解 SDE：(2−Xt)dt+e−tWtdWt(2-X_t)dt+e^{-t}W_tdW_t(2−Xt)dt+e−tWtdWt Solve the SDE 5 专题 Finance / 金融 难度 L4 来源 QuantQuestion 题目详情 Solve the stochastic differential equation dXt=(2−Xt)dt+e−tWtdWt.dX_{t} = (2 - X_{t})dt + e^{-t}W_{t}dW_{t}.dXt=(2−Xt)dt+e−tWtdWt. 解析 令 Yt=etXtY_t=e^{t}X_tYt=etXt，则 dYt=etXtdt+etdXt=2etdt+WtdWt.dY_t=e^{t}X_tdt+e^{t}dX_t=2e^{t}dt+W_t dW_t.dYt=etXtdt+etdXt=2etdt+WtdWt. 积分并用 ∫0tWsdWs=(Wt2−t)/2\int_0^t W_s dW_s=(W_t^2-t)/2∫0tWsdWs=(Wt2−t)/2，得 Yt=Y0+2(et−1)+Wt2−t2.Y_t=Y_0+2(e^{t}-1)+\frac{W_t^2-t}{2}.Yt=Y0+2(et−1)+2Wt2−t. 因此 Xt=X0e−t+2(1−e−t)+e−t2(Wt2−t).\boxed{X_t=X_0e^{-t}+2(1-e^{-t})+\frac{e^{-t}}{2}(W_t^2-t)}.Xt=X0e−t+2(1−e−t)+2e−t(Wt2−t).