$S_t^\alpha$ 何时是鞅

For which is a martingale

Suppose that $dS_{t} = \mu S_{t}dt + \sigma S_{t}dW_{t}$ . For which $\alpha$ is the process $S_{t}^{\alpha}$ a martingale?

解析

对 $dS_t=\mu S_tdt+\sigma S_tdW_t$ ，令 $Y_t=S_t^{\alpha}$ 。

Itô 得漂移项为

\alpha S_t^{\alpha}\left(\mu+\frac{\alpha-1}{2}\sigma^2\right)dt.

$Y_t$ 为鞅当且仅当漂移为 0：

\mu+\frac{\alpha-1}{2}\sigma^2=0 \quad\Longrightarrow\quad \boxed{\alpha=1-\frac{2\mu}{\sigma^2}}.

（ $\alpha=0$ 也给出平凡鞅常数 1。）