)

证 因为$Y^H$是中心高斯过程, 故只需证$\left\{ {Y_{th}^H \left( {a_1, a_2, \cdots a_m } \right)} \right\}$与

$\left\{ {Y_t^H \left( {a_1 h^{H_1 }, a_2 h^{H_2 }, \cdots a_m h^{H_m } } \right)} \right\}$

有相同的协方差函数.由定理2.1得

$\begin{eqnarray} {\hbox{Cov}}\left( {Y_{th}^H \left( {a_1 \cdots a_m } \right), Y_{sh}^H \left( {a_1 \cdots a_m } \right)} \right) = E\left( {Y_{th}^H \left( {a_1 \cdots a_m } \right)Y_{sh}^H \left( {a_1 \cdots a_m } \right)} \right) \nonumber\\ = \frac{1}{2}\sum\limits_{i = 1}^m {a_i^2 \left( {\left| {th} \right|^{2H_i } + \left| {sh} \right|^{2H_i } -\left| {th - sh} \right|^{2H_i } } \right)} \nonumber\\ =\frac{1}{2}\sum\limits_{i = 1}^m {a_i^2 h^{2H_i } \left( {\left| t \right|^{2H_i } + \left| s \right|^{2H_i }- \left| {t - s} \right|^{2H_i } } \right)} \nonumber\\ = E\left( {Y_t^H \left( {a_1 h^{2H_1 }, \cdots a_m h^{2H_m } } \right)Y_s^H \left( {a_1 h^{2H_1 }, \cdots a_m h^{2H_m } } \right)} \right) \nonumber\\ ={\hbox{ Cov}}\left( {Y_t^H \left( {a_1 h^{2H_1 }, \cdots a_m h^{2H_m } } \right), Y_s^H \left( {a_1 h^{2H_1 }, \cdots a_m h^{2H_m } } \right)} \right). \end{eqnarray}$

(2.2)

即得证.

定理2.3   $\forall (a_1, a_2 \cdots a_m ) \in \mathbb{R}^m, (a_1, a_2 \cdots a_m ) \ne (0, 0, \cdots 0)$与$H = \left( {H_1, H_2 \cdots H_m } \right) \in (0, 1)^m$, $\left( {Y_t^H (a_1, a_2, \cdots, a_m )} \right)_{t \in \mathbb{R}_ + }$不是马氏过程除非$\left( {H_1, H_2, \cdots H_m } \right) = \left( {\frac{1}{2}, \frac{1}{2}, \cdots \frac{1}{2}} \right).$

证 如果$Y^H$是马氏过程, 则对任意的$s < t < u$, 我们有

$\begin{equation} {\hbox{Cov}}\left( {Y_s^H, Y_u^H } \right){\hbox{Cov}}\left( {Y_t^H, Y_t^H } \right) ={\hbox{Cov}}\left( {Y_s^H, Y_t^H } \right){\hbox{Cov}}\left( {Y_t^H, Y_u^H } \right). \end{equation}$

(2.3)

又因为

$\begin{eqnarray} {\hbox{ Cov}}\left( {Y_s^H, Y_u^H } \right)Cov\left( {Y_t^H, Y_t^H } \right) = \frac{1}{2} \sum\limits_{i = 1}^m {a_i^2 \left( {s^{2H_i } + u^{2H_i } - \left| {s - u} \right|^{2H_i } } \right)} \cdot \frac{1}{2}\sum\limits_{i = 1}^m {a_i^2 \left( {2t^{2H_i } } \right)} \nonumber\\ = \frac{1}{2}\sum\limits_{i = 1}^m {a_i^2 \left( {s^{2H_i } + u^{2H_i } - \left| {s - u} \right|^{2H_i } } \right)} \cdot \sum\limits_{i = 1}^m {a_i^2 t^{2H_i } }, \end{eqnarray}$

(2.4)

且

$\begin{eqnarray} {\hbox{Cov}}\left( {Y_s^H, Y_t^H } \right){\hbox{Cov}}\left( {Y_t^H, Y_u^H } \right) = \frac{1}{2}\sum\limits_{i = 1}^m {a_i^2 \left( {s^{2H_i } + t^{2H_i } - \left| {s - t} \right|^{2H_i } } \right)} \nonumber \\ \cdot \frac{1}{2}\sum\limits_{i = 1}^m {a_i^2 \left( {u^{2H_i } + t^{2H_i } - \left| {u - t} \right|^{2H_i } } \right)}. \end{eqnarray}$

(2.5)

令$s = 1/2, t = 1, u = 3/2$, 代入既得

${\hbox{Cov}}\left( {Y_{1/2}^H, Y_{3/2}^H } \right){\hbox{Cov}}\left( {Y_1^H, Y_1^H } \right) ={\hbox{Cov}}\left( {Y_{1/2}^H, Y_1^H } \right){\hbox{Cov}}\left( {Y_1^H, Y_{3/2}^H } \right), $

等价于

$ \frac{1}{2}\sum\limits_{i = 1}^m {a_i^2 \left[{\left( {\frac{1}{2}} \right)^{2H_i } + \left( {\frac{3}{2}} \right)^{2H_i }-1} \right]} \sum\limits_{i = 1}^m {a_i^2 = } \frac{1}{4}\sum\limits_{i = 1}^m {a_i^2 } \left[{\left( {\frac{3}{2}} \right)^{2H_i } + 1-\left( {\frac{1}{2}} \right)^{2H_i } } \right]\sum\limits_{i = 1}^m {a_i^2 } \\ \Leftrightarrow 2\left[{\left( {\frac{1}{2}} \right)^{2H_i } + \left( {\frac{3}{2}} \right)^{2H_i }-1} \right] = \left( {\frac{3}{2}} \right)^{2H_i } + 1 - \left( {\frac{1}{2}} \right)^{2H_i }, {\rm{ }} i = 1, 2 \cdots m \\ \Leftrightarrow {\rm{3 + 3}}^{2H_i } {\rm{ - 3}} \cdot 2^{2H_i } = 0, {\rm{ }} i = 1, 2 \cdots m{\rm{ }}. $

可以看出对所有的$(H_1, H_2, \cdots H_m ) \ne \left( {\frac{1}{2}, \frac{1}{2}, \cdots \frac{1}{2}} \right)$

${\rm{3 + 3}}^{2H_i } {\rm{ - 3}} \cdot 2^{2H_i } \ne 0{\rm{ }} i = 1, 2 \cdots m{\rm{ }}, $

从而证明了$Y^H$不是马氏过程.

对于GMFBM增量间的的相关性, 有以下结论成立:

定理3.1   $\forall {\rm{ }}s, t, h \in\mathbb{R} _ +, 0 < h \le t - s, $有

$\begin{equation} \rho \left( {Y_{t + h}^H - Y_t^H, Y_{s + h}^H - Y_s^H } \right) = \frac{{\sum\limits_{i = 1}^m {a_i^2 \left[{(t-s + h)^{2H_i } + (t-s-h)^{2H_i } - 2(t - s)^{2H_i } } \right]} }}{{2\sum\limits_{i = 1}^m {a_i^2 h^{2H_i } } }}. \end{equation}$

(3.1)

证 由相关系数的定义可得

$\begin{equation} \rho \left( {Y_{t + h}^H - Y_t^H, Y_{s + h}^H - Y_s^H } \right) = \frac{{{\hbox{Cov}}\left( {Y_{t + h}^H - Y_t^H, Y_{s + h}^H - Y_s^H } \right)}}{{\sqrt {{\hbox{Var}}\left( {Y_{t + h}^H - Y_t^H } \right)} \sqrt {Var\left( {Y_{s + h}^H - Y_s^H } \right)} }}, \end{equation}$

(3.2)

其中

$ {\hbox{Var}}\left( {Y_{t + h}^H - Y_t^H } \right) = E\left( {Y_{t + h}^H - Y_t^H } \right)^2 = E\left( {\left( {Y_{t + h}^H } \right)^2 + \left( {Y_t^H } \right)^2 - 2Y_{t + h}^H Y_t^H } \right) \\ = \sum\limits_{i = 1}^m {a_i^2 \left( {\left( {t + h} \right)^{2H_i } + t^{2H_i } } \right) - } \sum\limits_{i = 1}^m {a_i^2 \left( {\left( {t + h} \right)^{2H_i } + t^{2H_i } - h^{2H_i } } \right)} \\ = \sum\limits_{i = 1}^m {a_i^2 h^{2H_i } }. $

故可得(3.2) 式右边项分母为

$\begin{equation} \sqrt {{\hbox{Var}}\left( {Y_{t + h}^H - Y_t^H } \right)} \sqrt {{\hbox{Var}}\left( {Y_{s + h}^H - Y_s^H } \right)} = \sum\limits_{i = 1}^m {a_i^2 h^{2H_i } }, \end{equation}$

(3.3)

(3.2) 式右边项分子为

$\begin{eqnarray} {\hbox{Cov}}\left( {Y_{t + h}^H - Y_t^H, Y_{s + h}^H - Y_s^H } \right) = E\left( {Y_{t + h}^H - Y_t^H } \right)\left( {Y_{s + h}^H - Y_s^H } \right)\nonumber \\ = E\left( {Y_{t + h}^H Y_{s + h}^H - Y_{t + h}^H Y_s^H - Y_t^H Y_{s + h}^H + Y_t^H Y_s^H } \right)\nonumber \\ = \frac{1}{2}\sum\limits_{i = 1}^m {a_i^2 \left[{(t-s + h)^{2H_i } + (t-s-h)^{2H_i } - 2(t - s)^{2H_i } } \right]}. \end{eqnarray}$

(3.4)

综合(3.3), (3.4) 式可得(3.1) 式.

定理3.2   $\forall (a_1, a_2, \cdots, a_m ) \in \mathbb{R}^m, (a_1, a_2, \cdots, a_m ) \ne (0, 0, \cdots, 0)$, 如果$1/2 < H_1, H_2, \cdots, $ $H_m < 1$, 则过程$\left( {Y_t^H (a_1, a_2, \cdots, a_m )} \right)_{t \in \mathbb{R}_ + }$的增量是正相关的; 如果$ 0< H_1, H_2, \cdots, H_m <1/2$, 则是负相关的.

证 我们知道, $\forall {\rm{ }}x \in \mathbb{R}_ +, \forall {\rm{ }}h > 0, $当$H > 1/2( < 1/2)$时, \begin{equation} (x + h)^{2H} -2x^{2H} + (x -h)^{2H} > 0(＜ 0). \end{equation}由定理3.1可得, 当${\rm{ }}H_i > 1/2 ({\hbox{resp.}}, = 1/2, < 1/2), i = 1, 2, \cdots, m$时,

$ \rho (Y_{t + h}^H - Y_t^H, Y_{s + h}^H - Y_s^H ) ＞ 0( = 0, ＜ 0).$

(3.5)

注3.3  由上述定理3.1, 定理3.2可得

(1) 当$H_1, H_2, \cdots, H_m > 1/2$ (resp., $H_1, H_2, \cdots, H_m < 1/2$且$a_i \ne 0, i = 1, \cdots, r - 1, r + 1, \cdots, m, a_{r_1 }, a_{r_2 }$为常数, $|a_{r_1 } | \le |a_{r_2 } |\left( {|a_{r_1 } | \ge |a_{r_2 } |} \right)$时, $\forall s, t, h \in \mathbb{R}_ +, 0 < h \le t - s$有当$H_1, H_2 \cdots H_m > 1/2$ (resp., $H_1, H_2, \cdots, H_m < 1/2), r = 1, 2, \cdots, m$时, 对于较小的$|a_r |$, 其$Y^H$增量间的相关性较小.

(2) 当$m$个$H$中有$k$个等于$1/2 (k = 1, 2, \cdots, m - 1 ), m-k$个小于$1/2$ (大于$1/2$)时, 不失一般性, 不妨假设前$k$个为$1/2$, 后$m-k$个为小于$1/2$ (大于$1/2$), 即$H_1 = \cdots = H_k =1/2, H_{k + 1}, \cdots H_m > 1/2( < 1/2)$则对于较小的(较大的) $|a_r |$, 其$Y^H$增量间的相关性较小.

(3) 其他情况下, 增量间的相关性不显著.

注3.4  在实际的模型中, 我们可以挑选满足上面条件的$H_1, \cdots, H_m, a_1, \cdots, a_m$, 使$\left\{ {Y_t^H \left( {a_1, a_2 \cdots a_m } \right);t \ge 0} \right\}$或为一个“好”模型.

定义3.5 ^[6]  设$\{ X_t, t \in \mathbb{R}_ + \}$为有平稳轨道的过程, $(r(n))_{n \in\mathbb{N}^ + }$定义为$r(n) = {\rm E}(X_{n + 1} X_n ), $ $\forall {\rm{ }}n \in\mathbb{N} ^ +$, 则过程$X$称为长(程)相依的充分必要条件为$\sum\limits_{n \in \mathbb{N}^ + } {r(n)} = + \infty$.因为$\{ X_t, t \in \mathbb{R}_ + \}$为有平稳轨道的过程, 则

$\begin{equation} r(n) = {\rm E}(X_{n + s} X_s ), \forall {\rm{s}} \in\mathbb{R} _ +, \forall {\rm{ }}n \in \mathbb{N}^ +. \end{equation}$

(3.6)

定理3.6  (1) 对任意的$a_i \in \mathbb{R}\backslash \{ 0\} {\rm{ }}, i = 1, 2, \cdots, m$, 过程$ \left( {Y_t^H (a_1, a_2, \cdots, a_m )} \right)_{t \in \mathbb{R}_ + }$的增量是长相依的充分必要条件为:其中至少有一个$j \in \{ 1, 2, \cdots, m\}$, 使$H_j > 1/2$.

(2) 不失一般性, 假设前$k$个$a$为0, 即$a_i = 0, i = 1, \cdots, k; a_{i + 1}, \cdots, a_m \in \mathbb{R}\backslash \{ 0\}$, 则过程$\left( {Y_t^H (a_1, a_2, \cdots, a_m )} \right)_{t \in \mathbb{R}_ + }$的增量是长相依的充分必要条件为:至少有一个$j \in \{ 1, 2, \cdots, m\}$, 使$H_j ＞ 1/2$.

证 对任意的$n \in \mathbb{N}^ +$,

$\begin{eqnarray} r(n) = {\rm E}((Y_{n + 1}^H - Y_n^H )Y_1^H ) = {\rm E}(Y_{n + 1}^H Y_1^H - Y_n^H Y_1^H )\nonumber \\ =\sum\limits_{i = 1}^m {a_i^2 [(n + 1)^{2H_i }-2n^{2H_i } + (n-1)^{2H_i }]} \nonumber \\ = \sum\limits_{i = 1}^m {a_i^2 H_i (2H_i - 1)2n^{2H_i - 2} + } n^{2H_i - 2} \gamma _i (n), \end{eqnarray}$

(3.7)

其中$\mathop {\lim }\limits_{n \to \infty } \gamma _i (n) = 0{\rm{ }}, {\rm{ }}i = 1, 2, \cdots, m$.从而可得$\sum\limits_{n \in \mathbb{N}^ + } {r(n)} = + \infty$的充分必要条件为$2H_1 - 2 > - 1$或$2H_2 - 2 > - 1$或$ \cdots {\rm{ }}2H_m - 2 > - 1$, 即$H_1 > 1/2$或$ \cdots H_m > 1/2$.

定理4.1  对任意的$T > 0$与$\gamma < \min \{ H_1, H_2, \cdots, H_m \}$, GMFBM在区间$[0, T]$上有一个样本轨道为H\"{o}lder连续的修正.

证 由高斯马尔科夫关于正则性的定理, 即要证$\forall \alpha > 0, \exists C_\alpha, \forall (s, t) \in [0, T]^2, $有

$\begin{equation} E\left( {\left| {Y_t^H - Y_s^H } \right|} \right) \le C_\alpha \left| {t - s} \right|^{\alpha h} . \end{equation}$

(4.1)

不失一般性, 令$\alpha > 0, s, t \in [0, T]^2, s < t.$利用过程$Y^H$的平稳性和混合自相似性, 有

$E\left( {\left| {Y_t^H - Y_s^H } \right|^\alpha } \right) = E\left( {\left| {Y_{t - s}^H } \right|^\alpha } \right) = E\left( {\left| {Y_1^H (a_1 (t - s)^{H_1 }, \cdots, a_m (t - s)^{H_m } )} \right|^\alpha } \right) .$

(1) 当$H_1 \le \min \left\{ {H_2, \cdots, H_m } \right\}$时, 设有依赖于$\alpha$的$m$个正实数$C_1^1, C_2^1, \cdots, C_m^1$, 有

$ E\left( {\left| {Y_t^H - Y_s^H } \right|^\alpha } \right) = E\left( {\left| {a_1 (t - s)^{H_1 } W_1^{H_1 } + a_2 (t - s)^{H_2 } W_1^{H_2 } + \cdots + a_m (t - s)^{H_m } W_1^{H_m } } \right|^\alpha } \right) \\ \le (t - s{\rm{)}}^{\alpha H_1 } E\left( {\left| {a_1 W_1^{H_1 } + a_2 (t - s)^{H_2 - H_1 } W_1^{H_2 } + \cdots + a_m (t - s)^{H_m - H_1 } W_1^{H_m } } \right|^\alpha } \right) \\ \le (t - s{\rm{)}}^{\alpha H_1 } \left[{C_1^1 \left| {a_1 } \right|^\alpha E\left( {\left| {W_1^{H_1 } } \right|^\alpha } \right) + C_2^1 \left| {a_2 } \right|^\alpha (t-s)^{\alpha (H_2-H_1 )} E\left( {\left| {W_1^{H_2 } } \right|^\alpha } \right) + } \right. \\ \left. { \cdots + C_m^1 \left| {a_m } \right|^\alpha (t-s)^{\alpha (H_m - H_1 )} E\left( {\left| {W_1^{H_m } } \right|^\alpha } \right)} \right] \\ \le C_\alpha (t - s{\rm{)}}^{\alpha H_1 }, $

其中

$ C_\alpha = C_1^1 \left| {a_1 } \right|^\alpha E\left( {\left| {W_1^{H_1 } } \right|^\alpha } \right) + C_2^1 \left| {a_2 } \right|^\alpha T^{\alpha (H_2 - H_1 )} E\left( {\left| {W_1^{H_2 } } \right|^\alpha } \right) + \\ \cdots + C_m^1 \left| {a_m } \right|^\alpha T^{\alpha (H_m - H_1 )} E\left( {\left| {W_1^{H_m } } \right|^\alpha } \right) . $

(2) 当$H_i \le \min \left\{ {H_1, \cdots, H_{i - 1}, H_{i + 1}, \cdots, H_m } \right\}$时, 设有依赖于$\alpha$的$m$个正实数$C_1^i, C_2^i, \cdots, C_m^i, i = 2, 3, \cdots, m $, 有

$ E\left( {\left| {Y_t^H - Y_s^H } \right|^\alpha } \right) = E\left( {\left| {a_1 (t - s)^{H_1 } W_1^{H_1 } + \cdots + a_i (t - s)^{H_i } W_1^{H_i } + \cdots + a_m (t - s)^{H_m } W_1^{H_m } } \right|^\alpha } \right) \\ \le (t - s{\rm{)}}^{\alpha H_i } E\left( {\left| {a_1 (t - s)^{H_1 - H_i } W_1^{H_1 } + \cdots a_i W_1^{H_i } + \cdots + a_m (t - s)^{H_m - H_i } W_1^{H_m } } \right|^\alpha } \right) \\ \le (t - s{\rm{)}}^{\alpha H_i } \left[{C_1^i \left| {a_1 } \right|^\alpha (t-s)^{H_1-H_i } E\left( {\left| {W_1^{H_1 } } \right|^\alpha } \right) + \cdots + C_i^i \left| {a_i } \right|^\alpha E\left( {\left| {W_1^{H_i } } \right|^\alpha } \right) + } \right. \\ \left. {{\rm{ }} \cdots + C_m^i \left| {a_m } \right|^\alpha (t-s)^{\alpha (H_m - H_i )} E\left( {\left| {W_1^{H_m } } \right|^\alpha } \right)} \right] \\ \le C_\alpha (t - s{\rm{)}}^{\alpha H_i }, $

其中

$ C_\alpha = C_1^i \left| {a_1 } \right|^\alpha T^{H_1 - H_i } E(\left| {W_1^{H_1 } } \right|^\alpha ) + \cdots + C_i^i \left| {a_i } \right|^\alpha E(\left| {W_1^{H_i } } \right|^\alpha ) + \\ \cdots + C_m^i \left| {a_m } \right|^\alpha T^{\alpha (H_m - H_i )} E(\left| {W_1^{H_m } } \right|^\alpha ). $

定义5.1 ^[5]  令$f$为定义在$[a, b]$上的连续函数.称$f$在点$t_0 \in [a, b]$右（左）分式$\alpha$ -可微(其中$\alpha \in (0, 1)$), 定义为

$\begin{equation} d_\sigma ^\alpha f(t_0 ) = \Gamma (1 + \alpha )\mathop {\lim }\limits_{t \to t_0^\sigma } \frac{{\sigma (f(t) - f(t_0 ))}}{{\left| {t - t_0 } \right|^\alpha }}, \end{equation}$

(5.1)

$\sigma = +$ (resp., $\sigma = - )$, $\Gamma$为欧拉函数.

定义5.2 ^[5]  令$f$为定义在$[a, b]$上的连续函数, $\alpha \in (0, 1)$.$f$为在$t_0 \in [a, b]$上$\alpha$ -可微的函数当且仅当$d_ + ^\alpha f(t_0 )$与$d_ - ^\alpha f(t_0 )$存在且相等时, $d ^\alpha f(t_0 )$为$f$在$t_0$点$\alpha $ -可微.

定理5.3  对任意$\alpha \in (0, h), h = \min \left\{ {H_1, H_2, \cdots H_m } \right\}$, GMFBM的样本轨道对所有的$t_0 \ge 0$为几乎处处$\alpha$ -可微, 且

$ \forall {\rm{ }}t_0 \ge 0, {\rm P}(d^\alpha Y_{t_0 }^H = 0) = 1.$

证  以下我们只用证明$\sigma = +$的情形，$\sigma = - $的证明同理可得.利用过程$Y^H$的平稳性和混合自相似性, 对所有的$0 \le t_0 < t$, 有

$ \begin{array}{l} \frac{{Y_t^H - Y_{t_0 }^H }}{{\left( {t - t_0 } \right)^\alpha }} \buildrel \Delta \over = \frac{{Y_{t - t_0 }^H }}{{\left( {t - t_0 } \right)^\alpha }} \buildrel \Delta \over = \left( {t - t_0 } \right)^{ - \alpha } Y_1^H (a_1 (t - t_0 )^{H_1 }, \cdots, a_m (t - t_0 )^{H_m } ) \\ \buildrel \Delta \over = \left( {t - t_0 } \right)^{ - \alpha } \left[{a_1 (t-t_0 )^{H_1 } W_1^{H_1 } + \cdots + a_m (t-t_0 )^{H_m } W_1^{H_m } } \right] \\ \buildrel \Delta \over = a_1 (t - t_0 )^{H_1 - \alpha } W_1^{H_1 } + \cdots + a_m (t - t_0 )^{H_m - \alpha } W_1^{H_m } \\ \end{array}$

即得; 如果$\alpha \in (0, h), h = \min \left\{ {H_1, H_2, \cdots H_m } \right\}$则

$\begin{equation} \begin{array}{l} {\rm P}(d^\alpha Y_{t_0 }^H = 0) = {\rm P}(\mathop {\lim }\limits_{t \to t_0 } \left( {t - t_0 } \right)^{ - \alpha } \left( {Y_t^H - Y_{t_0 }^H } \right) = 0) \\ = {\rm P}(\mathop {\lim }\limits_{t \to t_0 } a_1 (t - t_0 )^{H_1 - \alpha } W_1^{H_1 } + \cdots + a_m (t - t_0 )^{H_m - \alpha } W_1^{H_m } = 0) = 1. \\ \end{array} \end{equation}$

(5.2)

定理5.4  对任意的$\alpha \in (h, 1), h = \min \left\{ {H_1, H_2, \cdots H_m } \right\}$, GMFBM的样本轨道对所有的$t_0 \ge 0$几乎处处不$\alpha$ -可微.

证 对任意$r > 0$, 定义以下事件:

$\begin{equation} A(t) = \left\{ {\mathop {\sup }\limits_{0 \le s \le t} \left| {\frac{{Y_s^H (a_1, \cdots, a_m )}}{{s^\alpha }}} \right| > r} \right\}, \end{equation}$

(5.3)

对任意的递减序列$t_n \to 0$, 我们$A(t_{n + 1} ) \subset A(t_n )$, 即

$ {\rm P}\left( {\mathop {\lim }\limits_{n \to \infty } A(t_n )} \right) = \mathop {\lim }\limits_{n \to \infty } {\rm P}\left( {A(t_n )} \right), $

利用$Y^H$的混合自相似性(2.1), 有

${\rm P}\left( {A\left( {t_n } \right)} \right) \ge {\rm P}\left( {\left| {\frac{{Y_{t_n }^H (a_1, \cdots, a_m )}}{{t_n^\alpha }}} \right| > r} \right) = {\rm P}\left( {\left| {a_1 t^{H_1 - \alpha } W_1^{H_1 } + \cdots + a_m t^{H_m - \alpha } W_1^{H_m } } \right| > r} \right).$

(1) 如果$ H_i ＜ \min \left\{ {H_1, \cdots, H_{i - 1}, H_{i + 1}, \cdots, H_m } \right\}\left( {\alpha > H_i } \right)$, 则

${\rm P}\left( {A(t_n )} \right) \ge {\rm P}\left( {\left( {a_1 t_n ^{H_1 - H_i } W_1^{H_1 } + \cdots a_i W_1^{H_i } + \cdots + a_m t_n ^{H_m - H_i } W_1^{H_m } } \right) > rt_n^{\alpha - H_i } } \right), $

于是有

$ \begin{equation} \begin{array}{l} \mathop {\lim }\limits_{n \to \infty } {\rm P}\left( {\left( {a_1 t_n ^{H_1 - H_i } W_1^{H_1 } + \cdots a_i W_1^{H_i } + \cdots + a_m t_n ^{H_m - H_i } W_1^{H_m } } \right) > rt_n^{\alpha - H_i } } \right) \\ = {\rm P}\left( {\left| {a_i W_1^{H_i } } \right| \ge 0} \right) = 1. \end{array} \end{equation}$

(5.4)

(2) 不失一般性, 假设前$k$个$H$为0, 即$H_i = 0, i = 1, \cdots, k;H_{k + 1}, \cdots, H_m \in \mathbb{R}\backslash \{ 0\}$, 则

$ \begin{array}{l} {\rm P}\left( {A(t_n )} \right) \ge {\rm P}\left( {\left( {a_1 W_1^{H_1 } + \cdots + a_k W_1^{H_k } + a_{k + 1} t_n ^{H_{k + 1} - H_1 } W_1^{H_{k + 1} } + \cdots } \right.} \right.\\ \left. {+ a_m t_n ^{H_m - H_1 } W_1^{H_m } } \right)\left. { > rt_n^{\alpha - H_1 } } \right). \end{array} $

故

$\begin{equation} \begin{array}{l} \mathop {\lim }\limits_{n \to \infty } {\rm P}\left( {\left( {a_1 W_1^{H_1 } + \cdots + a_k W_1^{H_k } + a_{k + 1} t_n ^{H_{k + 1} - H_1 } W_1^{H_{k + 1} } + } \right.} \right. \left. { \cdots + a_m t_n ^{H_m - H_1 } W_1^{H_m } } \right)\\ \left. { ＞ rt_n^{\alpha - H_1 } } \right) \\ {\rm{ }} = {\rm P}\left( {\left| {a_1 W_1^{H_1 } + \cdots + a_k W_1^{H_k } } \right| \ge 0} \right) = 1. \end{array} \end{equation}$

(5.5)

由以上的结论我们得出对任意$\alpha \in (h, 1), t_0 \ge 0$, 有

$\begin{equation} {\rm P}\left( {\mathop {\lim \sup }\limits_{t \to t_o^ + } \left| {\frac{{Y_t^H (a_1, \cdots, a_m ) - Y_{t_0 }^H (a_1, \cdots, a_m )}}{{\left( {t - t_0 } \right)^\alpha }}} \right| = + \infty } \right) = 1. \end{equation}$

(5.6)

即得证.

以上结论中, 令$m=2$, 容易得到混合分数布朗运动的相应结论, 具体讨论参见文献[4].