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Topic Title: stratonovich integral
Created On Wed Oct 15, 03 02:34 PM
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munishg
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Wed Oct 15, 03 02:34 PM
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Hi,
I am student in computational finance.I am having some bad time understanding basically at what time values do we take teh integral as it says S[t]=summation S[{(Ti+1)+Ti}/2]*[B(Ti+1)-B(Ti)].Basically what algorithm is generated by this formula to compute the integral and How it is compared with Ito's integral.

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Munish
 
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Nonius
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Quote

Originally posted by: munishg
Hi,
I am student in computational finance.I am having some bad time understanding basically at what time values do we take teh integral as it says S[t]=summation S[{(Ti+1)+Ti}/2]*[B(Ti+1)-B(Ti)].Basically what algorithm is generated by this formula to compute the integral and How it is compared with Ito's integral.

Thanks

Munish



It has been a long time since I looked at the differences, but I am also slightly confused by the question. The diffs between the two formulations, from my understanding, revolve around whether you take left endpoints S(Ti) (Ito) or mid points (actually I thought left endpoints S(Ti+1) Stratonovic. At the discrete level, you have the above summation. To pass to the continuous limit, you need to define what you mean by integral. In Ito, you get Ito Isometry and the Ito Lemma. I actually don't know if you get the Ito isometry with Stratonovic, but I think the analog of Ito's lemma is, well, the plain vanilla Chain Rule. Ito makes more sense for Finance because your hedgin strategy depends on the knowledge of the price at the "left limit", ie you can't see into the future for an instantaneous increment of time. Having said that, coincidentally, in a previous life, I mused about using Stratonovic formulations to measure the risk on a Prop desk, under the assumption that somehow, miraculously, they could statistically "beat the market" by "seeing into" the future.


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ScottCaveny
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Edited: Sun Oct 19, 03 at 09:29 PM by ScottCaveny
 
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Nonius
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Originally posted by: ScottCaveny
munishg
Rest assured you aren't alone in the confusion here. I think that you've asked a *very* good question and I hope the resident experts will answer you in detail.

I suggest that you take this question over to the Quantitative FAQs project, because it seems to be one that belongs over there as well.

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Dude,
Funny you should bring up the idea of an arb. I wonder what other people have to say about that?


hmmm...I guess I should take that as an insult....oh well, guess I deserve it.

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ScottCaveny
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Edited: Sun Oct 19, 03 at 09:29 PM by ScottCaveny
 
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ScottCaveny
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Edited: Sun Oct 19, 03 at 09:29 PM by ScottCaveny
 
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Nonius
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Originally posted by: ScottCaveny
...the insult being what? I don't get it. What are you saying?


the implication being that I am not an expert and/or that I didn't not provide detail. anyway, you are right. nevermind....

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chiral3
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Refer to www.math.nyu.edu/phd students/frizpete

The idea is that when you re-write your equation as Stratonovich you are writing stuff that is first order Ito. In one of his papers he writes out the generators and the two different dX equations and you will see it

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ScottCaveny
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Edited: Sun Oct 19, 03 at 09:29 PM by ScottCaveny
 
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Nonius
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Originally posted by: ScottCaveny
Look dude, if you haven't figured it out yet, I'm a platypus. Part duck. Part beaver. I'm poisonous and if I get my g-friend knocked up she's gunna lay a bunch of eggs. I worry about platypus kinds of things which means I've got enough problems as it is.

But rest assurd, I was convinced a long time ago that you are some weird genetic recombinant of Evariste Galois, Henry Miller, George Soros and Attila the Hun.

Thus, any future misscommunications with you (or the rest of the cowboys and cowgirls in TV land) can be filed under the platypus effect; e.g., all faults being on the part of the platypus.


dude, I just keep thinking of Reed and the Acid thesis...totally cool school....by the way, before going to Berkeley, I went to Santa Cruz dude.

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ScottCaveny
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Edited: Sun Oct 19, 03 at 09:29 PM by ScottCaveny
 
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ScottCaveny
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Edited: Sun Oct 19, 03 at 09:29 PM by ScottCaveny
 
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Pat
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Stratnovich is the versin of white noise that comes up most often in modeling physical systems ... Suppose you have a physical system; it will have some natural time scale T. Suppose now this system has a driving term included, and the time scale t of this driving term is much, much faster than the natural time scale T. Intuitively, one believes that the system is going to respond to various averages of the driving term. If we model the driving term as a random function of time, then in the limit that the driving tems time scale becomes smaller and smaller (and the amplitude of the driving term becomes bigger and bigger) then one gets Stratonovich white noise. See papers by Varadhan, Papanicalous, Keller, and a host of others.

The Ito version of white noise is a different limiting process, literally defined by dS = a(t,S)dW meaning the limit of S(t0+nh+h) = S(t0+nh) + a(t0+nh,S(t0+nh))pW(t0+nh+h)-W(t0+nh)] as h goes to zero. It was defined to be as mathematically tractable as possible, but fortunately, there is an easy way to add a drift term to switch between S and I white noise.

Finance is the only application I know which is more naturally expressed in terms of the Ito calculus. If one tries to use Stratonovich calculus to derive it, then when one tries to put in "non-anticipating," one is forced to add a term which exactly changes Stratonovich back to Ito.
 
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scholar
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Here is a paper that analyzes BS within the Stratonovich calculus.
 
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munishg
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Thanks all for your replies.
I think Now I have lot better idea about stratovich integral.AFter discussion with Dr.Srdjan Stojanovic ,it comes like this.Basically we have to think db as dB=2*db and then take the alternate values of Brownian Values to get stratovich integral.This makes sense and helps to solve stochastic integral for any time.

Munish
 
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LongTheta
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Look up N van Kampen's book for a discussion of Ito vs Stratonovich. (I haven't read it myself, but I've always been refered to it -- I really should get to that someday.)

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ScottCaveny
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Edited: Sun Oct 19, 03 at 09:28 PM by ScottCaveny
 
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Marsden
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Probably someone else has already written this in longer form, but if I recall correctly, one way of stating the difference between Itô and Stratonovich is that where Itô uses σb(t)Δt in the series to which it seeks a limit, Stratonovich uses σb(t+Δt)Δt. Because we cannot know b(t+Δt) at time t, the usefulness of Stratonovich is limited.

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Edited: Fri Oct 17, 03 at 04:07 PM by Marsden
 
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elan
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From a modern perspective, the Stratonovich integral is part of "anticipating stochastic calculus". A nice presentation from this point of view is in Nualart's book on the Malliavin calculus. Another insightful (but quite technical) account is in Stroock's latest book. I find van Kampen's discussion not particularly interesting. Basically, it is a runt cursing mathematicians for their narrow mindedness without offering much real insight. As far as anything else is concerned, I agree with Pat's posting.

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chiral3
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http://www.math.nyu.edu/phd_students/frizpete/malliavin/mall.pdf

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