这是一份northeastern东北大学(美国) MATH 4683作业代写的成功案例
Suppose we write
$$
X(t)=\left(r-\frac{\sigma^{2}}{2}\right) t+\sigma Z(t)
$$
so that
$$
X(t)=\ln \frac{S(t)}{S_{0}} \quad \text { or } \quad S(t)=S_{0} e^{X(t)}
$$
Now, the respective partial derivatives of $S$ are
$$
\frac{\partial S}{\partial t}=0, \quad \frac{\partial S}{\partial X}=S \quad \text { and } \quad \frac{\partial^{2} S}{\partial X^{2}}=S .
$$
By the Ito lemma, we obtain
$$
d S(t)=\left(r-\frac{\sigma^{2}}{2}+\frac{\sigma^{2}}{2}\right) S(t) d t+\sigma S(t) d Z(t)
$$
or
$$
\frac{d S(t)}{S(t)}=r d t+\sigma d Z(t), \text { with } S(0)=S_{0}
$$
MATH 4683COURSE NOTES ：
Suppose ${X(t), t \geq 0}$ is the standard Brownian process, its corresponding reflected Brownian process is defined by
$$
Y(t)=|X(t)|, \quad t \geq 0 .
$$
Show that $Y(t)$ is also Markovian and its mean and variance are, respectively,
$$
E[Y]=\sqrt{\frac{2 t}{\pi}}
$$
and
$$
\operatorname{var}(Y)=\left(1-\frac{2}{\pi}\right) t
$$