We study the smoothness of the solution of the directed chain stochastic differential equations, where each process is affected by its neighborhood process in an infinite directed chain graph, introduced by Detering et al. (2020). Classic methods of Malliavin derivatives are not directly applicable, because of the auxiliary process in the chain-like structure. Namely, we cannot make a connection between the Malliavin derivative and the first order derivative of the state process. It turns out that the partial Malliavin derivatives can be used here to fix this problem. Furthermore, we apply the smoothness result to the filtering problem on the directed chain and derive the stochastic partial differential equations for the conditional densities.
Ming is a 4th year PhD student at University of California, Santa Barbara, advised by Professor Tomoyuki Ichiba. His research focus on stochastic analysis and applications in machine learning for sequential data.