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October 19th, 2013, 11:52 PM   #1
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Stochastic Calculus of Standard Deviations - An Introduction

In my new paper Stochastic Calculus of Standard Deviations: An Introduction downloadable at ... id=2337982 we have presented a new numerical method to evolve the densities of SDEs. The paper finds the solution of SDEs of the form dV(t)=V(t)^gamma*epsilon*dz(t). Then it continues to give the solution for densities of stochastic time integrals of the form Integrate[V(t)^beta*dt,{t,0,T}] called dt-integral in the paper, where evolution of V(t) is given by first SDE. Then it continues to solve for densities of stochastic integrals of the form Integrate[V(t)^alpha*dz(t),{t,0,T}] called dz-integral in the paper, where again evolution of V(t) is given by first SDE. Then it hints at the solution of densities of the SDEs of the form dV(t)=mu*V(t)^beta*dt+sigma*V(t)^gamma*dz(t).

The paper also contains experimental code that you could use to understand the method and play around with it.

Though I did not do a definitive study of the accuracy and efficiency of the method. Since I ran the program hundreds of time, here are my rough observations. For the first SDE which is supposed to be a martingale, when V(0)=1, the expected value of V(T) obtained from the densities hovers around the true expected value by a difference of 35-45 bp for 40% vol(sigma) for 1-4 years. The reason for this bias is that I used a simple Euler-like Evolution for variance which is known to have biases as we know from traditional monte carlo methods. The second reason is that numerical coefficients on dt-integral are only approximate.

The technique solely evolves the distribution of SDEs based on how the variance evolves and takes in absolutely no random numbers. Since in this method, we divide the doamin of normal distribution into standard deviation fractions, If we take 500-700 SD fractions and workload for each SD fraction is about 4 times Monte carlo Euler stepping, the computational complexity of the method when compared to monte carlo with Euler type discretization, is of the (500-700) X 4 monte carlo paths less the time taken to draw random numbers since this method does not need random numbers. I am sure that soon people will have super fast totally analytic techniques with variance derived from this method.

Though Noise in the technique is normal, but method only takes in variance and similar methods can be used with non-normal generating noises when the densities of evolution of thoise noise like arithmetic brownian motion are known.

We know that parabolic PDEs and SDEs are related by probablistic relationships. One way to solve the SDEs is to write the corresponding PDE and solve it with numerical PDE techniques but I am sure that once the fast derivatives of this new method are found, people will be solving parabolic PDEs with stochastics, the reverse of the current practice.

Here is the abstract of the paper.

Every density produced by an SDE which employs normal random variables for its simulation is either linear or non-linear transformation of the normal random variables. We find this transformation in case of a general SDE by taking into account how the variance evolves in that certain SDE. We map the domain of the normal distribution into the domain of the SDE by using the algorithm given in the paper which is based on how the variance grows in the SDE. We find the Jacobian of this transformation with respect to normal density and employ a change of variables formula for densities to get the density of simulated SDE. Briefly in our method, domain of the normal distribution is divided into equal subdivisions called standard deviation fractions that expand or contract as the variance increases or decreases such that probability mass within each SD fraction remains constant. Usually 300-500 SD fractions are enough for desired accuracy. Within each normal SD fraction, stochastic integrals are evolved/mapped from normal distribution to distribution of SDE based on change of local variance 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 and are independent of each other, this technique is much faster than the monte carlo simulation. Since this technique is very fast, we are confident that it will be the method of choice to evolve distributions of the SDEs as compared to the monte carlo simulations and the partial differential equations.
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