We present  a method for constructing new families of solvable one-dimensional diffusions with linear drift and nonlinear diffusion coefficient functions, whose transition densities are obtainable in analytically closed-form. Our approach is based on  the so-called diffusion canonical transformation method that allows us to uncover new multiparameter diffusions that are mapped onto various simpler underlying diffusions. We give a simple rigorous boundary classification and characterization of the newly constructed processes with respect to the martingale property. Specifically, we construct, analyse and classify three new families of nonlinear diffusion models with affine drift that arise from the squared Bessel process (the Bessel family), the CIR process (the confluent hypergeometric family), and the Ornstein-Uhlenbeck diffusion (the OU family) .
We present new extensions to a method for constructing several families of solvable one-dimensional time-homogeneous diffusions whose transition densities are obtainable in analytically closed-form. Our approach is based on a dual application of the so-called diffusion canonical transformation method that combines smooth monotonic mappings and measure changes via Doob-h transforms. This gives rise to new multi-parameter solvable diffusions that are generally divided into two main classes; the first is specified by having affine (linear) drift with various resulting nonlinear diffusion coefficient functions, while the second class allows for several specifications of a (generally nonlinear) diffusion coefficient with resulting nonlinear drift function. The theory is applicable to diffusions with either singular and/or non-singular endpoints. As part of the results in this paper, we also present a complete boundary classification and martingale characterization of the newly developed diffusion families .
A
new extensions to
R
new
several
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diffusions with linear drift and nonlinear diffusion coefficient functions,
time-homogeneous diffusions
A
a dual application of
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allows us to uncover new multiparameter
combines smooth monotonic mappings and measure changes via Doob-h transforms. This gives rise to new multi-parameter solvable
R
mapped onto various simpler underlying diffusions. We give a simple rigorous boundary classification and characterization of the newly constructed processes with respect to the martingale property. Specifically, we construct, analyse and classify three new families of nonlinear diffusion models with affine drift that arise from the squared Bessel process (the Bessel family), the CIR process (the confluent hypergeometric family), and the Ornstein-Uhlenbeck diffusion (the OU family)
generally divided into two main classes; the first is specified by having affine (linear) drift with various resulting nonlinear diffusion coefficient functions, while the second class allows for several specifications of a (generally nonlinear) diffusion coefficient with resulting nonlinear drift function. The theory is applicable to diffusions with either singular and/or non-singular endpoints. As part of the results in this paper, we also present a complete boundary classification and martingale characterization of the newly developed diffusion families