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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

July 20th, 2013, 9:53 am

I noticed that I could not download the attachment. When I uploaded the description of my work, I had downloaded a copy and it worked fine but I attach a second copy here again anyway.
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Integrals of Functions of Brownian Motion And Girsanov Theorem.zip
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Last edited by Amin on July 19th, 2013, 10:00 pm, edited 1 time in total.
 
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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

July 20th, 2013, 10:45 am

For those like me who cannot download off Wilmott, you can download from the following link.https://docs.google.com/file/d/0B1UoJb9 ... sp=sharing
Last edited by Amin on July 19th, 2013, 10:00 pm, edited 1 time in total.
 
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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

July 20th, 2013, 9:48 pm

Please notice a Typo. On Page 5 of the brief report, the First equation immediately following the calculation of dz-integrals. The sign before the second term on RHS is negative i.e Quadratic variations have a negative sign instead of positive. Here quadratic variations appear as a dt-integral.I hope to post results including Girsanov on SSRN in 2-3 days.
 
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cosmologist
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Deterministic Ito and other deterministic methods for stochastic processes

July 23rd, 2013, 7:39 am

Amin,When I unzipped the file the document comes out corrupted, one pager with half to be seen.Can you please check?thanks No worries, got it from googledocs. I guess, the interested ones will get it from google doc cheers.
Last edited by cosmologist on July 22nd, 2013, 10:00 pm, edited 1 time in total.
 
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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

July 26th, 2013, 1:17 pm

The next installment of Deterministic Ito-Girsanov is ready here. The work is nearing completion but here is the next version. You can read it herehttps://docs.google.com/file/d/0B1UoJb9Zaj22WWRTVWVybFA0Z0k/edit?usp=sharingIt is public on internet and is downloadable.
 
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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

July 27th, 2013, 1:28 am

For those who would be interested, the second formula on page ten is wrong. It should be the same as first formula on page eleven which is correct. I will fix this mistake in the next version.
 
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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

July 27th, 2013, 9:09 am

As it turns out, in deterministic Ito, everything goes almost like traditional Ito but with a few changes. 1. The formulas for dz-terms get a volatility of order dt (and higher orders) multiplied by standard deviation fraction size(that grows with increasing variance) and a multiple depending on standard deviation fraction. There is a quadratic variance adjustment in this volatility.2. dt-terms are simply multiplied by a dt and there is some volatility and quadratic variances depending on time.3. Once we got dt and dz-terms right, the Ito change of variable formula holds, in each SD fraction, just like that as in simulation. Only that dt and dz-terms are calculated according to the algorithm that I mentioned earlier in the first and latest description of my work.4. What is interesting once we have done deterministic Ito in each box, we can assign it the same probability that arithmetic Brownian motion has in that SD fraction, only rescaled by a one-dimensional Jacobian (pardon this misnomer) which can be easily calculated by simple finite difference numerically, so we can get the density easily.
 
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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

August 13th, 2013, 11:40 am

Here is the new version which can be read off google docs. I am about done for stochastic volatility as well and will release an update in 3-4 days. In the meantime I decided to give some interesting results on Girsanov which can be of interest to people in this area. Here is the link to google docshttps://docs.google.com/file/d/0B1UoJb9Zaj22SmZkQWN3MllDWTQ/edit?usp=sharingIf you find some error/typo, please mention that. Any remaining errors will be corrected in the next version.
 
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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

August 14th, 2013, 8:13 am

There was an error in last few lines of my Girsanov analysis. I have tried to fix it. Here is the new filehttps://docs.google.com/file/d/0B1UoJb9Zaj22U3N1aVZsOFphLWM/edit?usp=sharing
 
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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

August 14th, 2013, 5:39 pm

You have to use mathematica for solution of ODE given at the end of Girsanov section. Everything else stands. I will fix this error in next version. I will also give the order I am using for noise since the first order approximation to noise is poor at best.
 
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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

August 29th, 2013, 4:29 am

Here are some graphs showing comparison between monte carlo density and Analytic density of stochastic volatility for different values of mean reversion and exponent of stochastic volatility in noise term.The SDE isdV(t)=kappa(theta-V(t))dt+epsilon V(t)^gamma dz(t) We can see an instability on the left in the last graph. This is due to some derivatives being unbounded but can be very easily fixed. I am being a bit lazy to fix it right now but in the final paper, it will be shown as fixed. These graphs did not come from Girsanov but from another technique. As Stochastic calculus of standard deviations simply deals with variances, I found how the variances were changing with time and used that to evolve the stochastic volatility on the grid. I will write a detailed note in a day or two about it. Changing these distributions into correlated lognormal or correlated SABR asset prices should also be very simple.Please notice that Title on the second graph from bottom carries wrong parameters. Here kappa=3.65 instead of .65
Last edited by Amin on August 28th, 2013, 10:00 pm, edited 1 time in total.
 
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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

September 10th, 2013, 11:38 am

Here is a good introduction to stochastic calculus of standard deviations. I have given most formulas I have been using. I will soon be uploading a comprehensive work on SSRN with solution to SABR and Lognormal model with stochastic volatility. Stochastic Calculus could never have been so simpler, easier and faster. Here is the link to my work: https://docs.google.com/file/d/0B1UoJb9 ... sp=sharing
 
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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

September 11th, 2013, 1:45 am

I want to mention that last formula in section eight has an error. The first two terms have to be multiplied by n*DeltaX(0). The last term which is quadratic variation remains the same. This is a very elegant formula and comes from dz being square root of dt. This is how variance of dz integral evolves in time.
 
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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

September 12th, 2013, 7:48 am

Somebody asked me how I got the series solution for dz. Though I guessed it at the start. The right way to get is to writedz(t)=sqrt(t+dt)-sqrt(t)If we ask mathematica to expand the first term in series, we get the same series I wrote in the paper. This is how variance is related to dz in our framework. This is also basis for last formula in section eight that I mentioned in previous post. IF we know how the variance evolves we just need to multiply it with our n*DeltaX(0) to find the evolution of dz term.I will add this to explanation in the next version. Another point I failed to mention in the paper is that first increment is calculated with t in the middle of 0 and Delta_t(or numerical dt) otherwise it will obviously blow. In all the graphs, I presented the step size Delta_t=.01;But since there were only 200 branches of the tree, it was still very fast.
Last edited by Amin on September 11th, 2013, 10:00 pm, edited 1 time in total.
 
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Amin
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Deterministic Ito and other deterministic methods for stochastic processes

September 29th, 2013, 5:20 am

Here is new version of "Introduction to Stochastic Calculus of Standard Deviations." Now the evolution of distributions of SDEs is given in terms of change of variance of noise driving the sde. Here is the address to read or download it.https://docs.google.com/file/d/0B1UoJb9 ... haringHere is the abstract.Every density produced by an SDE which employs normal random variables for its simulation is a linear or mostly non-linear transformation of normal random variables. We find this transformation in case of a general SDE by taking into account how variance evolves in that certain SDE. We find Jacobian of this transformation with respect to normal density and employ change of variables formula for densities to get the density of simulated SDE. Briefly in our method, domain of the SDE is divided into standard deviation fractions that expand or contract as the variance increases or decreases such that probability mass within each SD fraction remains constant. Usually 200-400 SD fractions are enough for desired accuracy. Within each SD fraction stochastic integrals are evolved independently of other SD fractions. The work for each step is roughly the same as that of one step in monte carlo but since SD fractions are only a few hundred, this technique is much faster than monte carlo. Since this technique is very fast, we are confident that it will be the method of choice to evolve distributions of SDEs as compared to monte carlo simulations and partial differential equations.
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