We present a positivity preserving numerical scheme for the pathwise solution of nonlinear stochastic differential equations driven by a multi-dimensional Wiener process and governed by non-commutative linear and non-Lipschitz vector fields. This strong order one scheme uses: (i) Strang exponential splitting, an approximation that decomposes the stochastic flow separately into the drift flow, and the pure diffusion flow governed by the diffusion vector fields; (ii) an implicit Euler method to approximate the drift flow; and (iii) an implicit Milstein method to approximate the pure diffusion flow. The separate approximations for the drift and pure diffusion flows preserve positivity. Therefore the Strang exponential splitting approximation does also. We demonstrate the efficacy of our method by applying it to the Heston model and a variance curve model , and compare it against well-established positivity preserving schemes .
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 .
R
We present a positivity preserving numerical scheme for the pathwise solution of
For
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equations driven by a multi-dimensional Wiener process and governed by non-commutative linear and non-Lipschitz vector fields. This strong order one scheme uses: (i) Strang exponential splitting, an approximation that decomposes the stochastic flow separately into the drift flow, and the pure diffusion flow governed by the diffusion vector fields; (ii) an implicit Euler method to approximate the drift flow; and (iii) an implicit Milstein method to approximate the pure diffusion flow. The separate approximations for the drift and pure diffusion flows preserve positivity. Therefore the Strang exponential splitting approximation does also. We demonstrate the efficacy of our method by applying it to the Heston model and
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
R
, and compare it against well-established positivity preserving schemes
and the Heston model