<a href="https://colab.research.google.com/github/Jun-629/20MA573/blob/master/src/Hw6_Geometric%20Brownian%20Motion.ipynb" target="_parent"></a>

- __Example 1：__ 

Find $logS_t$ for $S \sim GBM(s,\mu,\sigma^2)$, where the dynamics $dS_t = \mu S_t dt + \sigma S_t dW_t$ with $S_0 = s.$

__Soln:__

By using Ito-Doeblin Formula, we have

\begin{equation}
\begin{aligned}
dlnS_t &= \frac{1}{S_t} dS_t - \frac {1}{2} \frac{1}{S_t^2} \cdot \sigma^2 S_t^2dW_tdW_t \\
&= \mu dt + \sigma dW_t - \frac{1}{2} \sigma^2 dt \\
&= (\mu -  \frac{1}{2} \sigma^2)dt + \sigma dW_t \\
\end{aligned}
\end{equation}

Then, transform this integral form into derivative form, we will get that

\begin{equation}
\begin{aligned}
lnS_t - lnS_0 &= \int_0^t (\mu - \frac{1}{2} \sigma^2) \, ds + \int_0^t \sigma dW_s \\
lnS_t &= lns +  (\mu - \frac{1}{2} \sigma^2) \cdot t + \sigma W_t. \\
\end{aligned}
\end{equation}

9e2678bd4a0059c4d14fdb71656887430a29e7f9
ipynb
Jupyter Notebook
src/Hw6_Geometric_Brownian_Motion.ipynb
Jun-629/20MA573
addad663d2dede0422ae690e49b230815aea4c70
MIT
src/Hw6_Geometric_Brownian_Motion.ipynb
Jun-629/20MA573
addad663d2dede0422ae690e49b230815aea4c70
MIT
src/Hw6_Geometric_Brownian_Motion.ipynb
Jun-629/20MA573
addad663d2dede0422ae690e49b230815aea4c70
MIT
2020-02-05T21:42:08.000Z
2020-02-05T21:42:08.000Z
Qwen/Qwen-72B
1. YES
2. YES