解 SDE：et(1+Wt2)e^{t}(1+W_t^2)et(1+Wt2) Solve the SDE 3 专题 Finance / 金融 难度 L4 来源 QuantQuestion 题目详情 Solve the stochastic differential equation dXt=et(1+Wt2)dt+(1+2etWt)dWtdX_{t} = e^{t}\left(1 + W_{t}^{2}\right)dt + \left(1 + 2e^{t}W_{t}\right)dW_{t}dXt=et(1+Wt2)dt+(1+2etWt)dWt with X0=0X_0 = 0X0=0 解析 令 Yt=etWt2Y_t=e^{t}W_t^2Yt=etWt2，则 dYt=et(1+Wt2)dt+2etWtdWt.dY_t=e^{t}(1+W_t^2)dt+2e^{t}W_tdW_t.dYt=et(1+Wt2)dt+2etWtdWt. 题目 SDE 为 dXt=et(1+Wt2)dt+(1+2etWt)dWt=dYt+dWt.dX_t=e^{t}(1+W_t^2)dt+(1+2e^{t}W_t)dW_t=dY_t+dW_t.dXt=et(1+Wt2)dt+(1+2etWt)dWt=dYt+dWt. 且 X0=0,Y0=0X_0=0,Y_0=0X0=0,Y0=0，因此 Xt=Yt+Wt=etWt2+Wt.\boxed{X_t=Y_t+W_t=e^{t}W_t^2+W_t}.Xt=Yt+Wt=etWt2+Wt.