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鞅表示：E[WT3∣Ft]\mathbb{E}[W_T^3\mid\mathcal{F}_t]

Find a stochastic process

专题
Finance / 金融
难度
L4

题目详情

Let Xt=E[WT3∣Ft],0≤t≤TX_{t} = \mathbb{E}\left[W_{T}^{3}\mid \mathcal{F}_{t}\right],0\leq t\leq T , where WtW_{t} is a Brownian motion. Find a stochastic process YtY_{t} such that Xt=∫0tYudWu,0≤t≤T.\begin{array}{r}X_{t} = \int_{0}^{t}Y_{u}dW_{u},0\leq t\leq T. \end{array}

解析

写 WT=Wt+(WT−Wt)W_T=W_t+(W_T-W_t)，其中增量与 Ft\mathcal{F}_t 独立且三阶矩为 0、二阶矩为 T−tT-t。

展开并取条件期望：

Xt=E[WT3∣Ft]=Wt3+3Wt(T−t).\boxed{X_t=\mathbb{E}[W_T^3\mid\mathcal{F}_t]=W_t^3+3W_t(T-t)}.

对 f(t,x)=x3+3x(T−t)f(t,x)=x^3+3x(T-t) 用 Itô 得

dXt=(3Wt2+3(T−t))dWt.dX_t=\bigl(3W_t^2+3(T-t)\bigr)dW_t.

因此可取

Yt=3Wt2+3(T−t)\boxed{Y_t=3W_t^2+3(T-t)}

使得

Xt=∫0tYu dWu.\boxed{X_t=\int_0^t Y_u\,dW_u}.