FR H-Volatility - page 14
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Thanks to Anton for this publicity, now I can be a "student of Shiryaev himself" and feel free to answer questions :)))
In general, the fact that a profitable TS cannot be built follows from the fact that the Stochastic Integral of any function on y has a mean zero (in the discrete case the stochastic integral turns into a sum and this fact is checked simply by the definition of a).
And finrise of any strategy is a Stochastic Integral of some function (implementing the strategy) on the price process.
P.S. If TC is a random process that depends on the price itself, then it is much more complicated...
Your informal explanation quite accurately conveys the point. Furthermore, it follows from the independence of past price values and future price increments that even in the case of rates depending on the (random) past, the situation will not change. In fact, the theory states even more generally that any strategy on a martingale yields a profit - martingale, and this is in fact a much stronger theorem than Dub's theorem.
to kniff
In general, the fact that a profitable TS cannot be built follows from the fact that the Stochastic Integral of any martingale function has a mean zero (in the discrete case the stochastic integral turns into a sum and this fact is checked simply by the definition of the martingale).
You should be beaten with a dirk, a good birch one :-). Economists at least harmless introduce notions of meaningless, and there is no mathematics behind it. You are mech-mathematician devils striped well know it, except that one thing - with all your mathematical research, often lose common sense.
Let's start in order.
Definition:
A martingale is a random process such that the best mean-squared prediction of how the process will behave in the future is its present state.
Great, now we're dancing.
You call the curve that everyone has on their monitor (quote stream) a Martingale. On what grounds? Prove it. You give me the link"What is a martingale?" it's more correct, but again it is always about the player (trading strategy) and not the streaming quotes. There is no doubling of bets, no profit and no loss on this curve, it's our inflamed brain that puts them there (or a trading system created).
Let's dance further.
You forget that any formula is a mathematical abstraction that tries to adequately reflect the real world around us. post here a TS that is martingale in its pure form. This TS should be since the inception of the Exchange, making 1 ... 50 deals a day have an average zero.
And now for a snack.
There was a great scientist Stratonovich. So the Stochastic Integral of any function on the martingale is one of the special cases of the Stochastic Integral Stratonovich. Yes, it is stupidly obvious, to 30 decimal places (as mathematicians like to do, you can't find the exact solution to his equation). But there is one little nuance to this equation, and it is being solved quite successfully. And solutions are found for practical important tasks. In our case it is a profitable TS. Open your eyes and look at the first 100 trading systems of the Championship, look at their total number of deals and try once again to statistically prove that it is impossible to create a profitable TS.
to Mathemat
Since returns is a random process with m.o. equal to zero (if the price is a martingale with independent increments) ....
You're confused, you know how to calculate m.o. from returns, and if it equals 0 all the time you can throw a stone at me.
Quote flow is not a martingale !!! but rather a superposition of Poisson flow and partial Bernoulli flows to put it strictly mathematically
And what does Dub.... have to do with it? Generally speaking, every other theorem in the theory of random processes is Dub's. And that it's referenced... As I understand it, they like to refer to sources with a lot of incomprehensible words and justify their delusional ideas ))))
Yes, they are old and unintelligent people who have been pounding the pond, but then a young, but very smart and quick kid came along and quickly kicked the crap out of everyone.
kniff , the above quote is pure boorishness. I refer you to the rules of this forum for the time being. Get to grips with them, with all the definitions and what follows from them. Perhaps this will somehow cool down your "mathematical" snobbery.
>> Arbitrage is a property of the market to produce a statistically reasonable return. It can be caused by e.g. stationarity of non-zero values of autocorrelation function, presence of deterministic or stochastic trends or something else. The main thing is that an arbitrage market allows to distinguish the situations with the probability of some outcome stably not equal to 0.5
Concerning the question of definitions - you probably want to say the following: "A market is arbitrage when at least one profitable TS exists in it.
A couple of questions then:
a) Is insider trading a TS?
b) Is it a TS on the news?
c) And on technical analysis (but this you certainly call a TS, fact - but in "a" and "b" I am not sure).
If you read Shiryaev's books thoughtfully, you can understand that it fits into the following, already quite strict definition:
"A market is Arbitrageous if it is NOT a martingale relative to a *** flow of sigma algebras". Where *** is substituted depending on what you want to consider a working TS of a,b,c - and what is not.
I said what I said. If I wanted to say otherwise, I would have done so. Your attempts to retell things in your own way are not objectionable to me, retell it however you want, it's just a result of your level of understanding. Perhaps you, "as a mathematician", think there is only one true form of any definition. That is your right. In reality, in mathematics, physics and the real world, any phenomenon or object can be defined in many ways. And they will all be true if they are equivalent, i.e. reducible to each other.
You've come up with your own definition of arbitrability and argue around it. Well, have fun with it. However, arbitrability has nothing to do with TC. Although, of course, the existence of at least one profitable TS undoubtedly proves the arbitrability of the market (in fact). Only one problem ! We do not know what a profitable TS is. You, as an expert in mathematics, should be ashamed to give a definition of one value through another one, which is also undefined.
Shiryaev's definition is undoubtedly correct, but it is poorly understood here. I, for example, do not know what a sigma algebra is, but I know that I have defined the concept of arbitrability correctly. And if you don't understand its meaning, it won't be hard for me to explain it to you.
No, I'm not a maths maniac - it's just that you're discussing seemingly clever things for 12 pages here, but in fact there's no clarity ))) If you play, then the full :-D
You young man unfortunately still do not understand what we are discussing here, even 12 pages was not enough. I can assure you, we are not discussing mathematical definitions. This thread just came up as a matter of course to reconcile the concepts.
It's also very unfortunate that you haven't noticed several quite clear questions I've posed. You, as a mathematician, could probably give clear answers to them. Especially if you have experience as a trader and know the practical side of the issue. If you don't have those answers, then your stormy entrance into the circle of debaters may be considered of no use at all.
I wanted to respond to your other statements, including my interlocutor, but rereading it, I got convinced that there was nothing to respond to. There is only noise, no signal. :-) Like in this one.
>> About Pastukhov's thesis, dispel your doubts - it's a good work. The mathematics there is elementary, and the main content of the work is the proof of theorems, which actually justify the method. For a person who wants to look at the market from a statistical point of view, it's a very useful experience. As a total ignoramus of mathematical statistics, this work has brought me up to the level where I know what I'm talking about. :-))
Did this work make you any money?
Don't you have any other questions about Pastukhov's work?
You are confused, how to calculate m.o. from returns you know, and if it equals 0 all the time you can throw a stone at me.
Quote flow is not a martingale!!! but rather a superposition of Poisson flow and Bernoulli partial flows to put it strictly mathematically
The latest posts are of course discussing martingale as a random process, not martingale as a strategy (let's not pick on the phonetics, OK?). And the theorems mentioned have as an essential assumption precisely that s.p. prices are martingale.
I have long suspected that price is not a martingale, although it is similar to one. This is why Doob Th. or its generalisation does not seem to me applicable to the flow of quotations.
But about the superposition of Poisson flow and Bernoulli partial flows - can you be more specific?
Yurixx
The market has no such property. It is a property of the trading system (the trader). The market (quotes flow) doesn't care about your or my income. Maybe this makes it clearer that you cannot apply this concept to the market.
My opinion, sorry, is not an authority, but the concept of efficiency is philosophical. Please try to explain it to me, as I did on shovels (see example above). But please don't refer to anyone else, and do not step on the same rake, don't attribute the same properties to quotes flow (the market) of a trading system that I understand physically (whether it (TS) brings me revenue, or washes money out of my pocket).
Sergey, if we prove the absence of arbitrage of the market, then we can calmly stop this silly struggle with windmills and do something more constructive. If, on the contrary, to prove the arbitrability of the market, then this struggle becomes meaningful. In particular, if arbitrability really exists, then we can already raise the question of where its source is, what its nature is, what is the mechanism of its manifestation. And from the solution of these questions to the TS, guaranteeing a golden rain, is close at hand. As long as neither is proven, we are left to argue and search.
Of course we can ignore the question of arbitrability and build the TS. And if it turns out to be non-functional - we can build another one. And then a third. But will not it be like building a perpetual motion machine? Wouldn't it be smarter to build a TS knowing where arbitrability lies and how to get it? Or is it better to point a finger in the sky - just in case I get it?
Don't get me wrong, I want to help you. You are right to say "that we can only talk about something if we agree on the concepts and the same language. Plus I would like to add that one can study (investigate) only those properties of an object (phenomenon) which it has. I just once a long time ago I was taught to conduct research and waved my dingle goodbye. There is one simple rule when you start a research, the first is to determine the physically understandable properties of the phenomenon (object, process ...), the purpose of research and how to achieve it. Second, you try to describe these properties mathematically and numerically. And the third brings a method (algorithm, formulas) so that the other researcher repeating all your calculations have the same results.
It's the cool, eternal principles of scientific inquiry. I'm all for it. So
1. Prove (or show) the presence or absence of arbitrage of the market of a specific currency pair, at least in a narrow sense - at the level of statistically significant (from a trader's point of view) time periods and volumes of historical data. To do it
2) Study statistical properties of a quote flow. Which ones and to what extent - it would be better if it were formulated by a specialist in mathematical statistics.
3 If arbitrage is detected, then find its source and mechanism of its manifestation.
4 Define a model of this manifestation.
5 Build the TS on the basis of this model.
I hope the goal and the way to achieve it are clear from this.
It is impossible to study the properties which do not exist! Let's say the inefficiency (efficiency, arbitrage to the heap) of the market now = 9, a minute ago it was 32, and yesterday it was -15. Gentlemen, let's have the formula. There is no formula - let philosophers deal with this concept. There is nothing to count, nothing to study and research IHMO an empty sound that does not bring me and you any closer to building a good TS.
If there are numbers, are they coming from somewhere ? Where do they come from? That's what I'm looking for - a numerical measure of arbitrability. If there isn't one, and this is just for the sake of example, then that's exactly what it makes sense to do - build such a measure and, accordingly, a formula (algorithm) for calculating it. So the sound will only be empty until we (or someone else) make sense of it.There was a great scientist called Stratonovich, so the Stochastic Integral of any function over a martingale is one of the special cases of Stratonovich's Stochastic Integral, and he also derived an equation named after him, which shows how to solve it all. Yes, bluntly, up to 30 decimal places (as mathematicians like to do, you can't find the exact solution to his equation). But there is one small nuance to this equation being solved and quite successfully. And solutions are found for practical important tasks. In our case, it is a profitable TS.
Apart from everything else that you've written, this piece makes you think that you are speaking in phrases and expressing terms, the meaning of which you don't understand. The Stratonovich integral has zero significance in financial mathematics problems because it "skips ahead" in time. In other words, by trying to model the TS as a Stratonovich integral, you are modeling tradingwith knowledge of the future price. Not very sensible, is it? That's why the only real integral in use (I stress: in financial mathematics) is the Ito integral, which lacks this disadvantage. That's what the whole theory of evaluating options and that sort of technique is based on.
Actually, this habit of playing with scientific terms without understanding their essence causes such a reaction from kniff, as well as from any other person who understands the subject at least somewhat. The conversation can be quite meaningful and not use special terminology, not turning the discussion into shamanism with the call of the spirits Stratonovich, Shiryaev, Pastukhov, etc.. Well or it is desirable to know this terminology.
Well, as to "the Great Scientist Stratonovich" I will confine myself to history. Once Stratonovich came to Shiryaev and said: "How strange it is in your probability theory that the integral of 2B dB is not equal to B^2. This is not the case in physics, in physics it should still be B^2." And he created the Stratonovich integral :)
It is possible to have a meaningful conversation without using special terminology, without turning the discussion into shamanism with the invocation of the spirits of Stratonovich, Shiryaev, Pastukhov, etc. Or it would be desirable to know this terminology.
I wonder how many people want to be censors of ideas, terminology, level of education ... Anything at all.
Do you think you have the right to tell even one person here what terminology to use and what not to use? Or do you have the right to judge who has that terminology and who doesn't? Are you a part-time atheist, part-time wrestler with shamanism? Is that why you came to this thread?
I'll tell you frankly, your flippancy (notflippancy) of a "disciple of Shiryaev himself" has reached a climax all too quickly.
It's a pity you have nothing to say on the topic of discussion.
I agree with you on most things, except these two phrases highlighted above.
I'm not trying to model a TS (trading system). I'm talking about the curve you see on the screen (streaming quotes), which is a completely different thing. It's important to correctly predict the "behaviour" of that curve, if we can do it correctly, only then maybe we will get a good TS.
But the second phrase I have to take back to you. There's no getting ahead of yourself there. I apologise, but you have a gap in your knowledge. Stochastic differential equations can be written both in the Ito and Stratonovich form. And there is an unambiguous relationship between these forms. Each has its own advantages and disadvantages. And the Stochastic Stratonovich integrals allow handling them according to the usual rules of mathematical analysis (replacement of variables, integration by parts, etc.), which requires special rules when dealing with ITO. And there are dissertation councils that do not allow to defend dissertations mentioning ITO, require an entry in the form of Stratonovich (IHMO correctly do, we must know our scientists and be proud of them).
Once again I apologize, but I have to recommend you a book. Yarlykov M.S. Connection of two forms of writing of equations of optimum nonlinear filtering for posterior probability distribution. - Izv. Vuzov SSR. Radioelectronics, 1978, vol.21, no.5, pp.33-37.
Yurixx
Would be very happy to get back to the discussion again, something that is really of interest. I'm sorry I'm a little out of line, I still have some questions.
Please let's all stop, we all have knowledge and there is no one who knows absolutely everything and his words are the absolute truth.
I have read an excerpt from Shiryaev's speech, it's interesting. Somehow these Kadzhi-Renko's concept reminded me of Mr Duk's: only what exceeds a certain threshold gets registered. It's about the same here. Also, interesting:
И почему Мандельброт взял эту тему, раз уж о фракталах так много говорят? По одной простой причине. Если мы описываем приращение C, алгоритм приращения C берем – естественно считать, что это нормальное распределение. Но опять-таки данные.. Тот же Мандельброт анализировал.. Выясняется, что есть пик в нуле. И хвосты тяжелые. А как это может получиться? Может получиться двумя способами по крайней мере. Или же считать, что это устойчивое распределение, а с устойчивым распределением очень трудно работать. А у них плотность распределения Коши именно имеет такой пик. Или же, так как не хочется отрываться от нормальности – нормальностью мы можем оперировать – заменить это приращением, которое зависимое. Вот так Мандельброт и пришел к своему понятию фронтального броуновского движения. Именно желание получить гаусовский нормальный процесс, но у которого корреляционная функция вот такая - распределение имеет пик в нуле - но тем не менее чтобы он оставался гаусовским.
P.S. By the way, Brownian motion is hardly frontal, it is rather fractal...