A 随机过程S_t在满足一下随机微分方程 (SDE)的情况下被认为遵循几何布朗运动：

http://upload.wikimedia.org/wikipedia/zh/math/0/0/9/009ae59e63d45218592e3528b032a83a.png

这里http://upload.wikimedia.org/wikipedia/zh/math/b/7/2/b72bb92668acc30b4474caff40274044.png ('百分比drift') 和http://upload.wikimedia.org/wikipedia/zh/math/9/d/4/9d43cb8bbcb702e9d5943de477f099e2.png ('百分比volatility')则是常量。

几何布朗运动的特性

给定初始值 S₀，根据伊藤积分，上面的 SDE有如下解:

http://upload.wikimedia.org/wikipedia/zh/math/2/c/7/2c7d3bf240b750bad2f1492a5efc96ed.png

对于任意值 t，这是一个 对数正态分布 随机变量，其 期望值 和方差分别是^[2]

http://upload.wikimedia.org/wikipedia/zh/math/9/d/e/9de4e6a91d990ab09287ece5f4b22246.png

http://upload.wikimedia.org/wikipedia/zh/math/3/b/3/3b33649de1d464a46e50107052491bef.png

也就是说S_t的概率密度函数是:

http://upload.wikimedia.org/wikipedia/zh/math/b/5/0/b501094c0277c92dd300a290b02cc926.png

根据伊藤引理，这个解是正确的。

When deriving further properties of GBM, use can be made of the SDE of which GBM is the solution, or the explicit solution given above can be used. 比如，考虑随机过程 log(S_t). 这是一个有趣的过程，因为在布莱克-舒尔斯模型中这和股票价格的对数回报率相关。对f(S) = log(S)应用伊藤引理，得到

http://upload.wikimedia.org/wikipedia/zh/math/5/6/a/56a348c1e06f5fc2d9bb1958d94bb78e.png

于是http://upload.wikimedia.org/wikipedia/zh/math/2/0/f/20f2e9dae0d55e5138df249168217ee8.png.

这个结果还有另一种方法获得：applying the logarithm to the explicit solution of GBM:

http://upload.wikimedia.org/wikipedia/zh/math/2/4/0/2403289d9756410dcafd90f14caa34a3.png

取期望值，获得和上面同样的结果: http://upload.wikimedia.org/wikipedia/zh/math/2/0/f/20f2e9dae0d55e5138df249168217ee8.png.

在金融中的应用

主条目：布莱克-舒尔斯模型

几何布朗运动在布莱克-舒尔斯定价模型被用来定性股票价格，因而也是最常用的描述股票价格的模型。

使用几何布朗运动来描述股票价格的理由:

• The expected returns of几何布朗运动are independent of the value of the process (stock price), which agrees with what we would expect in reality.^[3]
• 几何布朗运动process only assumes positive values, just like real stock prices.
• 几何布朗运动 process shows the same kind of 'roughness' in its paths as we see in real stock prices.
• 几何布朗运动过程计算相对简单。.

然而，几何布朗运动并不完全现实，尤其存在一下缺陷:

• In real stock prices, volatility changes over time (possibly stochastically), but in GBM, volatility is assumed constant.
• In real stock prices, returns are usually not normally distributed (real stock returns have higher 峰度 ('fatter tails'), which means there is a higher chance of large price changes).^[4]
  
  几何布朗运动推广

In an attempt to make GBM more realistic as a model for stock prices, one can drop the assumption that the volatility (http://upload.wikimedia.org/wikipedia/zh/math/9/d/4/9d43cb8bbcb702e9d5943de477f099e2.png) is constant. If we assume that the volatility is adeterministic function of the stock price and time, this is called a local volatility model. If instead we assume that the volatility has a randomness of its own—often described by a different equation driven by a different Brownian Motion—the model is called a 随机volatility model.

http://s14/middle/79883453tc2e34047d2cd&690