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### Introduction to Sinusoidal Processes

In the simplest of terms, a sinusoidal function describes a smooth, regular oscillation. Very few macroscopic processes in nature exhibit a pure sin(x) relationship, because there are usually additional factors to an equation that will prohibit a function from occurring in a regular pattern. This is due to extraneous factors that will interfere, damp, or even cause a function to oscillate without bound. As a result, the application of this theory is not without strict limits. Nonetheless, on a macroeconomic scale, and over long enough timeframes, certain metrics can be modeled or approximated by sinusoidal functions, or more completely, f(t)=A*sin(w*t+phaseshift)+mean(t).

### Stochastic Processes

On the other hand, instantaneous changes in the market can very rarely be modeled by this type of function, and more often exhibit Brownian type motion, typical of a martingale. The distinctive difference in the two models is that the outcome of a martingale, or stochastic process, will have an equal chance of occurring at any point in time (in other words, future outcomes do not depend on past events), whereas a sinusoidal or exponential function will, by definition, oscillate about a mean, or increase with time at a predictable rate, respectively.

The following is an opinion at MVS: Although on an instantaneous timeframe (and over the course of several hours) the market tends to behave like a Brownian motion, financial news data and market sentiment tend to govern its movement over a longer timeframe. On a macroscopic scale and over a timeframe greater than thirty days, market movement is governed more by earnings and financials than by market sentiment or stochastics. At this point, the effects of stochastics tend to be averaged out and directionality can be observed. Modeling this directionality is a central tenant of the business cycle. Creating a model from two underlying microeconomic trends is similar to the addition of two wave equations, which results in an interference pattern. More precisely, this illustrates how different cyclical functions complement or detract from each other in order to generate a final market curve.

### The Hybrid Model

This generates the final application of the Business Cycle: combining a Wiener Process with predictable sinusoidal and exponential processes to yield a probability density function (PDF) of the future value. The graph below depicts this hybrid model, where the variance (or spread) of the cumulative distribution function increases as the result of the random walk phenomenon. At the same time however, the CDF is subjected to a "force" that changes its mean value as a function of time. This secondary effect of the hybrid equation is the result of new information that would result in the dramatic change of future earnings. To keep the sequential graph simple, the graph is depicting "linear momentum," where the mean changes at a constant rate with respect to time. In a real situation, however, this market "momentum" is seldom linear and the mean value is better modeled by a differential equation.