References: Shreve Secs. 1.1, 1.2 Gardiner Secs. 2.1, 2.2 Grigoriu Secs , 2.10

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1 2011 Page 1 Random Variables Monday, January 31, :09 PM References: Shreve Secs. 1.1, 1.2 Gardiner Secs. 2.1, 2.2 Grigoriu Secs , 2.10 We will introduce the measure-theoretic formulation for random variables, which has the virtue of being a relatively clean way to represent the relationships between the variables in a complex system where enumerating all random variables (as in an undergrad class) could be cumbersome. Random variable is a measurable function from the sample space to a state space: Examples: A random variable could be the Exxon stock price at the close of trading today, and the corresponding state space would be, say, the nonnegative real numbers. Or we could take as a random variable the collection of all closing prices for S&P 500 stocks, and the corresponding state space would be Or we could take as a random variable the price trajectory of a given stock over tomorrow's trading day. For discrete time, the state space would be a space of sequences parameterized by the values of the stock price at every moment of time Or if one wants to treat time continuously, then one introduces "path spaces" as the state space for such trajectories. Which path space to choose is somewhat technical, but here's some examples: : state space of continuous functions on an interval of 7 (hours) One reason why one may choose not to use a path space of continuous functions is to model price shocks If the price shocks happen faster than I can react to them, then in practice they behave like

2 2011 Page 2 discontinuities as far as my financial considerations are concerned, so we should then really use a path space that allows discontinuities. for example, the space of piecewise continuous functions Or one could also work with Sobolev spaces, which are fancy Hilbert spaces for functions with varying degrees of smoothness What does measurable function mean? The state space S should also be equipped with its own sigma-algebra of measurable subsets; call it Typically this sigma-algebra is the Borel or Lebesgue sigma-algebra if the state space is Euclidean. Let's call the sigma-algebra of measurable subsets on the sample space For to be a measurable function means that the inverse image of any measurable set in state space is a measurable set in sample space. The reason we demand random variables are measurable functions on sample space is so the probability distribution for the random variable is well-defined mathematically: is a measure on the state space of X, and is defined in

3 2011 Page 3 is a measure on the state space of X, and is defined in terms of the original probability measure on the sample space by: So what does this mean in practice? A probability model for a complex system involves a generally high-dimensional probability triplet which is usually only implicitly defined and is rather abstract. Random variables can be thought of as more concrete reductions of the probability model into more managable (lower-dimensional) probability triplets One can indeed take this latter triplet as its own probability space. Working with random variables is more tractable, but usually random variables involve reductions of information, so one wants to keep the sample space in view to allow study of relationships between different random variables. Let's now focus on how to work with random variables. The raw prescription of a random variable in terms of the triplet is still a bit complicated because the probability distribution is a measure, meaning a function of sets. Can we relate the probability distribution to a simpler function? For concreteness, let's consider one-dimensional random variables

4 2011 Page 4 Let us further consider random variables X which are (absolutely) continuous with respect to Lebesgue measure: This is what one means in practice by a continuously distributed random variable, like usually, position in space or the value of an asset. This will be the main focus in this class. Absolute continuity of a probability distribution implies the existence of a probability density function (PDF) What is the intuitive meaning of the PDF? To understand this, consider a nice set which is a small neighborhood of x assuming the PDF is continuous (which it need not be). In particular, if we take

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