We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles

A central question in economics is how to discount future cash flows to obtain today's value of an asset. For instance, total wealth is the price of a claim to all future consumption (Lucas 1978). Similarly, the value of the aggregate stock market equals the sum of discounted future dividend payments (Gordon 1962). The major ity of the equity market literature has focused on the dynamics of the value of the aggregate stock market. However, in addition to studying the value of the sum of discounted dividends, exploring the properties of the individual terms in the sum, also called dividend strips, provides us with a lot of information about the way stock prices are formed. Analogous to zero-coupon bonds, which contain information about discount rates at different horizons for fixed income securities, having infor mation on dividend strips informs us about discount rates of risky cash flows at dif ferent horizons. Studying dividend strips can therefore improve our understanding of investors' risk preferences and the endowment or technology process in macro finance models. This paper is the first to empirically measure the prices of dividend strips to study the term structure of the equity premium. Our approach requires only no-arbitrage relations and does not rely on a specific model. We shed new light on the composition of the equity risk premium. The equity premium puzzle, identified by Mehra and Prescott (1985), Hansen and Singleton (1982), and Hansen and Singleton (1983), states that, for plausible values of the risk aversion coefficient, the difference in the expected rate of return on the stock market and the riskless rate of interest is too large, given the observed small variance in the growth rate in per capita consumption. When decomposing the index into dividend strips, a natural question that arises is whether dividends at different horizons contrib ute equally to the equity risk premium or whether either short- or long-term dividends contribute proportionally more than the other. We find that short-term dividends have

/pdf/on-the-timing-and-pricing-of-dividends-uf12chvisu.pdf

On the timing and pricing of dividends

We study a newly designed survey administered to a large panel of wealthy retail investors. The survey elicits beliefs that are important for macroeconomics and finance, and matches respondents with administrative data on their portfolio composition, their trading activity, and their login behavior. We establish five facts in these data. (i) Beliefs are reflected in portfolio allocations. The sensitivity of portfolios to beliefs is small on average, but varies significantly with investor wealth, attention, trading frequency, and confidence. (ii) Belief changes do not predict when investors trade, but conditional on trading, they affect both the direction and the magnitude of trades. (iii) Beliefs are mostly characterized by large and persistent individual heterogeneity. Demographic characteristics explain only a small part of why some individuals are optimistic and some are pessimistic. (iv) Expected cash flow growth and expected returns are positively related, both within and across investors. (v) Expected returns and the subjective probability of rare disasters are negatively related, both within and across investors. These five facts provide useful guidance for the design of macro-finance models.

/pdf/five-facts-about-beliefs-and-portfolios-1dipyzc5ry.pdf

Five Facts about Beliefs and Portfolios

We study the evolution of individual labor earnings over the life cycle, using a large panel data set of earnings histories drawn from US administrative records Using fully nonparametric methods, our analysis reaches two broad conclusions First, earnings shocks display substantial deviations from lognormality — the standard assumption in the literature on incomplete markets In particular, earnings shocks display strong negative skewness and extremely high kurtosis — as high as 30 compared with 3 for a Gaussian distribution The high kurtosis implies that, in a given year, most individuals experience very small earnings shocks, and a small but non-negligible number experience very large shocks Second, these statistical properties vary significantly both over the life cycle and with the earnings level of individuals We also estimate impulse response functions of earnings shocks and find important asymmetries: Positive shocks to high-income individuals are quite transitory, whereas negative shocks are very persistent; the opposite is true for low-income individuals Finally, we use these rich sets of moments to estimate econometric processes with increasing generality to capture these salient features of earnings dynamics

/pdf/what-do-data-on-millions-of-u-s-workers-reveal-about-life-tuyrs6auna.pdf

What Do Data on Millions of U.S. Workers Reveal About Life-Cycle Earnings Risk?

Misspecified Recovery: Misspecified Recovery

I study the long-run behavior of an economy with two types of agents who differ in their beliefs and are endowed with homothetic recursive preferences of the Duffie-Epstein-Zin type. Contrary to models with separable preferences in which the wealth of agents with incorrect beliefs vanishes in the long run, recursive preference specifications also lead to equilibria where both agents survive, or more incorrect agents dominate. In this respect, the market selection hypothesis is not robust to deviations from separability. I derive analytical conditions for the existence of nondegenerate long-run equilibria in which agents with differently accurate beliefs coexist in the long run, and show that these equilibria exist for broad ranges of plausible parameterizations when risk aversion is larger than the inverse of the intertemporal elasticity of substitution. The results highlight a crucial interaction between risk sharing, speculative behavior and the consumption-saving choice of agents with heterogeneous beliefs.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID2902280_code908611.pdf?abstractid=2023501&mirid=1

Survival and Long-Run Dynamics with Heterogeneous Beliefs under Recursive Preferences

I analytically characterize the long-run behavior of an economy with two types of agents who differ in their beliefs and are endowed with homothetic recursive preferences. Agents with more incorrec...

/pdf/survival-and-long-run-dynamics-with-heterogeneous-beliefs-tqnexkn2sn.pdf

Asset prices contain information about the probability distribution of future states and the stochastic discounting of those states as used by investors To better understand the challenge in distinguishing investors' beliefs from risk-adjusted discounting, we use Perron-Frobenius Theory to isolate a positive martingale component of the stochastic discount factor process This component recovers a probability measure that absorbs long-term risk adjustments When the martingale is not degenerate, surmising that this recovered probability captures investors' beliefs distorts inference about risk-return tradeoffs Stochastic discount factors in many structural models of asset prices have empirically relevant martingale components

Misspecified Recovery

We present a novel approach to depicting asset pricing dynamics by characterizing shock exposures and prices for alternative investment horizons. We quantify the shock exposures in terms of elasticities that measure the impact of a current shock on future cash-flow growth. The elasticities are designed to accommodate nonlinearities in the stochastic evolution modeled as a Markov process. Stochastic growth in the underlying macroeconomy and stochastic discounting in the representation of asset values are central ingredients in our investigation. We provide elasticity calculations in a series of examples featuring consumption externalities, recursive utility, and jump risk.

Risk-price dynamics

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Misspecified Recovery: Misspecified Recovery

Survival and Long-Run Dynamics with Heterogeneous Beliefs under Recursive Preferences

Survival and Long-Run Dynamics with Heterogeneous Beliefs under Recursive Preferences

Misspecified Recovery

Risk-price dynamics