14. Partial Differential Equation Approach Under Geometric Jump-Diffusion Process

$$\displaystyle{\frac{\partial f} {\partial x} = \frac{1} {x} \frac{\partial F} {\partial u},\qquad \frac{\partial ^{2}f} {\partial x^{2}} = - \frac{1} {x^{2}} \frac{\partial F} {\partial u} + \frac{1} {x^{2}} \frac{\partial ^{2}F} {\partial u^{2}} = \frac{1} {x^{2}}\left (\frac{\partial ^{2}F} {\partial u^{2}} -\frac{\partial F} {\partial u} \right ),}$$

$$\displaystyle\begin{array}{rcl} v^{2} -\frac{\beta _{1}^{2}} {4\alpha } & =& v^{2} -\frac{n\left (v^{2}\delta ^{4} - 2v\delta ^{2}\sigma \sqrt{\tau }(\gamma +\frac{\delta ^{2}} {2} ) +\sigma ^{2}\tau (\gamma +\frac{\delta ^{2}} {2} )^{2}\right )} {\delta ^{2}(\sigma ^{2}\tau +\delta ^{2}n)} {}\\ & =& \frac{\sigma ^{2}\tau v^{2} + 2n\sigma \sqrt{\tau }(\gamma +\frac{\delta ^{2}} {2} )v} {\sigma ^{2}\tau +\delta ^{2}n} - \frac{\sigma ^{2}\tau n(\gamma +\frac{\delta ^{2}} {2} )^{2}} {\delta ^{2}(\sigma ^{2}\tau +\delta ^{2}n)} {}\\ & =& \frac{\left (\sigma \sqrt{\tau }v + n(\gamma +\frac{\delta ^{2}} {2} )\right )^{2}} {\sigma ^{2}\tau +\delta ^{2}n} -\frac{n^{2}(\gamma +\frac{\delta ^{2}} {2} )^{2}} {\sigma ^{2}\tau +\delta ^{2}n} - \frac{\sigma ^{2}\tau n(\gamma +\frac{\delta ^{2}} {2} )^{2}} {\delta ^{2}(\sigma ^{2}\tau +\delta ^{2}n)}. {}\\ \end{array}$$

$$\displaystyle\begin{array}{rcl} v^{2} -\frac{\beta _{2}^{2}} {4\alpha } & =& \frac{\sigma ^{2}\tau v^{2} + 2\sigma \sqrt{\tau }n(\gamma -\frac{\delta ^{2}} {2} )v} {\sigma ^{2}\tau +\delta ^{2}n} - \frac{\sigma ^{2}\tau n(\gamma -\frac{\delta ^{2}} {2} )^{2}} {\delta ^{2}(\sigma ^{2}\tau +\delta ^{2}n)} {}\\ & =& \frac{\left (\sigma \sqrt{\tau }v + n(\gamma -\frac{\delta ^{2}} {2} )\right )^{2}} {\sigma ^{2}\tau +\delta ^{2}n} -\frac{n(\gamma -\frac{\delta ^{2}} {2} )^{2}} {\delta ^{2}}. {}\\ \end{array}$$