John A. D. Appleby

Stochastic stabilisation of functional differential equations

In this paper we investigate the problem of stochastic stabilisation for a general nonlinear functional differential equation. Given an unstable functional differential equation dx(t)/dt=f(t,xt), we stochastically perturb it into a stochastic functional differential equation dX(t)=f(t,Xt)dt+ΣX(t)dB(t), where Σ is a matrix and B(t) a Brownian motion while Xt={X(t+θ):-τ⩽θ⩽0}. Under the condition that f satisfies the local Lipschitz condition and obeys the one-side linear bound, we show that if the time lag τ is sufficiently small, there are many matrices Σ for which the stochastic functional differential equation is almost surely exponentially stable while the corresponding functional differential equation dx(t)/dt=f(t,xt) may be unstable.

ON EXPONENTIAL ASYMPTOTIC STABILITY IN LINEAR VISCOELASTICITY

This paper establishes results concerning the exponential decay of strong solutions of a linear hyperbolic integrodifferential equation in Hilbert space. Rather than the more commonly used assumptions that the relaxation function is non-negative, decreasing and convex, dissipation is modelled by requiring that the dynamic viscosity be a positive function. This restriction has a firm thermodynamic basis. Frequency domain methods are employed.

NON-EXPONENTIAL STABILITY AND DECAY RATES IN NONLINEAR STOCHASTIC DIFFERENCE EQUATION WITH UNBOUNDED NOISES

We consider the stochastic difference equation where f and g are nonlinear, bounded functions, is a sequence of independent random variables, and h>0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution . We also show that, for some natural choices of f and g, the rate of decay of is approximately polynomial: there exists such that decays faster than but slower than , for any . It turns out that, if decays faster than as , the polynomial rate of decay can be established precisely: tends to a constant limit. On the other hand, if g does not decay quickly enough, the approximate decay rate is the best possible result.

On local stability for a nonlinear difference equation with a non-hyperbolic equilibrium and fading stochastic perturbations

We consider the nonlinear stochastic difference equation Here, (ξ n ) n ∈ ℕ is a sequence of independent random variables with zero mean and unit variance and with distribution functions F n . The function f : ℝ → ℝ is continuous, f(0) = 0, xf(x)>0 for x ≠ 0. We establish a condition on the noise intensity σ and the rate of decay of the tails of the distribution functions F n , under which the convergence of solutions to zero occurs with probability zero. If this condition does not hold, and f is bounded by a linear function with slope 2 − γ, for γ ∈ (0, 2), all solutions tend to zero a.s. On the other hand, if f grows more quickly than linear function with slope 2 + γ, for γ>0, the solutions tend to infinity in modulus with arbitrarily high probability, once the initial condition is chosen sufficiently large. Such equations can still demonstrate local stability; for a wide class of highly nonlinear f, it is shown that solutions tend to zero with arbitrarily high probability, once the initial condition is chosen appropriately. Results which elucidate the relationship between the rate of decay of the noise intensities and the rate of decay of the tails, and the necessary condition for stability, are presented. The connection with the asymptotic dynamical consistency of the system, when viewed as a discretisation of an Itô stochastic differential equation, is also explored.

Almost Sure Asymptotic Stability of Stochastic Volterra Integro-Differential Equations with Fading Perturbations

Abstract In this note, we address the question of how large a stochastic perturbation an asymptotically stable linear functional differential system can tolerate without losing the property of being pathwise asymptotically stable. In particular, we investigate noise perturbations that are either independent of the state or influenced by the current and past states. For perturbations independent of the state, we prove that the assumed rate of fading for the noise is optimal.

Preserving positivity in solutions of discretised stochastic differential equations

We consider the Euler discretisation of a scalar linear test equation with positive solutions and show for both strong and weak approximations that the probability of positivity over any finite interval of simulation tends to unity as the step size approaches zero. Although a.s. positivity in an approximation is impossible to achieve, we develop for the strong (Maruyama) approximation an asymptotic estimate of the number of mesh points required for positivity as our tolerance of non-positive trajectories tends to zero, and examine the effectiveness of this estimate in the context of practical numerical simulation. We show how this analysis generalises to equations with a drift coefficient that may display a high level of nonlinearity, but which must be linearly bounded from below (i.e. when acting towards zero), and a linearly bounded diffusion coefficient. Finally, in the linear case we develop a refined asymptotic estimate that is more useful as an a priori guide to the number of mesh points required to produce positive approximations with a given probability.

Non-exponential stability of scalar stochastic Volterra equations

We study convergence rates to zero of solutions of the scalar equationwhere f, g, h are globally Lipschitz, xg(x)>0 for nonzero x, and k is continuous, integrable, positive and limt-->[infinity] k(t-s)/k(t)=1, for s>0. Thenfor nontrivial solutions satisfying limt-->[infinity] X(t)=0 on A, a set of positive probability.

Bubbles and crashes in a Black–Scholes model with delay

This paper studies the asymptotic behaviour of an affine stochastic functional differential equation modelling the evolution of the cumulative return of a risky security. In the model, the traders of the security determine their investment strategy by comparing short- and long-run moving averages of the security’s returns. We show that the cumulative returns either obey the law of the iterated logarithm, but have dependent increments, or exhibit asymptotic behaviour that can be interpreted as a runaway bubble or crash.

ON THE ALMOST SURE RUNNING MAXIMA OF SOLUTIONS OF AFFINE STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

This paper studies the large fluctuations of solutions of scalar and finite-dimensional affine stochastic functional differential equations with finite memory as well as related nonlinear equations. We find conditions under which the exact almost sure growth rate of the running maximum of each component of the system can be determined, both for affine and nonlinear equations. The proofs exploit the fact that an exponentially decaying fundamental solution of the underlying deterministic equation is sufficient to ensure that the solution of the affine equation converges to a stationary Gaussian process.

Geometric Brownian motion with delay: mean square characterisation

A geometric Brownian motion with delay is the solution of a stochastic differential equation where the drift and diffusion coefficient depend linearly on the past of the solution, i.e. a linear stochastic functional differential equation. In this work the asymptotic behavior in mean square of a geometric Brownian motion with delay is completely characterized by a sufficient and necessary condition in terms of the drift and diffusion coefficient.