Solution of stochastic boundary value problem in SDE using transition probability density function decomposition

Abstract A general framework for solving stochastic boundary value problems (SBVPs) in diffusion-type stochastic differential equations (SDEs) is developed based on the decomposition of the transition probability density function. The proposed approach relies on the Hermite polynomial expansion of the conditional transition density and its spatial derivatives, which allows for a direct construction of guided stochastic bridges connecting prescribed boundary points. The method unifies simulation, analytical approximation, and statistical reweighting in a single probabilistic representation that remains valid for nonlinear, non-Gaussian, and anisotropic diffusion systems. Several algorithms for guided bridge simulation are formulated and compared, including Gaussian-only approximation, Gaussian–Hermite correction, deterministic shooting, and MCMC-based path sampling. Each of these schemes is implemented using the Milstein discretization with local correction terms that preserve higher-order stochastic moments. The Hermite-corrected bridge, in particular, incorporates cubic and quartic terms of the transition density expansion, providing an enhanced representation of curvature and skewness in the conditional dynamics. A detailed numerical analysis is presented for both one-dimensional and two-dimensional diffusions with nonlinear drift functions and spatially varying diffusion matrices. The examples include models with polynomial, hyperbolic, arctangent, and tangent-type drifts that exhibit strong nonlinearity and coupling. Statistical diagnostics such as effective sample size (ESS), variance of log-weights, maximum normalized share, and perplexity are used to assess the efficiency of the importance reweighting. Residual-based QQ plots and histograms confirm that the Gaussian—Hermite bridge maintains near-Gaussian residual distributions and stable log-weight statistics across a broad range of time horizons. The obtained results demonstrate that the Gaussian–Hermite decomposition significantly improves the stability, accuracy, and efficiency of stochastic bridge simulations compared to Gaussian or shooting-based methods. The approach provides a theoretically consistent and computationally tractable framework for conditioned diffusion processes with low-regularity coefficients. Beyond SBVPs, the method can be extended to parameter estimation, inverse problems, and stochastic control formulations, where explicit transition densities or their functional decompositions are available.