Particle approximation for a conditional McKean--Vlasov stochastic differential equation

In this paper, we construct a type of interacting particle systems to approximate a class of stochastic different equations whose coefficients depend on the conditional probability distributions of the processes given partial observations. After proving the well-posedness and regularity of the particle systems, we establish a quantitative convergence result for the empirical measures of the particle systems in the Wasserstein space, as the number of particles increases. Moreover, we discuss an Euler–Maruyama scheme of the particle system and validate its strong convergence. A numerical experiment is conducted to illustrate our results.