Almost sure and moment stability for linear ito equations

Abstract

The asymptotic behavior of the linear stochastic differential equation in R^d

$$dx = Ax dt + \mathop \Sigma \limits_{i = 1}^m B_i x o dW_i (t), x(O) = x_O \ne O,$$

is studied. It is known (see [2]) in these Proceedings) that the projection of the solution x(t;x_o) onto the unit sphere has a unique invariant probability, while

$$\lambda = \mathop {\lim }\limits_{t \to \infty } \tfrac{1}{t} \log |x(t;x_O )|$$

exists a.s. and is essentially independent of chance and of x_o. Here we prove that

$$g(p) = \mathop {\lim }\limits_{t \to \infty } \tfrac{1}{t} \log E|x(t;x_O )|^P , p \in R,$$

exists and is independent of x_o. Further, g: R → R is convex and analytic with g(p)/p increasing (to γ, say) with g(O)=O and g' (O)=λ. The cases γ<∞ and γ=∞ are characterized. This answers the question of when sample stability (λ<O) implies moment stability (g(p)<O) for all p>O.

In case trace A = trace B_i=O for all i we have g(−d)=O, this enabling us to characterize the cases λ=O and λ>O by a simple criterion.