[QuantEcon]MATLAB混编FORTRAN语言

$$
\begin{array}{l}
\frac{\sigma_1 \,\sigma_2 }{2}+\frac{\sigma_1 \,{\mathrm{e}}^{\frac{t\,\sqrt{\gamma^2 -4\,{\omega_0 }^2 }}{2}} }{2}-\frac{\gamma \,\sigma_1 \,\sigma_2 }{2\,\sqrt{\gamma^2 -4\,{\omega_0 }^2 }}+\frac{\gamma \,\sigma_1 \,{\mathrm{e}}^{\frac{t\,\sqrt{\gamma^2 -4\,{\omega_0 }^2 }}{2}} }{2\,\sqrt{\gamma^2 -4\,{\omega_0 }^2 }}\\
\mathrm{}\\
\textrm{where}\\
\mathrm{}\\
\;\;\sigma_1 ={\mathrm{e}}^{-\frac{\gamma \,t}{2}} \\
\mathrm{}\\
\;\;\sigma_2 ={\mathrm{e}}^{-\frac{t\,\sqrt{\gamma^2 -4\,{\omega_0 }^2 }}{2}}
\end{array}
$$

$$
\mathrm{cosh}\left(\omega_0 \,t\,\sqrt{\zeta^2 -1}\right)\,{\mathrm{e}}^{-\omega_0 \,t\,\zeta }
$$

$$
\left(\begin{array}{c}
m\,\frac{\partial }{\partial t}\;\mathrm{Dxt}\left(t\right)-\frac{F\left(t\right)\,x\left(t\right)}{r}\\
g\,m+m\,\frac{\partial }{\partial t}\;\mathrm{Dyt}\left(t\right)-\frac{F\left(t\right)\,y\left(t\right)}{r}\\
-r^2 +{x\left(t\right)}^2 +{y\left(t\right)}^2 \\
\mathrm{Dxt}\left(t\right)-\frac{\partial }{\partial t}\;x\left(t\right)\\
\mathrm{Dyt}\left(t\right)-\frac{\partial }{\partial t}\;y\left(t\right)
\end{array}\right)
$$

$$
\left(\begin{array}{c}
m\,\mathrm{Dxtt}\left(t\right)-\frac{F\left(t\right)\,x\left(t\right)}{r}\\
g\,m+m\,\mathrm{Dytt}\left(t\right)-\frac{F\left(t\right)\,y\left(t\right)}{r}\\
-r^2 +{x\left(t\right)}^2 +{y\left(t\right)}^2 \\
\mathrm{Dxt}\left(t\right)-{\textrm{Dxt}}_1 \left(t\right)\\
\mathrm{Dyt}\left(t\right)-{\textrm{Dyt}}_1 \left(t\right)\\
2\,{\textrm{Dxt}}_1 \left(t\right)\,x\left(t\right)+2\,{\textrm{Dyt}}_1 \left(t\right)\,y\left(t\right)\\
2\,y\left(t\right)\,\frac{\partial }{\partial t}\;{\textrm{Dyt}}_1 \left(t\right)+2\,{{\textrm{Dxt}}_1 \left(t\right)}^2 +2\,{{\textrm{Dyt}}_1 \left(t\right)}^2 +2\,\mathrm{Dxt1t}\left(t\right)\,x\left(t\right)\\
\mathrm{Dxtt}\left(t\right)-\mathrm{Dxt1t}\left(t\right)\\
\mathrm{Dytt}\left(t\right)-\frac{\partial }{\partial t}\;{\textrm{Dyt}}_1 \left(t\right)\\
{\textrm{Dyt}}_1 \left(t\right)-\frac{\partial }{\partial t}\;y\left(t\right)
\end{array}\right)
$$

$$
\left(\begin{array}{cc}
\mathrm{Dytt}\left(t\right) & \frac{\partial }{\partial t}\;\mathrm{Dyt}\left(t\right)\\
\mathrm{Dxtt}\left(t\right) & \frac{\partial }{\partial t}\;\mathrm{Dxt}\left(t\right)\\
{\textrm{Dxt}}_1 \left(t\right) & \frac{\partial }{\partial t}\;x\left(t\right)\\
{\textrm{Dyt}}_1 \left(t\right) & \frac{\partial }{\partial t}\;y\left(t\right)\\
\mathrm{Dxt1t}\left(t\right) & \frac{\partial^2 }{\partial t^2 }\;x\left(t\right)
\end{array}\right)
$$

$$
\frac{d^2y}{dt^2} = (1-y^2)\frac{dy}{dt} - y
$$

1. Define Parameters of the Model Using Stochastic Differential Equations
$$
dX = \mu(t, X) dt + \sigma(t, X) dB(t)
$$

Manually integrate the right side by parts. The result is
$$\frac{\partial}{\partial X} \left(k(2)\frac{\partial v}{\partial X}\right) + k(3) -
\frac{\partial v}{\partial X} \frac{\partial k(2)}{\partial X}
$$

2. Apply Ito's Rule
Asset prices follow a multiplicative process. That is, the logarithm of the price can be described in terms of an SDE, but the expected value of the price itself is of interest because it describes the profit, and thus we need an SDE for the latter.
In general, if a stochastic process X is given in terms of an SDE, then Ito's rule says that the transformed process G(t, X) satisfies

$$
dG = \left(\mu \frac{dG}{dX} + \frac{\sigma^2}{2} \frac{d^2G}{dX^2} +
\frac{dG}{dt}\right) dt + \frac{dG}{dX} \sigma dB(t)
$$

3. Solve Feynman-Kac Equation
Before you can convert symbolic expressions to MATLAB function handles, you must replace function calls, such as diff(v(t, X), X) and v(t, X), with variables. You can use any valid MATLAB variable names.

$$
\frac{{\sigma \left(t,X\right)}^2 \,\frac{\partial^2 }{\partial X^2 }\;y\left(X\right)}{2}+\mu \left(t,X\right)\,\frac{\partial }{\partial X}\;y\left(X\right)=-1
$$
$$
{\left(\frac{X\,{\sigma_0 }^2 }{2}+X\,\mu_0 \right)}\,\frac{\partial }{\partial X}\;y\left(X\right)+\frac{X^2 \,{\sigma_0 }^2 \,\frac{\partial^2 }{\partial X^2 }\;y\left(X\right)}{2}=-1
$$
$$
\begin{array}{l}
\frac{2\,\mu_0 \,\sigma_5 \,\mathrm{log}\left(b\right)-2\,\mu_0 \,\sigma_4 \,\mathrm{log}\left(a\right)+a^{\sigma_1 } \,{\sigma_0 }^2 \,\sigma_5 \,\sigma_4 -b^{\sigma_1 } \,{\sigma_0 }^2 \,\sigma_5 \,\sigma_4 }{\sigma_3 }-\frac{\mathrm{log}\left(X\right)}{\mu_0 }+\frac{\sigma_2 \,{\left(\sigma_7 -\sigma_6 -a^{\sigma_1 } \,{\sigma_0 }^2 \,\sigma_5 +b^{\sigma_1 } \,{\sigma_0 }^2 \,\sigma_4 \right)}}{\sigma_3 }+\frac{X^{\sigma_1 } \,{\sigma_0 }^2 \,\sigma_2 }{2\,{\mu_0 }^2 }\\
\mathrm{}\\
\textrm{where}\\
\mathrm{}\\
\;\;\sigma_1 =\frac{2\,\mu_0 }{{\sigma_0 }^2 }\\
\mathrm{}\\
\;\;\sigma_2 ={\mathrm{e}}^{-\frac{2\,\mu_0 \,\mathrm{log}\left(X\right)}{{\sigma_0 }^2 }} \\
\mathrm{}\\
\;\;\sigma_3 =2\,{\mu_0 }^2 \,{\left(\sigma_5 -\sigma_4 \right)}\\
\mathrm{}\\
\;\;\sigma_4 ={\mathrm{e}}^{-\frac{\sigma_6 }{{\sigma_0 }^2 }} \\
\mathrm{}\\
\;\;\sigma_5 ={\mathrm{e}}^{-\frac{\sigma_7 }{{\sigma_0 }^2 }} \\
\mathrm{}\\
\;\;\sigma_6 =2\,\mu_0 \,\mathrm{log}\left(b\right)\\
\mathrm{}\\
\;\;\sigma_7 =2\,\mu_0 \,\mathrm{log}\left(a\right)
\end{array}
$$