An advanced numerical scheme for multi-dimensional stochastic Kolmogorov equations with superlinear coefficients. (English) Zbl 1523.60097
Summary: This work develops a novel approximation for a class of superlinear stochastic Kolmogorov equations with positive global solutions. On the one hand, most existing explicit methods that work for the superlinear stochastic differential equations (SDEs), e.g. various modified Euler-Maruyama (EM) methods, fail to preserve positivity of the solution. On the other hand, methods that preserve positivity are mostly implicit, or fail to cope with the multi-dimensional scenario. This work aims to construct an advanced numerical method which is not only naturally structure preserving but also cost effective. A strong convergence framework is then developed with an almost optimal convergence rate of order arbitrarily close to \(1/2\). To make the arguments concise, we elaborate our theory with the generalised stochastic Lotka-Volterra model, though the method is applicable to a wide bunch of multi-dimensional superlinear stochastic Kolmogorov systems in various fields including finance and epidemiology.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 34F05 | Ordinary differential equations and systems with randomness |
| 92D25 | Population dynamics (general) |
Keywords:
stochastic differential equation; Kolmogorov equation; structure preserving numerical method; exponential Euler-Maruyama method; convergence rateReferences:
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