Cónall Kelly

Cónall Kelly, Gabriel J. Lord

We introduce a class of adaptive timestepping strategies for stochastic differential equations with non-Lipschitz drift coefficients. These strategies work by controlling potential unbounded growth in solutions of a numerical scheme due to the drift. We prove that the Euler-Maruyama scheme with an adaptive timestepping strategy in this class is strongly convergent. Specific strategies falling into this class are presented and demonstrated on a selection of numerical test problems. We observe that this approach is broadly applicable, can provide more dynamically accurate solutions than a drift-tamed scheme with fixed stepsize, and can improve MLMC simulations.

Ricardo Baccas, Cónall Kelly, Alexandra Rodkina

We consider the stochastically perturbed cubic difference equation with variable coefficients \[ x_{n+1}=x_n(1-h_nx_n^2)+ρ_{n+1}ξ_{n+1}, \quad n\in \mathbb N,\quad x_0\in \mathbb R. \] Here $(ξ_n)_{n\in \mathbb N}$ is a sequence of independent random variables, and $(ρ_n)_{n\in \mathbb N}$ and $(h_n)_{n\in \mathbb N}$ are sequences of nonnegative real numbers. We can stop the sequence $(h_n)_{n\in \mathbb N}$ after some random time $\mathcal N$ so it becomes a constant sequence, where the common value is an $\mathcal{F}_\mathcal{N}$-measurable random variable. We derive conditions on the sequences $(h_n)_{n\in \mathbb N}$, $(ρ_n)_{n\in \mathbb N}$ and $(ξ_n)_{n\in \mathbb N}$, which guarantee that $\lim_{n\to \infty} x_n$ exists almost surely (a.s.), and that the limit is equal to zero a.s. for any initial value $ x_0\in \mathbb R$.

Cónall Kelly, Alexandra Rodkina, Eeva Maria Rapoo

We consider the use of adaptive timestepping to allow a strong explicit Euler-Maruyama discretisation to reproduce dynamical properties of a class of nonlinear stochastic differential equations with a unique equilibrium solution and non-negative, non-globally Lipschitz coefficients. Solutions of such equations may display a tendency towards explosive growth, countered by a sufficiently intense and nonlinear diffusion. We construct an adaptive timestepping strategy which closely reproduces the a.s. asymptotic stability and instability of the equilibrium, and which can ensure the positivity of solutions with arbitrarily high probability. Our analysis adapts the derivation of a discrete form of the Itô formula from Appleby et al (2009) in order to deal with the lack of independence of the Wiener increments introduced by the adaptivity of the mesh. We also use results on the convergence of certain martingales and semi-martingales which influence the construction of our adaptive timestepping scheme in a way proposed by Liu & Mao (2017).