For nonlinear stochastic differential systems, we develop strong fully implicit positivity preserving numerical methods in the case that the zero boundary is non-attracting. These methods are implicit in the diffusion vector fields. They thus apply to a restricted class, namely those with sublinear form. This however, still includes most Langevin derived processes typical of volatility models in finance and molecular simulation in physics. When the  zero boundary is attracting and attainable, we specialize to a prototypical model, namely the mean-reverting Cox--Ingersoll--Ross process. We thus consider the non-central chi-squared transition density with fractional degrees of freedom. We prove that we can sample from this density by simulating Poisson distributed sums of powers of generalized Gaussian random variables. Further we prove that Marsaglia's polar method extends to the generalized Gaussian distribution, providing an exact and efficient method for generalized Gaussian sampling. We apply our methods to a variance curve model and the Heston model .
In the Heston stochastic volatility model, the transition probability of the variance process can be represented by a non-central chi-square density. We focus on the case when the number of degrees of freedom is small and the zero boundary is attracting and attainable, typical in foreign exchange markets. We prove a new representation for this density based on sums of powers of generalized Gaussian random variables. Further we prove Marsaglia's polar method extends to this distribution, providing an exact method for generalized Gaussian sampling. The advantages are that for the mean-reverting square-root process in the Heston model and Cox-Ingersoll-Ross model, we can generate samples from the true transition density simply, efficiently and robustly .
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For nonlinear stochastic differential systems, we develop strong fully implicit positivity preserving numerical methods in the case that the zero boundary is non-attracting. These methods are implicit in the diffusion vector fields. They thus apply to a restricted class, namely those with sublinear form. This however, still includes most Langevin derived processes typical of volatility models in finance and molecular simulation in physics. When
In the Heston stochastic volatility model,
A
transition probability of the variance process can be represented by a non-central chi-square density. We focus on the case when the number of degrees of freedom is small and the
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we specialize to a prototypical model, namely the mean-reverting Cox--Ingersoll--Ross process. We thus consider the non-central chi-squared transition density with fractional degrees of freedom. We prove that we can sample from this density by simulating Poisson distributed
typical in foreign exchange markets. We prove a new representation for this density based on
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that
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the generalized Gaussian
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and efficient
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We apply our methods to a variance curve model and the Heston model
The advantages are that for the mean-reverting square-root process in the Heston model and Cox-Ingersoll-Ross model, we can generate samples from the true transition density simply, efficiently and robustly