We present the Kolmogorov–Lindblad mapping (KLM), which transforms nonlinear stochastic differential equations into half-density stochastic Schrödinger dynamics and, after averaging, an exact random-unitary Lindblad channel whose diagonal recovers the probability density. Structure-preserving discretizations then provide quantum algorithms for estimating statistical observables and multi-time correlations without postselection. We discuss regimes in which additional structure may yield provable quantum speedups..