| Summary: | Root’s solution (Root [1969]) to the Skorokhod embedding problem can be described as the first hitting time of a space-time process (Xt, t) on a so-called barrier, charac- terised by certain properties, such that the stopped underlying process X has a given distribution. Recent work of Dupire [2005] and Carr and Lee [2010] has highlighted the importance of understanding the Root’s solution for the model-independent hedging of variance options. We consider the problem of finding Root’s solutions when the underlying process is a time-homogeneous diffusion with a given initial distribution in one dimension. We are interested in constructing Root’s solution by partial differential equations. We begin by showing that, under some mild conditions, constructing Root’s solution is equiv- alent to solving a specialized parabolic free boundary problem in the case where the underlying process is a Brownian motion starting at 0. This result is then extended to time-homogeneous diffusions. Replacing some conditions needed in the free boundary construction, we then also consider the construction of Root’s solutions by variational inequalities. Finally we consider the optimality and applications of Root’s solutions. Unlike the existing proof of optimality (Rost [1976]), which relies on potential theory, an alternative proof is given by finding a path-wise inequality which has an impor- tant application for the construction of subhedging strategies in the financial context. In addition, we also consider these questions, construction and optimality, for Rost’s solution, which is also known as the reverse of the Root’s solution.
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