Multidimensional subdiffusion model: Arbitrage-free market

To capture the subdiffusive characteristics of financial markets, the subordinated process, directed by the inverse α-stale subordinator S_α(t) for 0 < α <1, has been employed as the model of asset prices. In this article, we introduce a multidimensional subdiffusion model that has a bond and K correlated stocks. The stock price process is a multidimensional subdiffusion process directed by the inverse α-stable subordinator. This model describes the period of stagnation for each stock and the behavior of the dependency between multiple stocks. Moreover, we derive the multidimensional fractional backward Kolmogorov equation for the subordinated process by Laplace transform technique. Finally, using martingale approach, we prove that the multidimensional subdiffusion model is arbitrage-free, and also gives an arbitrage-free pricing rule for contingent claims associated with the martingale measure.