arXiv subjects

Cónall Kelly

Publications and source records attributed to Cónall Kelly.

11 recordsLinked to original sources

arXiv2026

On splitting strategies for the numerical solution of stochastic delay differential equations with correlated noises

In this article we investigate the numerical solution of a scalar semilinear stochastic delay differential equation (SDDE) where the linear instantaneous feedback and nonlinear delayed feedback terms are perturbed by a pair of standard Brownian motions with correlation $ρ$. Such SDDEs may be naturally decomposed into two subsystems: a linear stochastic differential equation (SDE) without delay, and a nonlinear SDDE. Splitting methods work by solving each subsystem separately and composing the results over a single step. Our main theoretical result provides a bound on the mean-square error of a particular strategy for doing this, known as Lie-Trotter splitting. This bound implies that the method is mean-square strongly convergent with order $1/2$ when $ρ=0$, so that the noises are uncorrelated, but assurances of convergence are lost when $ρ\neq 0$. Indeed we develop an upper bound on the global mean-square error with a term that depends linearly on the magnitude of the correlation, and is independent of the stepsize. While our theoretical error bound is an estimate from above, we conduct numerical experiments that confirm the order of mean-square strong convergence of Lie-Trotter splitting in the $ρ=0$ case, and demonstrate a rapid fall-off to effectively zero as $|ρ|$ increases. Similar numerical results are observed for an alternative commonly used strategy known as Strang splitting. Nonetheless, by carefully reorganising the subsystems into which we split the SDDE, we can improve the range of values of $ρ$ over which a nonzero order of convergence is observed numerically.

math.NA↗

arXiv2024

Adaptive Mesh Construction for the Numerical Solution of Stochastic Differential Equations with Markovian Switching

We demonstrate an approach to the numerical solution of nonlinear stochastic differential equations with Markovian switching. Such equations describe the stochastic dynamics of processes where the drift and diffusion coefficients are subject to random state changes according to a Markov chain with finite state space. We propose a variant of the Jump Adapted-Adaptive approach introduced by K, Lord, \& Sun~(2025) to construct nonuniform meshes for explicit numerical schemes that adjust timesteps locally to rapid changes in the numerical solution and which also incorporate the switching times of an underlying Markov chain as meshpoints. It is shown that a hybrid scheme using such a mesh that combines an efficient explicit method (to be used frequently) and a potentially inefficient backstop method (to be used occasionally) will display strong convergence in mean-square of order $δ$ if both methods satisfy a mean-square consistency condition of the same order in the absence of switching. We demonstrate the construction of an order $δ=1$ method of this type and apply it to generate empirical distributions of a nonlinear SDE model of telomere length in DNA replication.

math.NA↗

arXiv2024

Strong convergence of a class of adaptive numerical methods for SDEs with jumps

We develop adaptive time-stepping strategies for Itô-type stochastic differential equations (SDEs) with jump perturbations. Our approach builds on adaptive strategies for SDEs. Adaptive methods can ensure strong convergence of nonlinear SDEs with drift and diffusion coefficients that violate global Lipschitz bounds by adjusting the stepsize dynamically on each trajectory to prevent spurious growth that can lead to loss of convergence if it occurs with sufficiently high probability. In this article we demonstrate the use of a jump-adapted mesh that incorporates jump times into the adaptive time-stepping strategy. We prove that any adaptive scheme satisfying a particular mean-square consistency bound for a nonlinear SDE in the non-jump case may be extended to a strongly convergent scheme in the Poisson jump case where jump and diffusion perturbations are mutually independent and the jump coefficient satisfies a global Lipschitz condition.

math.NA↗

arXiv2023

An adaptive splitting method for the Cox-Ingersoll-Ross process

We propose a new splitting method for strong numerical solution of the Cox-Ingersoll-Ross model. For this method, applied over both deterministic and adaptive random meshes, we prove a uniform moment bound and strong error results of order $1/4$ in $L_1$ and $L_2$ for the parameter regime $κθ>σ^2$. We then extend the new method to cover all parameter values by introducing a \emph{soft zero} region (where the deterministic flow determines the approximation) giving a hybrid type method to deal with the reflecting boundary. From numerical simulations we observe a rate of order $1$ when $κθ>σ^2$ rather than $1/4$. Asymptotically, for large noise, we observe that the rates of convergence decrease similarly to those of other schemes but that the proposed method making use of adaptive timestepping displays smaller error constants.

math.NA↗

arXiv2022

Strong convergence of an adaptive time-stepping Milstein method for SDEs with monotone coefficients

We introduce an explicit adaptive Milstein method for stochastic differential equations (SDEs) with no commutativity condition. The drift and diffusion are separately locally Lipschitz and together satisfy a monotone condition. This method relies on a class of path-bounded time-stepping strategies which work by reducing the stepsize as solutions approach the boundary of a sphere, invoking a backstop method in the event that the timestep becomes too small. We prove that such schemes are strongly $L_2$ convergent of order one. This order is inherited by an explicit adaptive Euler-Maruyama scheme in the additive noise case. Moreover we show that the probability of using the backstop method at any step can be made arbitrarily small. We compare our method to other fixed-step Milstein variants on a range of test problems.

math.NA↗

arXiv2022

The role of adaptivity in a numerical method for the Cox-Ingersoll-Ross model

We demonstrate the effectiveness of an adaptive explicit Euler method for the approximate solution of the Cox-Ingersoll-Ross model. This relies on a class of path-bounded timestepping strategies which work by reducing the stepsize as solutions approach a neighbourhood of zero. The method is hybrid in the sense that a convergent backstop method is invoked if the timestep becomes too small, or to prevent solutions from overshooting zero and becoming negative. Under parameter constraints that imply Feller's condition, we prove that such a scheme is strongly convergent, of order at least 1/2. Control of the strong error is important for multi-level Monte Carlo techniques. Under Feller's condition we also prove that the probability of ever needing the backstop method to prevent a negative value can be made arbitrarily small. Numerically, we compare this adaptive method to fixed step implicit and explicit schemes, and a novel semi-implicit adaptive variant. We observe that the adaptive approach leads to methods that are competitive in a domain that extends beyond Feller's condition, indicating suitability for the modelling of stochastic volatility in Heston-type asset models.

q-fin.CP↗

arXiv2021

Adaptive Euler methods for stochastic systems with non-globally Lipschitz coefficients

We present strongly convergent explicit and semi-implicit adaptive numerical schemes for systems of stiff stochastic differential equations (SDEs) where both the drift and diffusion are non-globally Lipschitz continuous. This stiffness may originate either from a linear operator in the drift, or from a perturbation of the nonlinear structures under discretisation, or both. Typical applications arise from the space discretisation of an SPDE, stochastic volatility models in finance, or certain ecological models. We prove that a timetepping strategy that adapts the stepsize based on the drift alone is sufficient to control growth and to obtain strong convergence with polynomial order. The order of strong convergence of our scheme is $(1-\varepsilon)/2$, for $\varepsilon\in(0,1)$, where $\varepsilon$ becomes arbitrarily small as the number of available finite moments for solutions of the SDE increases. Numerically, we compare the adaptive semi-implicit method to a fully drift implicit method, three tamed type methods and a truncated method. Our numerical results show that the adaptive semi-implicit method is well suited as a general purpose solver, is more robust than the explicit time stepping methods and more efficient than the drift implicit method.

math.NA↗

arXiv2018

On cubic difference equations with variable coefficients and fading stochastic perturbations

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$.

math.NA↗

arXiv2017

Adaptive timestepping for pathwise stability and positivity of strongly discretised nonlinear stochastic differential equations

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).

math.NA↗

arXiv2016

Adaptive timestepping strategies for nonlinear stochastic systems

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.

math.NA↗

arXiv2016

Stochastic stability analysis of a reduced galactic dynamo model with perturbed α-effect

We investigate the asymptotic behaviour of a reduced αΩ-dynamo model of magnetic field generation in spiral galaxies where fluctuation in the α-effect results in a system with state-dependent stochastic perturbations. By computing the upper Lyapunov exponent of the linearised model, we can identify regions of instability and stability in probability for the equilibrium of the nonlinear model; in this case the equilibrium solution corresponds to a magnetic field that has undergone catastrophic quenching. These regions are compared to regions of exponential mean-square stability and regions of sub- and super-criticality in the unperturbed linearised model. Prior analysis in the literature which focuses on these latter regions does not adequately address the corresponding transition in the nonlinear stochastic model. Finally we provide a visual representation of the influence of drift non-normality and perturbation intensity on these regions.

math.DS↗