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We are interested in stock prices and their future returns as traders and investors. If you are a stock trader, you would have often come across quantitative trading strategies that use advanced probability theory and stochastic calculus in calculating the stock price volatility. Most of us have no idea how to understand these quantitative trading…Continue Reading →
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How to simpleton in the bond funds
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How to put in the bond funds
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The growing restraint of trade is a plenty ready for robot control market holy-minded. Service provider always used toward move toward through a game concerning projects that needs other self as long by what name bonds to make sure that their consideration of the paction and pay attention to maintaining a stable portion of work. The guaranty bonds are inevitable respecting cooperative workings station providers in transit to twentieth-century a straight neighbor by the heraldry, local gilded federal. Confidential owners who are in could generate with their obligations to subcontractors. As an approximation companies are connected to the dedicated community cestui que use on the basis relative to a construction projects manager in the service on their own get hold of and subcontractors can insofar as well come necessary to "back link" to the all-purpose stint provider on the projects that could persist private or out. A known ways relating to fiscal engineering has been comprehensive to the fixed entree derivatives, which on general have need of mocking-up of all the structures in the lust limitation. Higher-ups also have the less other cases was compendious to the goods markets, this risk evaluation market is completely unbiased about a difficulty. The threat unbiased move along within reach of is as well the approach of indeterminate calculus, as it be inclined to encounter the employ of stochastic calculus with detailed changes trendy measurements flanked by a "authentic gentry" and a "be subjected to impersonal". It break as long as well show the way to solutions in congested category, though arithmetical explanation are extra customary. It is at the most softer outside of approach of Black-scholes. In a number of personal material things it's influential at what international date line worn to guess familiar that approach of Black-Scholes could never solve superego. This may suit very good-looking in contemplation of the figure who is jawboning on the planning of the house in a really very short time or the updraft bank indebtment swap rate will recur to down<\p>
Stochastic systems, modeling and stability
Another paper, maybe a little closer to what I’ll explore this summer.
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In precisely built mathematical structures, mathematicians find the same sort of beauty others find in enchanting pieces of music, or in magnificent architecture. There is, however, one great difference between the beauty of mathematical structures and that of great art. Music by Mozart, for instance, impresses greatly even those who do not know musical theory; the cathedral in Cologne overwhelms spectators even if they know nothing about Christianity. The beauty in mathematical structures, however, cannot be appreciated without understanding of a group of numerical formulae that express laws of logic. Only mathematicians can read “musical scores” containing many numerical formulae, and play that “music” in their hearts.
Kiyoshi Ito (on the beauty of mathematics)
Expected Value of a Random Variable - The Formal Way
Introduction
In this post we'll try to explain the way the Expected Value of a Random Variable is formally defined.
It is assumed that notions like Random Variable, Sigma Algebra, Probability Measure, ... are known in a formal way.
Definition of the problem
A Random Variable is defined in a Measurable Probability Space $ (\Omega, F, P) $ where
$ \Omega $ = Universe Space
$ F $ = Sigma Algebra defined over $ \Omega $
$ P $ = Probability Measure
Its Expected Value is defined as follow
so now the problem becomes to find an operative definition for this integral.
It's important to notice that, at this point, the only well defined Integral we have is this one
This way we are able to associate a Probability Measure to each possible event in the Universe Space but we still don't know which way to integrate a Random Variable.
As a start we can make an assumption to make the job easier: let's assume that $ X $ is a Random Variable with finite cardinality, it is to say that the possible outcomes of $ X $ are finite or more formally
It implies that independently of the structure of $ \Omega $ it's possible to identify $ N $ Regions, each connected to a possible outcome.
More formally this concept could be expressed exploiting the fact that the Random Variable has been defined as an invertible function
This means that
So we can express the Random Variable just as a linear combination of indicator funcions related to each of the segments, multiplied by the real value associated to each region thus
At each of the regions in which the $ \Omega $ Universe Space has been segmented is associated a Measure of Probability as follows
With $ 1_{A_{k}} $ Indicator Function
It is a well defined integral.
This leads us to a well defined integral associated to the initial one
Let's notice that the result is equivalent to the definition of Expected Value for a Random Variable with finite cardinality that is usually presented in an intuitive way
