Various types of models have been recently suggested to fit the Lagrangian high-Re turbulence data by the groups at Cornell and ENS-Lyon, and DNS by Mordant et al., Yeung, and Gotoh and Fukayama. The random intensity of noise approach to the 1D Laval-Dubrulle-Nazarenko model, based on the Navier-Stokes equation, is used to describe Lagrangian acceleration of a fluid particle in developed turbulence. This leads to consideration of a nonlinear Langevin equation for the acceleration $a$ with coupled additive and multiplicative noises. The stationary PDF associated to this equation is calculated exactly for model white-in-time Gaussian noises. The additive noise intensity and the cross correlation are assumed to depend on velocity fluctuations $u$ in an exponential way. The resulting conditional acceleration PDF $P(a|u)$, variance, and mean are found to be in a good agreement with the recent high-precision Lagrangian data by Mordant, Crawford, and Bodenschatz (2003).