Speeding up the Euler scheme for killed diffusions
针对线性扩散过程在边界被杀死时的期望计算，标准欧拉格式的弱收敛速度会降至1/√N。本文提出一种漂移隐式欧拉方法，将收敛速度恢复至1/N，即无杀死时的最优速率。
Abstract Let $X$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> be a linear diffusion taking values in $(\ell ,r)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:math> and consider the standard Euler scheme to compute an approximation to $\mathbb{E}[g(X_{T}){\mathbf{1}}_{\{T<\zeta \}}]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> <mml:mo>[</mml:mo> <mml:mi>g</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>T</mml:mi> </mml:msub> <mml:mo>)</mml:mo> <mml:msub> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>T</mml:mi> <mml:mo><</mml:mo> <mml:mi>ζ</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:msub> <mml:mo>]</mml:mo> </mml:math> for a given function $g$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> and a deterministic $T$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> , where $\zeta =\inf \{t\geq 0: X_{t} \notin (\ell ,r)\}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>inf</mml:mo> <mml:mo>{</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>∉</mml:mo> <mml:mo>(</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> <mml:mo>}</mml:mo> </mml:math> . It is well known since Gobet (Stoch. Process. Appl. 87:167–197, 2000) that the presence of killing introduces a loss of accuracy and reduces the weak convergence rate to $1/\sqrt{N}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mi>N</mml:mi> </mml:msqrt> </mml:math> with $N$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> being the number of discretisations. We introduce a drift-implicit Euler method to bring the convergence rate back to $1/N$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>N</mml:mi> </mml:math> , i.e., the optimal rate in the absence of killing, using the theory of recurrent transformations developed in Çetin (Ann. Appl. Probab. 28:3102–3151, 2018). Although the current setup assumes a one-dimensional setting, multidimensional extension is within reach as soon as a systematic treatment of recurrent transformations is available in higher dimensions.