Analytical solution of the Dupire-like equation in calibration to the generalized stochastic volatility jump-diffusion model for option pricing

In this paper, we succeed to solve the open problem that was about finding the analytical solution of the Dupire-like equation and that we have considered as a result of our Corollary 4.1 (Jraifi et al. in J Adv Math Stud 11(2):282–294, 2018) and which was defined by, \(\begin{aligned} \displaystyle \partial _T \omega +\displaystyle \sum _{i=1}^n F_i (T,K) \partial _{K_i} \omega +(r+\displaystyle \sum _{i=1}^n F_i^{\prime })\omega= & \frac{1}{2}\displaystyle \sum _{i, j=1}^n \left( \displaystyle \int _{\mathbb {R}^d}(GG^T )_{ij}\Psi (T,K,{\bar{y}}) d{\bar{y}}\right) \\ & +\mathcal {I} (\Psi )(T,K,{\bar{y}}) \end{aligned}\) $$\begin{array}{cc}\hfill {\displaystyle {\partial }_T\omega +\sum _{i=1}^n{F}_i(T,K){\partial }_{K_i}\omega +(r+\sum _{i=1}^n{F}_i^{\prime })\omega =}& {\displaystyle \frac{1}{2}\sum _{i,j=1}^n\left({\displaystyle {\int }_{{\mathbb{R}}^d}{(G{G}^T)}_{\mathit{ij}}\mathrm{\Psi }(T,K,\overline{y})d\overline{y}}\right)}\hfill \\ \hfill & +\mathcal{I}(\mathrm{\Psi })(T,K,\overline{y})\hfill \end{array}$$ with taking as example \(F_i = \alpha _{i} K_{i}\) $F_i={\alpha }_i{K}_i$ , \(K\in \mathbb {R}^{+n}\) $K\in {\mathbb{R}}^{+n}$ , \(T>t^* \) $T>t^{\ast }$ , and in the presence of the initial condition \(\omega |_{T=t^{*}}=h(x^{*})\) $\omega |_{T=t_{}^{\ast }}^{}=h(x_{}^{\ast })$ . Since we are taking a generalized form of stochastic volatility jump-diffusion models for option pricing, such new results also represent an important step of contribution to the theory of stochastic control problems of jump diffusions as more precised in our conclusion.