Before revision:  The process effectively interpolates the dynamic of the asset price between the two dates while recovering the local volatility function. This approach leads to a convex optimisation problemand it is numerically solved ] via an augmented Lagrangian method and the alternative direction method of multipliers (ADMM) algorithm  .
After revision:  Inspired by the seminal work of Benamou and Brenier ] via an augmented Lagrangian method and the alternative direction method of multipliers (ADMM) algorithm .