Divergence with probability one of the multilevel Monte Carlo Euler method

Let $T\in(0,\infty)$, let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space with a normal filtration $(\mathcal{F}_t)_{t\in[0,T]}$, let $W^{l,k}$ be a family of independent one-dimensional standard Brownian motions, and let $\xi^{l,k}$ be an independent identically distributed family of initial values with finite moments of every order. Let $\mu,\sigma\colon\mathbb{R}\to\mathbb{R}$ be continuous, and let $X$ solve

$$X_t=\xi^{0,1}+\int_0^t\mu(X_s)\,ds+\int_0^t\sigma(X_s)\,dW_s^{0,1}.$$

Define the Euler approximations by $Y_0^{N,l,k}=\xi^{l,k}$ and

$$Y_{n+1}^{N,l,k}=Y_n^{N,l,k}+\mu(Y_n^{N,l,k})\frac{T}{N}+\sigma(Y_n^{N,l,k})\left(W_{\frac{(n+1)T}{N}}^{l,k}-W_{\frac{nT}{N}}^{l,k}\right).$$

Assume that $\alpha,c\in(1,\infty)$ satisfy

$$\frac{|x|^\alpha}{c}\leq|\mu(x)|+|\sigma(x)|\leq c|x|^c$$

for all $x\in\mathbb{R}$ with $|x|\geq c$, and assume either $\mathbb{P}[\sigma(\xi^{0,1})\neq0]>0$ or that there exists $\beta\in(1,\infty)$ such that $\mathbb{P}[|\xi^{0,1}|\geq x]\geq\beta^{-x^\beta}$ for all $x\in[1,\infty)$. Let $f\colon\mathbb{R}\to\mathbb{R}$ be measurable and satisfy $c^{-1}|x|^{1/c}-c\leq f(x)\leq c(1+|x|^c)$. Divergence with probability one of the multilevel Monte Carlo Euler method. Then

$$\lim_{\substack{N\to\infty\\ \operatorname{ld}(N)\in\mathbb{N}}}\left|\mathbb{E}[f(X_T)]-\frac1N\sum_{k=1}^Nf(Y_1^{1,0,k})-\sum_{l=1}^{\operatorname{ld}(N)}\frac{2^l}{N}\left(\sum_{k=1}^{N/2^l}f(Y_{2^l}^{2^l,l,k})-f(Y_{2^{l-1}}^{2^{l-1},l,k})\right)\right|=\infty$$

$\mathbb{P}$-almost surely. The conjecture asserts almost-sure divergence of the multilevel Monte Carlo Euler estimator for nonlinear stochastic differential equations whose coefficients grow superlinearly. The paper presents this as a pathwise divergence phenomenon motivated by numerical experiments; the supplied text does not establish its resolution.