Intuitive Methods in Economic Mathematics: Probability and Statistics Reading Notes

Summary

Economic mathematics is the study of mathematical methods of analysis in economics. It refers to the application of mathematical methods to economics, the study of resource allocation and decision-making by society and individuals, in order to model and analyze mathematically in order to gain deeper insights.

Economic mathematics is used to model theories of supply and demand or by applying mathematical approaches to a variety of other economic problems, such as pricing, cost reduction, production optimization, investment, portfolio theory, and financial market theory.

Economic mathematics may require advanced mathematical techniques and statistical methods, requiring strong knowledge of both mathematics and economics. Those approaches can also provide hints for machine learning in general. Furthermore, combining theoretical insights in economics with quantitative analysis can lead to more valid conclusions and decisions.

In this section, we will address this economic mathematics based on “Intuitive Methods in Economic Mathematics: Probability and Statistics“.

This issue discusses the reading notes.

Intuitive Methods in Economic Mathematics: Probability and Statistics Reading Notes

In this book, you will intuitively understand the essence of highly developed economic mathematics with the help of 70 figures and graphs. In this book, “Probability and Statistics Edition,” you will learn the mechanism by which the normal distribution curve is formed, touch upon the wonder of the central limit theorem, which is considered the most important principle in probability and statistics theory, and learn Black-Scholes theory as an educational tool.

Chapter 1: Elementary

1. fundamental ideas for understanding probability statistics

2. obstacles that appear in learning probability statistics
God does not roll the dice
God’s fingerprints” were found in probability statistics during the Gaussian era
What kind of thought has been used by human beings for “error correction” until now?
Gauss studied the “error theory
What is the mechanism of error in astronomical observations?
Traditional Methods to Maximize Hit Efficiency
How have people corrected errors?
Correct the fleet’s batteries until you “get oleander” and leave the rest to luck
The first important point
The nature of error consists of two parts
Error that goes out in a certain direction
The object of correction
Errors that occur equally in the + and – directions
Errors governed purely by probability theory
Second important point
Errors in this world are created in a multistage structure
Error is a message game
Simplification of the problem
Normalization (±1 error)
Transposition of order
Basic idea of probability statistics
The “Creator’s Conveyor Belt” Creates Individual Differences
What distribution pattern does the variability of things in the world follow?
Why does the variation of things in the world follow a normal distribution?

2. how the probability statistics of our world was established

The greatest crossroads in the theory of probability statistics
At its core, is there a fingerprint of God or no fingerprint of God?
Complex in form but expressed in simple mathematical equations
The mechanism that gives rise to the normal distribution curve
The mechanism of a simple process in a multi-stage structure
How statistics evolved in the parallel world of “without God’s fingerprints
Two sets of data “tell the whole story”
How many balls are in any position can be determined with two multiplications (length of the base and the center position (average))
The “standard deviation” of a surprisingly simple parallel world
The standard deviation corresponds to the width of the base
How to attack probability statistics with this method
What does “deviation” mean?
Determine how much you deviate from the mean
The size of the deviation/length of the base
Parallel World formula that exactly matches the textbook
Calculate with the center value as 50
Standard deviation in the real world
D just became sigma
What is “deviation 70” in a parallel world?
How to use “deviation” in a parallel world
If the standard deviation is 60 or higher, you will be in the top 1/8th, and if the competition is about 8 times higher, you will be in the passing line.
Standard deviation in our world
How to specify the “width” of the normal distribution curve
Normal distribution spreads to infinity.
The “width of two special points” appearing there
Since the overall width cannot be defined, it is defined by the distance between the inflection points
Standard deviation σ

3. supplementary basic knowledge

Actual normal distribution curve
When the number of spheres is small, at most a few dozen, the pattern is a “binomial distribution
Rough shape of the curve
Shape 2
Why does “squared” appear in the calculation of standard deviation?
How to find the base d in a parallel world
When each data is xi and the center line of the triangle is 0m
Standard deviation in the real world
Reference: How to find the formula for standard deviation in a parallel world
Contrast with the standard method of finding the standard deviation
Conclusion

Chapter 2 Intermediate

1. essence of the least squares method

Essence of the Least Squares Method
When data obtained from a survey or observation are scattered with errors and the true value is not known, the least-squares method is used.
A method for determining the true value and minimizing the error
To deduce the location of the center line of the normal distribution curve hidden behind the data
Parallel World’s “Least (Absolute) Squares” method
Estimating the position of the triangle hidden behind the distribution graph of the data
The center line is the real data
How to deduce the position of the center line
Case 1
Case 2
Case 3
Why square instead of one?
Using squares gives better accuracy
Parallel Worlds and Normal Distribution
Conclusion

2. the wonder of the central limit theorem

The Deep World Guided by the Idea of Normal Distribution
There are various probability distributions in the world
When they are mixed and synthesized, we get a beautiful normal distribution
Mechanism of the Central Limit Theorem
Constituent Factors of Variability
The part that appears in one direction and can be corrected by humans
The part of the probability that is subject to bias
The regularity of the bias can produce any number of distribution patterns
Part that appears in left-right (+-) direction and can only be handled probabilistically
Normal distribution
Example
Example 1
Example 2
Example 3
Various statistical distributions arise from variations of the normal distribution
Only binomial distribution can be considered as the initial state of normal distribution
Infinite times binomial distribution becomes normal distribution
The binomial distribution is the normal distribution reduced to a finite number of times.
Examples of Various Distributions
Poisson distribution
A distribution that examines how an event that occurs only rarely at a single time would curve when viewed over a very long time period.
A distribution of a rare phenomenon with a probability of only about 1 in 10,000 per occurrence over a period of years.
Example: traffic accident
In general, the probability of a construction accident occurring is quite low, but if the time scale is taken long enough, it appears in a visible magnitude.
A curve in the form of a large scale transformation of both the per-occurrence probability and the time scale of a normal distribution.
Predistribution
Biased with a certain regularity from the structural side
The Central Limit Theorem, which was rather natural
The Central Limit Theorem was originally conceived as a story of binomial to normal distribution
How gratifying it is for economic forecasting
Adding a wide variety of probability distributions returns to a simple normal distribution, just as adding light gives white.
The “last laugh” is the normal distribution

3. brownian motion and black scholes theory

Brownian motion and normal distribution
Drunken Walk” = Random Walk
Normal distribution extended to a 2-dimensional plane
Brownian motion is a variation of the normal distribution
Clearing up some misconceptions
Brownian motion diffuses spatially over time
Where does it correspond to the normal distribution?
The standard deviation σ expands with time
In a real system, the diffusion is finite, so it looks a little different from the normal distribution.
The expansion of the radius of diffusion in Brownian motion is caused by an increase in the “number of zigzags” rather than time
Diffusion cannot occur without the basic setting that “the number of zigzags increases in proportion to time.
In the example of a pachinko machine, the number of nails is fixed, so diffusion is limited.
If the number of nails increases, diffusion will occur.
What is the temporal increase in coin tossing?
How many times can you toss a coin with +1 point for the front side of the coin and -1 point for the back side of the coin, and how many points will the total score be?
Close to zero if you do it long enough
What is the difference from the model of diffusion of Brownian motion?
A set of 50 tosses is considered a set.
In Brownian motion, imagine an increasing number of sets (500, 5000)
If the prize is absolute, it increases steadily over time
Black-Scholes theory
Two types of offsetting mechanisms
If you collect a large number of samples, the variation cancels out to zero
There are two types of countervailing mechanisms
Depends on the “numerical size” variety of the sample itself
Variations in the size of the numbers, such as “3” or “7”, cancel out each other’s unevenness
Two-curvature offsetting of + and –
Variety of gas molecules
In the absolute value game, the second mechanism is turned off, so it increases
Science-based approach How does that change over time?
Intuitive picture of it being √t
Temporal diffusion of position is proportional to the square root of time (√t)
Mechanism of Brownian motion
First on a circle of radius r
Repeats zigzag motion on parallel path r
Moves within a radius r
When we consider a circle of radius r, the outer fan is farther away (+) and the inner fan is closer to the outer fan (-).
Since the outer fan is larger in area, it is more probable to move outward
Spreading outward
As r increases, the r/R ratio decreases
Movement becomes slower
As the circle of R becomes larger, the ratio of outside to inside is closer to 1
Outward spreading becomes slower
From a vectorial viewpoint, the mechanism of seepage is concentrated in (1), (2), and the average of (1) and (2).
The seepage mechanism is concentrated in (3).
It can be simplified if only (3) and (4) are repeated.
Divide into two groups: (1) and (2) and (3) and (4).
Groups (1) and (2) are +/- zero
Observe the vector change in only ③.
R1 = √(R02 + r2)
R12 = R02 + r2
R22 = R12 + r2 = R02 + r2 + r2
Rt after T seconds is
Rt2 = R02 +tr2
Rt =√t ・r
proportional to √t
Diffusion has the same form in 2D and 3D
Mapping image
When talking about one dimension (absolute value) in the economy
How to find the absolute value of a number
Square the number once and take its square root
Expand the absolute value by √t
Example: Toss a coin 3 times
A = a1 + a2 + a3
|A| = √(a12 + a22 +a32)
|A| = √(a12 + a22 + a32 +2(a1a2 + a2a3 + a3a1))
when a is +a or -a.
The first three are all a2
The last three are + or – in combination, and randomness is mixed in.
The same is true when N is increased
As N increases, the random term is close to zero
Therefore|A| = √t-a
Wiener process
Relation to the Central Limit Theorem
Connecting zigzags
For N zigzags, the number of zigzags is √n times larger
Standard deviation is √n・σ
Zigzag = many probability distributions
Standard deviation is √n・σ
Approach from the humanities side – How to create an absolute value game
If you can find the absolute value game in the real world of stocks and bonds, your profits will increase unilaterally.
How to avoid losing money on wartime bonds
Risk hedging
Decide how to sell or buy something that is linked, rather than doing it statically
Choice for aircraft manufacturers
If the linkage is an upward curve, you can create an “absolute value game”
Basic principle that risk disappears
Risk-free portfolio
The key is “curvilinear linkages”
Linkage is not an unknown quantity
If the stochastic part can be divided into a mechanism-driven part and a pure stochastic part
The stochastic part can be set up as an absolute value game
Essence of Black-Scholes Theory
Central Limit Theorem says that if you collect a lot of data, the distribution will be close to the normal distribution
Understanding this theory in a panorama diagram
Organization of understanding
Trend” is the part that moves in a certain direction
Polarity” is the part that moves randomly
How it is used in practice

Black-Scholes theory as an educational tool

Necessity of knowing this as general knowledge
Discomfort of this story and this similarity
Agricultural Economics
Wealth cannot be created if the weather is bad
Trade Economy
Wealth is created if there is a price difference between regions even if the weather is bad
Commercial vs. agricultural countries
The Surprising Role of Black-Scholes Theory in the “End of Capitalism”
A New Vision of Capitalism in the Future
Will there be a shift to a “polarity-type” economy?
Capitalism and the Edo Economy as Seen through Black-Scholes Theory
Islamic Finance as Viewed by Black-Scholes Theory
Intruding Mechanism of Exponential Growth
Linear or Exponential?
The Real Value of This Story

Chapter 3: Advanced

1. Ito’s Lemma and Stochastic Differential Equations

Stochastic Differential Equations Inheriting the Basic Idea of Gauss
Stochastic differential equation
𝒅𝓍 = 𝑨𝒅𝓉 + 𝑩𝒅𝓌
X(t) = A-t + Bw(t)
The motion of an object can be divided into two parts: the part that moves in a constant direction and can be predicted by humans, and the part that moves randomly in ± either direction and can only be bounded by probability
𝒅𝓍
Amount of micro-change
𝑨𝒅𝓉
Movement in micro time 𝒅𝓉
𝑩𝒅𝓌
Follows the pattern of a normal distribution
What exactly does this study want to do?
x is an independent variable, there is a quantity y that moves under the influence of x, and we want to know how y behaves over time
Y = F(x)
dy = F(dx)
The time behavior of the independent variable x is dx =Adt
The modification when a zigzag motion is included
dx = A1dt + B1dw
dy = F(dx)
dy = F(A1dt + B1dw)
If we remove F and use dt, the equation becomes complicated and dt and dw get mixed up.
Ito’s Lemma is the one that neatly divides the two.
Purpose of Ito’s Lemma from a humanities perspective
dx = A1dt + B1dw
A1dt
For stocks, the part of the trend that analysts can predict as a “trend”
B1dw
The part of “polarity” that moves randomly
Taylor expansion in economics
Taylor series
F(x) = a0 + a1x + a2x2 +…
a1 and a2 are constants obtained by differentiating F once and twice, respectively
If the value of a function F at x0 is F(x0)
The value at the point “x0 +dx”, which is shifted by dx from there, is
Since dx is minute, parts such as dx2 and dx3 are close to zero and can be ignored.
Cannot be used if F is unknown
The trick of Taylor expansion
Ignore terms after dx2 and dx3
Confirmation by figure
The value of F at the point x0 is F(x0)
Value of F at x0+dx shifted by dx F(x0+dx)
The height of the triangle is the height of the base dx multiplied by the slope ratio
Slope ratio
dF(x0)/dx
Image of block stacking of a multi-stage Taylor series
Which order terms can be discarded?
In actual calculations, terms of the third order (dx3) and beyond are often ignored.
Cases where the second order term (1/2d2F/dx2/dx2) comes into play
Cases where the first order term (first order term) cannot be used well in some way
In Ito’s Lemma and Black-Scholes theory, the second order term plays a major role
Only the second term becomes very important
In what cases can we ignore it as small?
When adding terms of different orders (orders of magnitude), small terms can be ignored
Ito’s Lemma and the x2 term
Linkage with Y is given by y=F(x) or dy=F(dx)
If it is written dx=Adt+Bdw, then
we can write dy=F(Adt+Bdw)
Using Taylor series, we can write
For dt, we can consider order and ignore terms after the second order
For dw, it is different from dt
The final equation
Clarification of some questions
Remaining questions
Expression of Ito’s Lemma
Significance of Ito’s Lemma (1) Innovative tool for problems with zigzag motion
Significance of Ito’s Lemma (2) How to create a risk-free portfolio
Organization of the current talk
Consistency with the intermediate level talk
Issues between science and humanities
Ideological problems that also exist in the fundamentals
Derivatives can only be used in a continuous manner
Application of Lebesgue integrals
1. ideological foundations of stochastic differential equations

2. actual Black-Scholes theory

Differences from actual Black-Scholes theory
Black-Scholes equation
Difficulties with the Black-Scholes formula
Two difficulties in the mathematical aspect of the Black-Scholes formula
(1) Why Fourier series appears in this problem?
(2) What does the symbol N(*) in the Black-Scholes formula mean?
Finally – The Roots of Probability Statistics

Chapter 4: Measures and Lebesgue Integrals

Measures and the Lebesgue integral

Required for stochastic processes with motion in probability
Measure theory is an extension of the concept of phase space

How these concepts were required

The “phase” in macroeconomics
The idea of “set” is used to introduce a virtual “distance” between goods that can only be expressed in vague terms, such as how “close” one good is to another.
About “measurement
To use the concept of “set” to define well when dealing with ambiguous items such as area and volume instead of distance.
Integral handling of discontinuities
Example: Acceleration of colliding molecules
The point that pops out is infinitesimal
Measurements are needed as a tool to deal with the area of infinitesimals
Integral using them is Lebesgue integral
How to deal with non-smooth functions that jump from 0 to 1 discontinuously and abruptly
Differentiation is not possible because it is not continuous
In probability, it is important to “calculate the expected value
Calculating expected value requires integration

Analog and digital quantities

In physics, most things are analog, but in economics, they are digital
Even in the world of probability, the world is digitally expressed as “a certain problem occurs/ does not occur
In the world of probability, integration and other calculations are performed through digital-to-analog conversion

Expected value of an analog quantity and its integral calculation

Example: Expected value of a lottery ticket
The area of a bar graph is the expected value
Example 2: Expected value of a lottery ticket
There are various winners with the same probability (the horizontal axis of the bar is the same).
Similarly, the expected value is the area of the whole

Concept of “measure

Prepare a function that substitutes sets instead of numbers
Substitute a set of numbers to display “length” and so on.
Example: dice
Think of an event or event side as a kind of set
The event of throwing a dice and getting a 1
Set A1
The event that the dice rolls a 2
Set A2
The set function P(A) transforms abstract sets such as A1 and A2 into an analog value of “probability value 1/6
Abstraction is limited to some limited properties
This is true in the case of dice.
Probability of 1 + Probability of 2 = Probability of 1 or 2
P(A1) + P(A2) = P(A1 ∪ A2)
A1 ∩ A2 = 𝟇(empty set)

What is the Lubek integral?

An integral using a measure
Conventional Integral
Riemann integral
Example
Consider the function “f = 1 at points of rational numbers but f = 0 at other places of irrational numbers
Rational number
A rational number has a finite number of decimal places, or a seemingly infinite number of decimal places, but is a circulating decimal number.
Impossible numbers
An infinite number of decimal places that do not have a circulating decimal point
Most points on a straight line are irrational numbers
Rational numbers are only special points
If you integrate it, does it equal zero, does it have a constant finite value, or does it diverge to infinity?
There are infinitely many 1’s in the interval of rational numbers
Since the individual microscopic widths are infinitesimally small, we don’t know if the integral will cancel them both out when multiplied together
Using the Lebesgue integral, we know this is zero.