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Gordon's Answer is nice (+1). I want to add the other solution.

Remark 1

Let $X=(X_1,X_2,...,X_n)$ where the component $X_i$ has a stochastic differential of the form $$dX_i(t)=\mu_i(t)dt+\sum_{j=1}^{d}\sigma_{ij}(t)dW_j(t)$$ where $dW_k(t)dW_j(t)=\rho_{kj}dt$, for all $k,j\in\{1,2,...,d\}$. Let $f:\mathbb{R}^+\times\mathbb{R}^n\to\mathbb{R}\in\mathbb{C}^{1,2}.$ By application of Ito's lemma, we have $$df(t,X_1,...,X_n)=\frac{\partial f}{\partial t}dt+\sum_{i=1}^{n}\frac{\partial f}{\partial x_i}dX_i+\frac 12\sum_{i=1}^{n}\sum_{l=1}^{n}\frac{\partial ^2f}{\partial x_i\partial x_l}dX_idX_l\tag 1$$


Remark 2

Assume that the short rate $r_t$ follows the Ito process as described by the following stochastic differential equation $$dr_t=\mu(t,r_t)dt+\sigma(t,r_t)dW_t$$ and $P(t,T)$ denotes the zero-coupon bond price with maturity $T$. We can show $$\frac{\partial P}{\partial t}+\mu(t,r_t)\frac{\partial P}{\partial r}+\frac{1}{2}\sigma^2(t,r_t)\frac{\partial^2 P}{\partial r^2}-r_tP=0\tag 2$$


Remark 3

Let $$\qquad dS_t=rS_t+\sigma S_tdW_1(t)\\ \quad\qquad\quad dr_t=\kappa(\theta-r_t)dt+\Sigma dW_2(t)\\ dW_1(t)dW_2(t)=0\tag 3$$

Now we form a portfolio consisting of one option $V=V(t,S,r,T,K)$ (Short position), $\Delta_1$ units of the stock (long position) and $\Delta_2$ units of the $T-$zero-coupon bond price (long position). The portfolio has value $$\Pi=\Delta_1S_t+\Delta_2 P(t,T)-V(t,S,r,T,K)$$ therefore $$d\Pi=\Delta_1dS_t+\Delta_2dP(t,T)-dV(t,S,r,T,K)\tag 4$$ By application of Ito's lemma, we have $$dV_t=\frac{\partial V}{\partial t}dt+\frac{\partial V}{\partial s}dS+\frac{\partial V}{\partial r}dr+\frac 12\left(\sigma^2\frac{\partial^2 V}{\partial s^2}+\Sigma^2\frac{\partial^2 V}{\partial r^2}\right)dt\tag 5$$ and $$dP=\frac{\partial P}{\partial t}dt+\frac{\partial P}{\partial r}dr+\frac{1}{2}\Sigma^2\frac{\partial^2 P}{\partial r^2}dt\tag 6$$ $(4)\,,\,(5)\,,(6)$ and $(3)$ $$d\Pi =-\left( \frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}} \right)dt+\left( \Delta _1-\frac{\partial V}{\partial S} \right)dS\\+\left( \Delta _2\frac{\partial P}{\partial r}\,-\frac{\partial V}{\partial r} \right)dr+\Delta _2\,\left( \frac{\partial P}{\partial t}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}P}{\partial {{r}^{2}}} \right)dt\,$$ We then find the values of $\Delta_1$ and $\Delta_2$ that makes the portfolio riskless. Indeed, we set $$ \Delta_1=\frac{\partial V}{\partial S}\\ \Delta_2=\frac{\frac{\partial V}{\partial r}}{\frac{\partial P}{\partial r}}$$ thus $$d\Pi =-\left( \frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}_{t}}^{2}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}V}{\partial {{r}^{2}}} \right)dt+\Delta {{}_{2}}\,\left( \frac{\partial P}{\partial t}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}P}{\partial {{r}^{2}}} \right)dt\tag 7$$ $(2)$ and $(7)$ $$d\Pi =-\left( \frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}V}{\partial {{r}^{2}}} \right)dt+\Delta {{}_{2}}\,\left( {{r}_{t}}\,P-\kappa \left( \theta -r \right)\,\frac{\partial P}{\partial r} \right)dt\tag 8$$ The condition that the portfolio earn the risk-free rate, $r$, implies that the change in portfolio value is $$d\Pi=r\Pi dt$$ in other words $$d\Pi =(\Delta _1r_t\,S_t+\Delta_2r_tP-r_tV)dt\tag 9$$ $(8)$ and $(9)$ $$\frac{\partial V}{\partial t}+{{r}_{t}}\,{{S}_{t}}\frac{\partial V}{\partial S}+\kappa \left( \theta -r \right)\frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}V}{\partial {{r}^{2}}}-{{r}_{t}}V=0$$

Gordon's Answer is nice (+1). I want to add other solution.

Remark 1

Let $X=(X_1,X_2,...,X_n)$ where the component $X_i$ has a stochastic differential of the form $$dX_i(t)=\mu_i(t)dt+\sum_{j=1}^{d}\sigma_{ij}(t)dW_j(t)$$ where $dW_k(t)dW_j(t)=\rho_{kj}dt$, for all $k,j\in\{1,2,...,d\}$. Let $f:\mathbb{R}^+\times\mathbb{R}^n\to\mathbb{R}\in\mathbb{C}^{1,2}.$ By application of Ito's lemma, we have $$df(t,X_1,...,X_n)=\frac{\partial f}{\partial t}dt+\sum_{i=1}^{n}\frac{\partial f}{\partial x_i}dX_i+\frac 12\sum_{i=1}^{n}\sum_{l=1}^{n}\frac{\partial ^2f}{\partial x_i\partial x_l}dX_idX_l\tag 1$$


Remark 2

Assume that the short rate $r_t$ follows the Ito process as described by the following stochastic differential equation $$dr_t=\mu(t,r_t)dt+\sigma(t,r_t)dW_t$$ and $P(t,T)$ denotes the zero-coupon bond price with maturity $T$. We can show $$\frac{\partial P}{\partial t}+\mu(t,r_t)\frac{\partial P}{\partial r}+\frac{1}{2}\sigma^2(t,r_t)\frac{\partial^2 P}{\partial r^2}-r_tP=0\tag 2$$


Remark 3

Let $$\qquad dS_t=rS_t+\sigma S_tdW_1(t)\\ \quad\qquad\quad dr_t=\kappa(\theta-r_t)dt+\Sigma dW_2(t)\\ dW_1(t)dW_2(t)=0\tag 3$$

Now we form a portfolio consisting of one option $V=V(t,S,r,T,K)$ (Short position), $\Delta_1$ units of the stock (long position) and $\Delta_2$ units of the $T-$zero-coupon bond price (long position). The portfolio has value $$\Pi=\Delta_1S_t+\Delta_2 P(t,T)-V(t,S,r,T,K)$$ therefore $$d\Pi=\Delta_1dS_t+\Delta_2dP(t,T)-dV(t,S,r,T,K)\tag 4$$ By application of Ito's lemma, we have $$dV_t=\frac{\partial V}{\partial t}dt+\frac{\partial V}{\partial s}dS+\frac{\partial V}{\partial r}dr+\frac 12\left(\sigma^2\frac{\partial^2 V}{\partial s^2}+\Sigma^2\frac{\partial^2 V}{\partial r^2}\right)dt\tag 5$$ and $$dP=\frac{\partial P}{\partial t}dt+\frac{\partial P}{\partial r}dr+\frac{1}{2}\Sigma^2\frac{\partial^2 P}{\partial r^2}dt\tag 6$$ $(4)\,,\,(5)\,,(6)$ and $(3)$ $$d\Pi =-\left( \frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}} \right)dt+\left( \Delta _1-\frac{\partial V}{\partial S} \right)dS\\+\left( \Delta _2\frac{\partial P}{\partial r}\,-\frac{\partial V}{\partial r} \right)dr+\Delta _2\,\left( \frac{\partial P}{\partial t}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}P}{\partial {{r}^{2}}} \right)dt\,$$ We then find the values of $\Delta_1$ and $\Delta_2$ that makes the portfolio riskless. Indeed, we set $$ \Delta_1=\frac{\partial V}{\partial S}\\ \Delta_2=\frac{\frac{\partial V}{\partial r}}{\frac{\partial P}{\partial r}}$$ thus $$d\Pi =-\left( \frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}_{t}}^{2}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}V}{\partial {{r}^{2}}} \right)dt+\Delta {{}_{2}}\,\left( \frac{\partial P}{\partial t}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}P}{\partial {{r}^{2}}} \right)dt\tag 7$$ $(2)$ and $(7)$ $$d\Pi =-\left( \frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}V}{\partial {{r}^{2}}} \right)dt+\Delta {{}_{2}}\,\left( {{r}_{t}}\,P-\kappa \left( \theta -r \right)\,\frac{\partial P}{\partial r} \right)dt\tag 8$$ The condition that the portfolio earn the risk-free rate, $r$, implies that the change in portfolio value is $$d\Pi=r\Pi dt$$ in other words $$d\Pi =(\Delta _1r_t\,S_t+\Delta_2r_tP-r_tV)dt\tag 9$$ $(8)$ and $(9)$ $$\frac{\partial V}{\partial t}+{{r}_{t}}\,{{S}_{t}}\frac{\partial V}{\partial S}+\kappa \left( \theta -r \right)\frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}V}{\partial {{r}^{2}}}-{{r}_{t}}V=0$$

Gordon's Answer is nice (+1). I want to add the other solution.

Remark 1

Let $X=(X_1,X_2,...,X_n)$ where the component $X_i$ has a stochastic differential of the form $$dX_i(t)=\mu_i(t)dt+\sum_{j=1}^{d}\sigma_{ij}(t)dW_j(t)$$ where $dW_k(t)dW_j(t)=\rho_{kj}dt$, for all $k,j\in\{1,2,...,d\}$. Let $f:\mathbb{R}^+\times\mathbb{R}^n\to\mathbb{R}\in\mathbb{C}^{1,2}.$ By application of Ito's lemma, we have $$df(t,X_1,...,X_n)=\frac{\partial f}{\partial t}dt+\sum_{i=1}^{n}\frac{\partial f}{\partial x_i}dX_i+\frac 12\sum_{i=1}^{n}\sum_{l=1}^{n}\frac{\partial ^2f}{\partial x_i\partial x_l}dX_idX_l\tag 1$$


Remark 2

Assume that the short rate $r_t$ follows the Ito process as described by the following stochastic differential equation $$dr_t=\mu(t,r_t)dt+\sigma(t,r_t)dW_t$$ and $P(t,T)$ denotes the zero-coupon bond price with maturity $T$. We can show $$\frac{\partial P}{\partial t}+\mu(t,r_t)\frac{\partial P}{\partial r}+\frac{1}{2}\sigma^2(t,r_t)\frac{\partial^2 P}{\partial r^2}-r_tP=0\tag 2$$


Remark 3

Let $$\qquad dS_t=rS_t+\sigma S_tdW_1(t)\\ \quad\qquad\quad dr_t=\kappa(\theta-r_t)dt+\Sigma dW_2(t)\\ dW_1(t)dW_2(t)=0\tag 3$$

Now we form a portfolio consisting of one option $V=V(t,S,r,T,K)$ (Short position), $\Delta_1$ units of the stock (long position) and $\Delta_2$ units of the $T-$zero-coupon bond price (long position). The portfolio has value $$\Pi=\Delta_1S_t+\Delta_2 P(t,T)-V(t,S,r,T,K)$$ therefore $$d\Pi=\Delta_1dS_t+\Delta_2dP(t,T)-dV(t,S,r,T,K)\tag 4$$ By application of Ito's lemma, we have $$dV_t=\frac{\partial V}{\partial t}dt+\frac{\partial V}{\partial s}dS+\frac{\partial V}{\partial r}dr+\frac 12\left(\sigma^2\frac{\partial^2 V}{\partial s^2}+\Sigma^2\frac{\partial^2 V}{\partial r^2}\right)dt\tag 5$$ and $$dP=\frac{\partial P}{\partial t}dt+\frac{\partial P}{\partial r}dr+\frac{1}{2}\Sigma^2\frac{\partial^2 P}{\partial r^2}dt\tag 6$$ $(4)\,,\,(5)\,,(6)$ and $(3)$ $$d\Pi =-\left( \frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}} \right)dt+\left( \Delta _1-\frac{\partial V}{\partial S} \right)dS\\+\left( \Delta _2\frac{\partial P}{\partial r}\,-\frac{\partial V}{\partial r} \right)dr+\Delta _2\,\left( \frac{\partial P}{\partial t}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}P}{\partial {{r}^{2}}} \right)dt\,$$ We then find the values of $\Delta_1$ and $\Delta_2$ that makes the portfolio riskless. Indeed, we set $$ \Delta_1=\frac{\partial V}{\partial S}\\ \Delta_2=\frac{\frac{\partial V}{\partial r}}{\frac{\partial P}{\partial r}}$$ thus $$d\Pi =-\left( \frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}_{t}}^{2}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}V}{\partial {{r}^{2}}} \right)dt+\Delta {{}_{2}}\,\left( \frac{\partial P}{\partial t}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}P}{\partial {{r}^{2}}} \right)dt\tag 7$$ $(2)$ and $(7)$ $$d\Pi =-\left( \frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}V}{\partial {{r}^{2}}} \right)dt+\Delta {{}_{2}}\,\left( {{r}_{t}}\,P-\kappa \left( \theta -r \right)\,\frac{\partial P}{\partial r} \right)dt\tag 8$$ The condition that the portfolio earn the risk-free rate, $r$, implies that the change in portfolio value is $$d\Pi=r\Pi dt$$ in other words $$d\Pi =(\Delta _1r_t\,S_t+\Delta_2r_tP-r_tV)dt\tag 9$$ $(8)$ and $(9)$ $$\frac{\partial V}{\partial t}+{{r}_{t}}\,{{S}_{t}}\frac{\partial V}{\partial S}+\kappa \left( \theta -r \right)\frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}V}{\partial {{r}^{2}}}-{{r}_{t}}V=0$$

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user16651

Gordon's Answer is nice (+1). I want to add other solution.

Remark 1

Let $X=(X_1,X_2,...,X_n)$ where the component $X_i$ has a stochastic differential of the form $$dX_i(t)=\mu_i(t)dt+\sum_{j=1}^{d}\sigma_{ij}(t)dW_j(t)$$ where $dW_k(t)dW_j(t)=\rho_{kj}dt$, for all $k,j\in\{1,2,...,d\}$. Let $f:\mathbb{R}^+\times\mathbb{R}^n\to\mathbb{R}\in\mathbb{C}^{1,2}.$ By application of Ito's lemma, we have $$df(t,X_1,...,X_n)=\frac{\partial f}{\partial t}dt+\sum_{i=1}^{n}\frac{\partial f}{\partial x_i}dX_i+\frac 12\sum_{i=1}^{n}\sum_{l=1}^{n}\frac{\partial ^2f}{\partial x_i\partial x_l}dX_idX_l\tag 1$$


Remark 2

Assume that the short rate $r_t$ follows the Ito process as described by the following stochastic differential equation $$dr_t=\mu(t,r_t)dt+\sigma(t,r_t)dW_t$$ and $P(t,T)$ denotes the zero-coupon bond price with maturity $T$. We can show $$\frac{\partial P}{\partial t}+\mu(t,r_t)\frac{\partial P}{\partial r}+\frac{1}{2}\sigma^2(t,r_t)\frac{\partial^2 P}{\partial r^2}-r_tP=0\tag 2$$


Remark 3

Let $$\qquad dS_t=rS_t+\sigma S_tdW_1(t)\\ \quad\qquad\quad dr_t=\kappa(\theta-r_t)dt+\Sigma dW_2(t)\\ dW_1(t)dW_2(t)=0\tag 3$$

Now we form a portfolio consisting of one option $V=V(t,S,r,T,K)$ (Short position), $\Delta_1$ units of the stock (long position) and $\Delta_2$ units of the $T-$zero-coupon bond price (long position). The portfolio has value $$\Pi=\Delta_1S_t+\Delta_2 P(t,T)-V(t,S,r,T,K)$$ therefore $$d\Pi=\Delta_1dS_t+\Delta_2dP(t,T)-dV(t,S,r,T,K)\tag 4$$ By application of Ito's lemma, we have $$dV_t=\frac{\partial V}{\partial t}dt+\frac{\partial V}{\partial s}dS+\frac{\partial V}{\partial r}dr+\frac 12\left(\sigma^2\frac{\partial^2 V}{\partial s^2}+\Sigma^2\frac{\partial^2 V}{\partial r^2}\right)dt\tag 5$$ and $$dP=\frac{\partial P}{\partial t}dt+\frac{\partial P}{\partial r}dr+\frac{1}{2}\Sigma^2\frac{\partial^2 P}{\partial r^2}dt\tag 6$$ $(4)\,,\,(5)\,,(6)$ and $(3)$ $$d\Pi =-\left( \frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}} \right)dt+\left( \Delta _1-\frac{\partial V}{\partial S} \right)dS\\+\left( \Delta _2\frac{\partial P}{\partial r}\,-\frac{\partial V}{\partial r} \right)dr+\Delta _2\,\left( \frac{\partial P}{\partial t}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}P}{\partial {{r}^{2}}} \right)dt\,$$ We then find the values of $\Delta_1$ and $\Delta_2$ that makes the portfolio riskless. Indeed, we set $$ \Delta_1=\frac{\partial V}{\partial S}\\ \Delta_2=\frac{\frac{\partial V}{\partial r}}{\frac{\partial P}{\partial r}}$$ thus $$d\Pi =-\left( \frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}_{t}}^{2}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}V}{\partial {{r}^{2}}} \right)dt+\Delta {{}_{2}}\,\left( \frac{\partial P}{\partial t}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}P}{\partial {{r}^{2}}} \right)dt\tag 7$$ $(2)$ and $(7)$ $$d\Pi =-\left( \frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}V}{\partial {{r}^{2}}} \right)dt+\Delta {{}_{2}}\,\left( {{r}_{t}}\,P-\kappa \left( \theta -r \right)\,\frac{\partial P}{\partial r} \right)dt\tag 8$$ The condition that the portfolio earn the risk-free rate, $r$, implies that the change in portfolio value is $$d\Pi=r\Pi dt$$ in other words $$d\Pi =(\Delta _1r_t\,S_t+\Delta_2r_tP-r_tV)dt\tag 9$$ $(8)$ and $(9)$ $$\frac{\partial V}{\partial t}+{{r}_{t}}\,{{S}_{t}}\frac{\partial V}{\partial S}+\kappa \left( \theta -r \right)\frac{\partial V}{\partial t}+\frac{1}{2}{{\sigma }^{2}}{{S}^{2}}\frac{{{\partial }^{2}}V}{\partial {{S}^{2}}}+\frac{1}{2}{{\Sigma }^{2}}\frac{{{\partial }^{2}}V}{\partial {{r}^{2}}}-{{r}_{t}}V=0$$