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计算 E[Wte−λ∫0tsdWs]\mathbb{E}[W_t e^{-\lambda\int_0^t s dW_s}]

Calculate in two different ways

专题
Finance / 金融
难度
L4

题目详情

Calculate

E[W(t)e−λ∫0tsdW(s)]\mathbb{E}\left[W(t)e^{-\lambda \int_{0}^{t}sdW(s)}\right]

in two different ways: using Girsanov's theorem and without Girsanov's theorem.

解析

设 It=∫0ts dWsI_t=\int_0^t s\,dW_s。

用 Girsanov（取 θ(s)=−λs\theta(s)=-\lambda s）可得在新测度下

W^t=Wt+λt22\widehat W_t=W_t+\frac{\lambda t^2}{2}

为布朗运动，且

dP^dP=exp⁡(−λIt−λ22∫0ts2ds)=exp⁡(−λIt−λ2t36).\frac{d\widehat{\mathbb{P}}}{d\mathbb{P}}=\exp\left(-\lambda I_t-\frac{\lambda^2}{2}\int_0^t s^2ds\right)=\exp\left(-\lambda I_t-\frac{\lambda^2 t^3}{6}\right).

于是

E[Wte−λIt]=EP^[W^t−λt22]eλ2t3/6=−λt22eλ2t3/6.\mathbb{E}\left[W_t e^{-\lambda I_t}\right]=\mathbb{E}_{\widehat{\mathbb{P}}}\left[\widehat W_t-\frac{\lambda t^2}{2}\right]e^{\lambda^2 t^3/6} =-\frac{\lambda t^2}{2}e^{\lambda^2 t^3/6}.

即

E[Wte−λ∫0tsdWs]=−λt22exp⁡(λ2t36).\boxed{\mathbb{E}\left[W_t e^{-\lambda\int_0^t s dW_s}\right]=- \frac{\lambda t^2}{2}\exp\left(\frac{\lambda^2 t^3}{6}\right)}.