Abstract
In this chapter, we discuss numerical methods for SDEs with coefficients of polynomial growth. The nonlinear growth of the coefficients induces instabilities, especially when the nonlinear growth is polynomial or even exponential. For stochastic differential equations (SDEs) with coefficients of polynomial growth at infinity and satisfying a one-sided Lipschitz condition, we prove a fundamental mean-square convergence theorem on the strong convergence order of a stable numerical scheme in Chapter 5.2. We apply the theorem to a number of existing numerical schemes. We present in Chapter 5.3 a special balanced scheme, which is explicit and of half-order mean-square convergence. Some numerical results are presented in Chapter 5.4. We summarize the chapter in Chapter 5.5 and present some bibliographic notes on numerical schemes for nonlinear SODEs. Three exercises are presented for interested readers.
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Notes
- 1.
As Brownian motion can take values in \(\mathbb{R}\), the (numerical) solution may reach any value at a certain time step. We can assume that we compute the solutions from such a step and denote it as the zeroth step.
- 2.
Assuming additional smoothness of a(t, x), we can get an estimate for \(\mathbb{E}\tilde{\rho }(t,x)\) of order h 2 but this will not improve the result of this lemma for the balanced Euler scheme (5.3.1).
- 3.
However, (5.4.1) is not applicable when diffusion grows faster than a linear function.
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Zhang, Z., Karniadakis, G.E. (2017). Balanced numerical schemes for SDEs with non-Lipschitz coefficients. In: Numerical Methods for Stochastic Partial Differential Equations with White Noise. Applied Mathematical Sciences, vol 196. Springer, Cham. https://doi.org/10.1007/978-3-319-57511-7_5
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