We present a simulation-based method for solving discrete-time portfolio choice problems involving non-standard preferences, a large number of assets with arbitrary return distribution, and, most importantly, a large number of state variables with potentially path-dependent or non-stationary dynamics. The method is flexible enough to accommodate intermediate consumption, portfolio constraints, parameter and model uncertainty, and learning. We first establish the properties of the method for the portfolio choice between a stock index and cash when the stock returns are either iid or predictable by the dividend yield. We then explore the problem of an investor who takes into account the predictability of returns but is uncertain about the parameters of the data generating process. The investor chooses the portfolio anticipating that future data realizations will contain useful information to learn about the true parameter values.Except for the nonparametric approach of Brandt (1999), they assume unrealistically simple return distributions. All of the methods rely on CRRA preferences, or its extension by Epstein and Zin (1989), to eliminate the dependence of the portfolio policies on wealth and thereby make the problem path-independent. Most importantly, the methods cannot handle the large number of state variables with complicated dynamics which arise in many realistic portfolio choice problems. A partial exception is Campbell, Chan, and Viceira (2003), who use log-linearization to solve a problem with many state variables but linear dynamics.Our simulation method overcomes these limitations. The first step of the method entails simulating a large number of hypothetical sample paths of asset returns and state variables. We simulate these paths from the known, estimated, or bootstrapped joint dynamics of the returns and state variables. Alternatively, we simulate the sample paths from the investor's posterior belief about the joint distribution of the returns and state variables to incorporate parameter and model uncertainty in a Bayesian context. The key feature of the simulation stage is that the joint dynamics of the asset returns and state variables can be high-dimensional, arbitrarily complicated, path-dependent, and even non-stationary.Given the set of simulated paths of returns and state variables, we solve for the optimal portfolio (and consumption) policies recursively in a standard dynamic programming fashion. Starting one period before the end of the investor's horizon, at time T − 1, we compute for each simulated path the portfolio weights that maximize a Taylor series expansion of the investor's expected utility. This approximated problem has a straightforward (semi-) closedform solution involving conditional (on the state variables) moments of the utility function, its derivatives, and the asset returns. We compute these conditional moments with leastsquares regressions of the realized utility, its derivatives, and the asset returns at time T on basis functions of the realizations of the state variable...

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This paper provides a methodology for computing optimal filtering distributions in discretely observed continuous-time jump-diffusion models. Although it has received little attention, the filtering distribution is useful for estimating latent states, forecasting volatility and returns, computing model diagnostics such as likelihood ratios, and parameter estimation. Our approach combines time-discretization schemes with Monte Carlo methods to compute the optimal filtering distribution. Our approach is very general, applying in multivariate jump-diffusion models with nonlinear characteristics and even non-analytic observation equations, such as those that arise when option prices are available. We provide a detailed analysis of the performance of the filter, and analyze four applications: disentangling jumps from stochastic volatility, forecasting realized volatility, likelihood based model comparison, and filtering using both option prices and underlying returns.

We propose a model for non-stationary spatiotemporal data. To account for spatial variability, we model the mean function at each time period as a locally weighted mixture of linear regressions. To incorporate temporal variation, we allow the regression coef®cients to change through time. The model is cast in a Gaussian state space framework, which allows us to include temporal components such as trends, seasonal effects and autoregressions, and permits a fast implementation and full probabilistic inference for the parameters, interpolations and forecasts. To illustrate the model, we apply it to two large environmental data sets: tropical rainfall levels and Atlantic Ocean temperatures.

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