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Alan
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January 16th, 2012, 3:50 pm

The normative Ito sde is dX = b dt + a dW, where W(t) is a Brownian motion.However, you believe that there exists an alternative sde, the Fermion sde, dX = b dt + a dZ, where Z(t) is some non-Gaussianprocess. You also believe this alternative sde leads to the standard Fokker-Planck equation: p_t = 0.5 (a^2 p)_xx - (b p)_x So, you are looking for a 'gap' in the standard theory that allows your alternative theory to exist. I believe you are quite misguided in this and no such sensible alternative theory with those properties exists. As I said before, the only well-established sde associated to the _standard_ FPE is the Ito theory. That's why you can't find your alternative theory in any textbook. All you can find are "pre deltaT->0" versions where Z(t) is non-Gaussian before the continuum limit.At this point, you are looking for gaps in Levy's characterization of a Brownian motion and higher moments conditions that allows for your theory to exist.Fine, maybe some other board member will explore those issues with you. In the end, you have to make a belief choice.Belief #1 is that you have discovered some new thing, but are not mathematically equipped to quite 'prove it'.Belief #2 is that the standard Ito theory is really all there is -- given that the standard FPE is the target.My position is clear.
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frenchX
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January 16th, 2012, 5:09 pm

There are FPE for non gaussian increment such as this Fractional Fokker?Planck Equation forNonlinear Stochastic Differential EquationsDriven by Non-Gaussian Levy Stable NoisesOk the process is not continuous but this one should be FOKKER-PLANCK-KOLMOGOROV EQUATIONS ASSOCIATEDWITH TIME-CHANGED FRACTIONAL BROWNIAN MOTIONAnd to relax a bit the atmosphere here a little anecdote from Laurent Schwart from Ecole Polytechnique. "Three weeks ago Paul Levy was asked to teach probability at Ecole Polytechnique but he claimed that he knew nothing to probability since it was not his main speciality. He told :"3 weeks is not enough for me to read the whole litterature but it's enough for me to rediscover everything". " Then he became the probabilist you all know.
 
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Alan
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January 16th, 2012, 5:19 pm

sure, but the discussion is about what kind of sde is compatible with p_t = 0.5 (a^2 p)_xx - (b p)_x (I have now edited my Mon Jan 16, 12 04:50 PM remarks to make this clearer to those who don't have the patience to read the whole, (tedious!), thread)
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croot
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January 16th, 2012, 5:20 pm

In the deltaT>0 approximating processes consider adding a third condition (L3) lim(DeltaT->0) E[1_{X_{t+deltaT}^{deltaT} \notin B(X_{t}^{deltaT},\epsilon)} ]/DeltaT = 0 ?This subject is in textbook form since Gihkman & Skorokhod Vol.3 Ch.2 Sec.3, try taking a look.
 
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Alan
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January 16th, 2012, 5:52 pm

I only have Vol. 2. Maybe you can use the latex and also elaborate.
 
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Fermion
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January 16th, 2012, 9:06 pm

QuoteOriginally posted by: AlanThe normative Ito sde is dX = b dt + a dW, where W(t) is a Brownian motion.However, you believe that there exists an alternative sde, the Fermion sde, dX = b dt + a dZ, where Z(t) is some non-Gaussianprocess. You also believe this alternative sde leads to the standard Fokker-Planck equation: p_t = 0.5 (a^2 p)_xx - (b p)_x So, you are looking for a 'gap' in the standard theory that allows your alternative theory to exist. No. You clearly haven't understood what I have written since your post of Jan 14. I completely accept the idea that the process I am discussing is discrete and has the moment behavior (for the first two moments) that you described there. That makes complete sense to me. If you are fixated on believing what you just wrote, then it is no wonder that you keep trying to talk past me.QuoteAll you can find are "pre deltaT->0" versions where Z(t) is non-Gaussian before the continuum limit.As I said, I pretty much accepted that many posts ago -- except for one minor detail which is if we are only interested in stuff that doesn't depend on an sde (e.g. moments, FPE), does delta Z necessarily become Gaussian in the continuum limit? I don't have an opinion on that, just guessing that if it doesn't, then the distribution may not be identical for all t. And if I am wrong, then what does happen to delta Z in the limit deltaT->0 if it is incompatible with a Gaussian (except in the steady state or the limit t->infinity).QuoteAt this point, you are looking for gaps in Levy's characterization of a Brownian motion and higher moments conditions that allows for your theory to exist.Yes -- once we are quite clear that I am only speculating on an idea and do not confuse it with your view of what "my theory" is.QuoteFine, maybe some other board member will explore those issues with you. Hopefully so.QuoteIn the end, you have to make a belief choice.Belief #1 is that you have discovered some new thing, but are not mathematically equipped to quite 'prove it'.Belief #2 is that the standard Ito theory is really all there is -- given that the standard FPE is the target.Not really. I am more interested in extending my knowledge beyond the Ito/Gaussian box in a way that is relevant to my project. As it happens, thanks to responses from you, polter and frenchX in other threads, none of this thread is critical for my work so I am not forced to make a choice. I am just curious in a way that you seem to find too much of a stretch.
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Alan
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January 16th, 2012, 9:29 pm

Well, if you say I don't understand, so be it -- I gave it my best shot. Maybe others can resolve your conundrums.
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January 16th, 2012, 11:49 pm

" What I have found is that the issue lies with \sqrt{\delta t} making the infinitessimal limit problematic. A Taylor series, however, that always has terms (\delta t)^n, does not face this problem. /// If you consder a solution of SDE, S ( t ) then you can apply Taylor formula to f ( S ( t + h )) which will contain W in 0.5 *n powors. In general, only quantities that do not involve fractional powers of \delta t can be easily converted to differential equations in the infinitessimal limit without worrying about Ito. It's true if we study expectations like E f ( S ( t + h )). The PDEs could be obtained but always if S is a Markov process.The process Z should be specified in desrete or continuous time. otherwise in your question you use a letter no one knows what is it.if Z depends on parameter n and for example converges to W when n goes to infinity then in continuos time we can write Kolmogorov equations for the limit process and we could not say anything about prelimit process.
 
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croot
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January 17th, 2012, 1:00 am

Quoteif we are only interested in stuff that doesn't depend on an sde (e.g. moments, FPE).You need to write down this discrete FPE please!
 
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January 17th, 2012, 1:21 am

list:Suppose we have a discrete processdel x = a(x,t) del t + b(x,t) del Z a(x,t), b(x,t) are analytic in t and del Z is independent of x.Now suppose that the probability density of x is analytic in t and satisfies the FORWARD Kolmogorov Eqn (FKE). Suppose also that all the finite moments of del Z are analytic in t. By writing a Taylor expansion for a small increment del t, we can take the limit(1) lim(del t -> 0) E[(del Z)^n]/(del t) = m_n(t)where m_n(t) is analytic in t , then the moments of del x will also be analytic in t and, in fact, will be solutions of the BACKWARD Kolmogorov Eqn (BKE) and so is m_n(t) (when we choose a = 0, b = 1).At no point have I written any sde or claimed anything about any quantity involving a fractional power of del t.Normalizing, we can always choose where E[del Z] -> 0 and E[(del Z)^2] /del t -> 1 as del t -> 0. But (1) still holds for n > 2.What else can we say about del Z? Well, I think we expect that in the limit t->infinity it will become Gaussian even if non-Gaussian for general t. (It could be Gaussian for all t -- in which case we know we can write Ito's sde.) But if the limits (1) exist for n > 2 and are not what we would expect for a Gaussian, what can we say about del Z in greater generality? This is the crucial thing I am curious about.An alternative possibility that we might consider (and one I suspect Alan would prefer to discuss) is that we can find an alternative del Z->Gaussian whenever del t->0 by re-defining a(x,t) and b(x,t). I have just realized that by allowing del Z to be t-dependent as well as a(x,t) and b(x,t) we may have a multiple set of equivalent discrete processes.)
 
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croot
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January 17th, 2012, 1:38 am

Quote(1) lim(del t -> 0) E[(del Z)^n]/(del t) = m_n(t)Really this normalization? Not other powers of del t on the denumerator?
Last edited by croot on January 16th, 2012, 11:00 pm, edited 1 time in total.
 
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croot
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January 17th, 2012, 2:09 am

Many things I don't understand in your text:"suppose that the probability density of x is analytic in t and satisfies the FORWARD Kolmogorov Eqn (FKE). "Why should the law of x_{t} have a density? Why should I follow you along to suppose it is analytic in t? What is the meaning of a FKE for a discrete process (2nd time of asking )."the moments of del x will also be analytic in t and, in fact, will be solutions of the BACKWARD Kolmogorov Eqn (BKE) "Same question, what is the meaning of a discrete time BKE? Some kind of recursion involving expectations or integrals??"What else can we say about del Z? Well, I think we expect that in the limit t->infinity it will become Gaussian even if non-Gaussian for general t."The object Z you have introduced to define from it the object X. You can suppose what you like about Z. Since Z depends both on t and delta t, maybe in your mind it is a process with two parameters. From what you have written down, there is nothing preventing Z from being the (deterministic) process Z_{t}=position in the alphabet of the [integral part of t]th letter in Hamlet. In this particular case, since Hamlet is finite well for large enough t one has dZ=0, so ""we expect that in the limit t->infinity it will become Gaussian "" is clearly wrong. If analytic moments are the key then del Z=anything smooth early on, and 0 after a while: limit of Z can be any law. Also, I don't understand why t->\infty is coming into the picture, it was already complicated enough having delta t->0 and looking over the interval t \in [0 T]."It could be Gaussian for all t -- in which case we know we can write Ito's sde"There is no such thing as a Gaussian dZ in Ito's theory, well there kind of is in the framework of tangent vectors on the space of probability measures but goshdarnit let's not go there it's complicated, I mean dZ is just shorthand in continuous time.
 
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January 17th, 2012, 7:39 am

QuoteOriginally posted by: crootMany things I don't understand in your text:"suppose that the probability density of x is analytic in t and satisfies the FORWARD Kolmogorov Eqn (FKE). "Why should the law of x_{t} have a density? Why should I follow you along to suppose it is analytic in t? I can choose to consider such a case. Whether or not you wish to follow along is up to you.QuoteWhat is the meaning of a FKE for a discrete process (2nd time of asking ).Read the thread through. I don't want to have to write a whole book repeating myself with every post. This has been explained by Alan as well as myself. We're talking about taking the lim del t -> 0 of expressions (e.g. for moments) that can be written in terms of integer powers of del t (such as a Taylor series for instance). The Kolmogorov eqns likewise can be obtained as del t -> 0 limits of discrete relationships involving first and second differences (see PWOQF chap 10 of the 1st edition for a proof). Quote"the moments of del x will also be analytic in t and, in fact, will be solutions of the BACKWARD Kolmogorov Eqn (BKE) "Same question, what is the meaning of a discrete time BKE?Same answer.Quote Some kind of recursion involving expectations or integrals??"What else can we say about del Z? Well, I think we expect that in the limit t->infinity it will become Gaussian even if non-Gaussian for general t."The object Z you have introduced to define from it the object X. You can suppose what you like about Z. Since Z depends both on t and delta t, maybe in your mind it is a process with two parameters. From what you have written down, there is nothing preventing Z from being the (deterministic) process This whole thread is about del Z being drawn from a distribution.QuoteZ_{t}=position in the alphabet of the [integral part of t]th letter in Hamlet. In this particular case, since Hamlet is finite well for large enough t one has dZ=0, so ""we expect that in the limit t->infinity it will become Gaussian "" is clearly wrong.If analytic moments are the key then del Z=anything smooth early on, and 0 after a while: limit of Z can be any law. Also, I don't understand why t->\infty is coming into the picture, it was already complicated enough having delta t->0 and looking over the interval t \in [0 T].It's another choice. Why in the limit t->infinity? Because in that limit we expect the t-dependence to disappear, del t becomes insignificant and it is reasonable (in fact I think it is pretty standard practice) to consider processes that become Ito-like in that limit. But, as I said, it's only a choice.
Last edited by Fermion on January 16th, 2012, 11:00 pm, edited 1 time in total.
 
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January 17th, 2012, 8:59 am

Let me make some preliminary corrections in your text.Suppose we have a discrete processdel x = a(x,t) del t + b(x,t) del Z /// it should be written in summation form x ( t _k ) = sum { j =<k : a( x ( t_j )) del t_j + b( x ( t_j )) del Z ( t_j ) del Z ( t_j ) is independent of x ( t_j ) /// this can be supposed for each t_jNow suppose that the probability density OF x ( t ) satisfies the FORWARD Kolmogorov Eqn (FKE) /// this cannot be supposed as far as 1) it is condition on x but x also depend on Z i.e it is condition on Z and x simultaneously. { Imagine that one write noun+verb and it is clear the sense of each words and correspondent their sum. Then one try to say that their sum should be interpreted in other sense while each is interpreted in its standard sense}.2) the Kolm. equation is related to continuous time but it was written in discrete form.
 
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January 17th, 2012, 10:07 am

Quote (see PWOQF chap 10 of the 1st edition for a proof). I believe you want to explore avenues where one does not havedeltaZ=O(sqrt(delta T) , the condition which PWOQF requires "otherwise the diffusive properties of the problem are lost."I guess ...- IF you are interested in making deltaZ much bigger than O(sqrt(delta T) in the limit, for there to be some chance that the m_{N}(t) are non-zero (for some N>2),THEN all the derivatives between 2 and N-1 must vanish because they are multiplied by lesser powers of deltaZ so when dividing by delta T before making it tend to 0 the result would be infinite were the derivatives not zero. This includes the first derivative wrt t for instance, so now the solution functions are constants.- IF you are interested in making deltaZ much smaller than O(sqrt(delta T) in the limit, then, ( once again expanding the analytic function p in an effort to find a PDE it might solve :flyingpig in the limit ), THEN EITHER delZ=O(delta T) and you get a first order linear PDE with coeff whatever limit of delZ/delT is , OR delZ is not O(delta T) and then all terms drop out and p(y,t,y',t') is constant wrt t .More generally, the exercise would be: 1)suppose a formal relationship between dZ and dt (maybe one is the power of the other)(relation valid in the limit as these d's go to 0, not necessarily for non-zero size d's)2)write down a Taylor expansion of an analytic function of z and t3)apply whatever cancellations are applicable in the limit as d's go to 0 using relations in 14) obtain a PDE describing this state of affairs.Is this the kind of thing you want to pursue? If you want to go on to5) build a coherent universe where dZ represents a random quantity, you will need to be able to have infinitely many of these random quantities at hand simultaneously when taking the limit as dt->0 , so dZ can no longer be absolutely anything you wish. Rest yourself, this does not mean it need be Gaussian.Rgds. P.S.Quote in that limit we expect the t-dependence to disappear, del t becomes insignificant Is this expectation somehow related to "the central limit theorem"?
Last edited by croot on January 16th, 2012, 11:00 pm, edited 1 time in total.
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