Dissipative cascaded multistep backward stochastic differential equation (BSDE) with polynomial growth

This work investigates a cascaded multistep backward stochastic differential equation in which an inner backward component feeds the outer backward dynamics through both the terminal condition and the generator. The construction yields a hierarchical stochastic response where an intermediate adapted quantity serves as a latent driver for a second backward layer. The analysis is formulated under local Lipschitz conditions together with integrability and monotonicity requirements that are compatible with polynomial growth. The inner layer is posed with a locally Lipschitz generator and admits a square-integrable solution under standard growth bounds. The outer layer allows a non-Lipschitz polynomial dependence in the backward variable while retaining local Lipschitz regularity in the martingale integrand and a dissipativity structure expressed through a one-sided monotonicity inequality. A priori estimates quantify how dissipation controls higher-order moments of the outer component and how perturbations in the inner layer propagate through the cascade. In the Markovian regime, the multistep BSDE admits a deterministic representation that leads to a hierarchical system of semilinear parabolic equations with a cubic damping term in the outer equation. The results provide a rigorous basis for regression-based Monte Carlo discretizations that rely on conditional expectation approximations and localization.