The Cut-Off Phenomenon in Random Walks on Finite Groups

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J.P. McCarthy: Math Page

Last year's maths is easy, this year's maths is hard and next year's maths is impossible.

The Cut-Off Phenomenon in Random Walks on Finite Groups

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February 12, 2015 at 12:29 pm

StudentHi J.P.,

I was a student in your Maths Studies class last year. I am currently doing a MSc in Financial Economics and my thesis is aimed at examining the reliability of mathematical models as stock market predictors. One of the methods I am applying is Markov Chains.

I remembered you mentioned that your area of study involves this method. I was wondering would you be able to clear up a question i have regarding the Markov property.

I fully understand that it states that the future state of a subject is determined by the current state and so the past has no effect on the future state. However, seeing as yesterdays state determined the current state, does this not suggest that the past does indeed have an indirect effect on the future?

i.e. If the stock price of a company today affects the price tomorrow. Do past prices in turn affect future prices?

I would really appreciate your help,

Thanks.

February 12, 2015 at 1:03 pm

J.P. McCarthyI think you are getting caught up in semantics, and I think I can help you using Dynamical Systems (who would have thunk it?)

Consider,

Past Current Future

as

Yesterday Today Tomorrow

You state that

“

tomorrow’s state is determined by today’s state.”I think you are thinking of the state as the price of the stock. For ease of explanation, we will suppose the stock price is a whole number so now the price of a stock can be modelled by a natural number so, in this framework, the set of states is . If tomorrow’s state (price) is always determined by today’s state (price) then there is a rule that tells you how to go from one state (price) to another:

Yesterday Today Tomorrow

That is the price of the stocks is a dynamical system as to get from one state to the next you apply a rule (function).

Now, denote yesterday’s price by , today’s by and tomorrow’s by . What you are inferring is that

and so that

so that tomorrow’s price depends on yesterday’s and so how is this Markov… clearly this is a terrible model of financial stocks. For the remainder of this we will fix an initial date and define

stock price on morning of initial date

stock price at close of trading after days of trading

The behaviour of the stocks is given by an orbit:

Consider stocks that go as

so a fall followed by a bounce-back, a mini-set back and then growth. This simple behaviour can’t be captured by a dynamical system because we have

AND ,

and this is a contradiction because we can’t have the rule, , sending nine to 7 AND 11: a function will always send an input to the same output. So this model is not only unrealistic but empirically impossible.

The problem you are having is understanding “determined by today’s state”… or rather the set of states. Let us model a stock again. Define

stock price on morning of initial date

stock price at close of trading after days of trading.

Now there are a whole range of things that determines the price of the stock, . Let us call this whole range of things the probability space or the

universe. Now the stock price depends on the circumstances of the universe at that time so we have.

Now the thing is that we do not and can not know the entire state of the ‘universe’. So is a

randomvariable. We can only talk about the probability of it being equal to something. So for example, for a stock that has an orbit:we can’t say what is going to be equal to instead we can talk about the probability of being, e.g. more than 14, less than 14, equal to 14, between 12 and 16, even, odd, prime, etc.

All of these outcomes are events and we can talk about the probability of an event. Probability then is a function:

,

from the set of events, , to a number between 0 and 1 (i.e. the event’s probability).

Now, in particular, you are interested in events such as , a specific number, e.g. — the probability that is 13. You can make a(n infinite) row vector of these probabilities:

,

and this is called the

distributionof the random variable .Now, enter conditional probability. When you are talking about , you already

knowthe stock price on the first nine closes… now the probability distribution of isconditionedby the fact that you know the history up to that point… so rather than looking at the distribution , you look at theconditional distribution(this means “given that I know”)Now… the Markov property doesn’t say that the past

can’taffect the future exactly — it just says that the distribution of depends on only… and knowing about doesn’t tell you any extra. We write this as, for a given ,So the proper statement you should have for the Markov Property is

“

The.”distributionof the next state is determined by the present stateThis is more correct and the states are the random variables . This is the answer to the question you asked.

There is actually a way though to make a dynamical system out of this where the set of states isn’t the stock prices but their

distributions. E.g. the states are sequences, i.e. the sequence is the distribution of . Then you can define a function called astochastic operatorthat takes one distribution to the next:Then the orbit of the distributions of the stock prices

is a dynamical system. The set of states is the set of distributions and the iterator function is this . For more on this you can look in my MSc thesis or indeed at Q.2 of the homework I gave ye (https://jpmccarthymaths.files.wordpress.com/2012/02/homework1.pdf).

Now what this means for you is your project is that you need to swallow the following. Suppose you have a (time-homogenous (Google this)) Markov stock that has the following orbit:

.

Now, because , because the stock is Markov, you have to accept that

because the

distribution of thenext state only depends on the present state: it doesn’t matter that is after a slump and after a rise. Is this a good model for stock prices? This is your project.Regards,

J.P.