El Vasicek modelo de tasa corta es $$dr_t=\kappa(\theta-r_t)dt+\sigma dW_t$$ Definir los procesos $x_t$ y $f(x,t)$ $$x_t=\frac{r_t}{\kappa}(1-e^{-\kappa(T-t)})+\int_0^tr_sds$$ $$f(x,t)=e^{a(T-t)-x_t}$$ Tenga en cuenta que $a(t)$ es una función.
Pregunta 1: Encontrar $df(x,t)$ usando el Lema de Ito.
Mi intento: \begin{align} dx_t&=\frac{1-e^{-\kappa(T-t)}}{\kappa}dr_t+\frac{r_t}{\kappa}(-e^{-\kappa(T-t)})\kappa dt\\ &=\frac{1-e^{-\kappa(T-t)}}{\kappa}(\kappa(\theta-r_t)dt+\sigma dW_t)-r_te^{-\kappa(T-t)}dt\\ &=(1-e^{-\kappa(T-t)})(\theta-r_t)dt+\frac{\sigma}{\kappa}(1-e^{-\kappa(T-t)})dW_t-r_te^{-\kappa(T-t)}dt\\ &=\Big ( (1-e^{-\kappa(T-t)})\theta-r_t+r_te^{-\kappa(T-t)}-r_te^{-\kappa(T-t)}\Big)dt+\frac{\sigma}{\kappa}(1-e^{-\kappa(T-t)})dW_t\\ &=\Big( (1-e^{-\kappa(T-t)})\theta-r_t\Big)dt+\frac{\sigma}{\kappa}(1-e^{-\kappa(T-t)})dW_t \end{align} $$(dx_t)^2=\frac{\sigma^2}{\kappa^2}(1-e^{-\kappa(T-t)})^2dt$$
$$\ln{f(x,t)}=a(T-t)-x_t$$ $$d\ln{f(x,t)}=d_ta(T-t)-dx_t=d_ta(T-t)-\Big( (1-e^{-\kappa(T-t)})\theta-r_t\Big)dt-\frac{\sigma}{\kappa}(1-e^{-\kappa(T-t)})dW_t$$ $$(d\ln{f(x,t)})^2=(dx_t)^2=\frac{\sigma^2}{\kappa^2}(1-e^{-\kappa(T-t)})^2dt$$ La aplicación de Ito Lema a $f(x,t)$ \begin{align} d(e^{\ln{f(x,t)}})&=f(x,t)d\ln{f(x,t)}+\frac{1}{2}f(x,t)(d\ln{f(x,t)})^2\\ &=f(x,t)\Bigg(d_ta(T-t)-\Big ( (1-e^{-\kappa(T-t)})\theta-r_t\Big)dt-\frac{\sigma}{\kappa}(1-e^{-\kappa(T-t)})dW_t\Bigg)+\frac{\sigma^2}{2\kappa^2}(1-e^{-\kappa(T-t)})^2dt\\ &=f(x,t)\Bigg(\frac{d}{dt}a(T-t)-\Big ( (1-e^{-\kappa(T-t)})\theta-r_t\Big)+\frac{\sigma^2}{2\kappa^2}(1-e^{-\kappa(T-t)})^2\Bigg)dt-\frac{\sigma}{\kappa}(1-e^{-\kappa(T-t)})dW_t\\ \end{align} Pregunta 2: Encontrar la dinámica de la función $(t)$ tal que el proceso de $f(x,t)$ es una martingala.
Mi intento: $$\frac{d}{dt}a(T-t)-\Big( (1-e^{-\kappa(T-t)})\theta-r_t\Big)+\frac{\sigma^2}{2\kappa^2}(1-e^{-\kappa(T-t)})^2=0$$ $$\frac{d}{dt}a(T-t)=\Big( (1-e^{-\kappa(T-t)})\theta-r_t\Big)-\frac{\sigma^2}{2\kappa^2}(1-e^{-\kappa(T-t)})^2$$ $$d_ta(T-t)=\Bigg(\Big( (1-e^{-\kappa(T-t)})\theta-r_t\Big)-\frac{\sigma^2}{2\kappa^2}(1-e^{-\kappa(T-t)})^2\Bigg)dt$$
¿Alguien puede confirmar la veracidad de mis intentos? Cualquier ayuda es muy apreciada.
