  Guyon and Lekeufack recently proposed a path-dependent volatility model and
documented its excellent performance in fitting market data and capturing
stylized facts. The instantaneous volatility is modeled as a linear combination
of two processes, one is an integral of weighted past price returns and the
other is the square-root of an integral of weighted past squared volatility.
Each of the weightings is built using two exponential kernels reflecting long
and short memory. Mathematically, the model is a coupled system of four
stochastic differential equations. Our main result is the wellposedness of this
system: the model has a unique strong (non-explosive) solution for all
parameter values. We also study the positivity of the resulting volatility
process and the martingale property of the associated exponential price
process.
