We consider the dynamics of a nonlinear partial differential equation perturbed by additive noise. Assuming that the underlying deterministic equation has an unstable equilibrium, we show that the nonlinear stochastic partial differential equation exhibits essentially linear dynamics far from equilibrium. More precisely, we show that most trajectories starting at the unstable equilibrium are driven away in two stages. After passing through a cylindrical region, most trajectories diverge from the deterministic equilibrium through a cone-shaped region which is centered around a finite-dimensional subspace corresponding to strongly unstable eigenfunctions of the linearized equation, and on which the influence of the nonlinearity is surprisingly small. This abstract result is then applied to explain spinodal decomposition in the stochastic Cahn-Hilliard-Cook equation on a domain $G$ . This equation depends on a small interaction parameter $\varepsilon >0$ , and one is generally interested in asymptotic results as $\varepsilon \to 0$ . Specifically, we show that linear behavior dominates the dynamics up to distances from the deterministic equilibrium which can reach $\varepsilon ^{-2+\dim G / 2}$ with respect to the $H^2(G)$ -norm.

We consider a stochastic partial differential equation (Swift-Hohenberg equation) on the real axis with periodic boundary conditions that arises in pattern formation. If the trivial solution is near criticality, and if the stochastic forcing and the deterministic (in)stability are of a comparable magnitude, a so called stochastic Landau equation can be derived in order to describe the dynamics of the bifurcating solutions. Here we establish attractivity and approximation results for this equation.

Utility and risk are two often competing measurements on the investment success. We show that efficient trade-off between these two measurements for investment portfolios happens, in general, on a convex curve in the two-dimensional space of utility and risk. This is a rather general pattern. The modern portfolio theory of Markowitz (1959) and the capital market pricing model Sharpe (1964), are special cases of our general framework when the risk measure is taken to be the standard deviation and the utility function is the identity mapping. Using our general framework, we also recover and extend the results in Rockafellar et al. (2006), which were already an extension of the capital market pricing model to allow for the use of more general deviation measures. This generalized capital asset pricing model also applies to e.g., when an approximation of the maximum drawdown is considered as a risk measure. Furthermore, the consideration of a general utility function allows for going beyond the “additive” performance measure to a “multiplicative” one of cumulative returns by using the log utility. As a result, the growth optimal portfolio theory Lintner (1965) and the leverage space portfolio theory Vince (2009) can also be understood and enhanced under our general framework. Thus, this general framework allows a unification of several important existing portfolio theories and goes far beyond. For simplicity of presentation, we phrase all for a finite underlying probability space and a one period market model, but generalizations to more complex structures are straightforward.