Non-globally Lipschitz Counterexamples for the stochastic Euler scheme
Non-globally Lipschitz Counterexamples for the stochastic Euler scheme
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发表时间：
2009-05
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通讯作者：
Martin Hutzenthaler;Arnulf Jentzen
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作者：
Martin Hutzenthaler;Arnulf Jentzen
已知的随机欧拉格式收敛于具有全局Lipschitz系数的随机微分方程解，甚至具有至多线性增长的系数。对于超线性增长系数，在强意义下和在数值弱意义下的收敛仍然是一个悬而未决的问题。本文证明了对于许多具有超线性增长系数的随机微分方程，欧拉逼近既不是在强L意义下收敛，也不是在数值弱意义下收敛到精确解。更糟糕的是，精确解和数值近似的差在强L意义下和在数值弱意义下发散到无穷远。
The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation with globally Lipschitz coefficients and even with coefficients which grow at most linearly. For super-linearly growing coefficients convergence in the strong and numerically weak sense remained an open question. In this article we prove for many stochastic differential equations with super-linearly growing coefficients that Euler’s approximation does not converge neither in the strong L-sense nor in the numerically weak sense to the exact solution. Even worse, the difference of the exact solution and of the numerical approximation diverges to infinity in the strong L-sense and in the numerically weak sense.