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dX=X3/2dWdX=X^{3/2}dW 的均值与方差

Consider the stochastic process

专题
Probability / 概率
难度
L4

题目详情

Consider the stochastic process defined by X0=x0>0X_{0} = x_{0} > 0 and dXt=Xt32dWtdX_{t} = X_{t}^{\frac{3}{2}}dW_{t} . What are the mean and variance of XtX_{t} for fixed tt ?

解析

由

Xt=x0+∫0tXs3/2dWsX_t=x_0+\int_0^t X_s^{3/2}dW_s

可知随机积分期望为 0，因此

E[Xt]=x0.\boxed{\mathbb{E}[X_t]=x_0}.

但该过程的高阶矩会爆炸：对 m≥2m\ge 2 用 Itô 可得

E[Xtm]=x0m+(m2)∫0tE[Xsm+1]ds,\mathbb{E}[X_t^m]=x_0^m+\binom{m}{2}\int_0^t \mathbb{E}[X_s^{m+1}]ds,

从而对任意 t>0t>0，E[Xtm]=+∞\mathbb{E}[X_t^m]=+\infty。

因此

Var⁡(Xt)=+∞.\boxed{\operatorname{Var}(X_t)=+\infty}.