From: Roland Franzius Subject: Re: Stochastic integration Date: Mon, 29 May 2000 18:05:49 +0200 Newsgroups: sci.math,sci.math.research,sci.stat.math Summary: [missing] Hi, let`s see if its possible (caution: physicists shortcut to reality) Normally integration of a differential equation means: Take a time intervall (0,t), divide in intervalls ds_i, i=1..n. Given the velocity v(x,t), calculate x(t) = x(0) + sum_i v(x(s_i + h_i),s_i) ds_i for short time intervalls ds_i The result happens to be independent of partition of the time intervall (0,t) into subintervalls ds_i and the choice of the evaluation point in the limit max ds_i->0 abs(h_i)< abs (ds_i), given a smooth velocity function. This process has no limit if the the terms dX_t = v dt + "something wild" are not properly approximating 0 in the limit dt->0. As a standard process vioalating the riemann/lebesgue integral smoothness requirements consider a white noise generator G_t. At each point of time t it gives an output signal of gaussian distribution with mean 0 and variance 1. Signals at different times are uncorrelated. Expectation((G_t)^(2n+1)) = 0, Expectation(G_t G_s) = 0 : t\ne s, Expectation((G_t)^2n)=(2n-1)!! Now suppose you want to sum these signals in an invariant way. Define a Wiener differential dW_t = G_t \sqrt(dt) Then, the sum of independent Gaussian random variables dW_t with mean 0 and variance dt define a Gaussian random variable IW_t = sum dW_(t_i) = sum \sqrt(dt_i) G_(t_i) with mean 0 and variance sum dt_i. This means, the sum has a limit W_t as a random variable with mean 0 and correlation function Expection(W_t)=0, Expectation(W_t W_s)= min(t,s) since in products the overlap is 1:1 correlated with expectation of length of time interval. The solutions W_t happen to be random paths of brownian motion, continous with probability one, but nowhere diffrentiable in the usual sense. In the sense of differential equations we made the solution unique by choosing the evaluation point of the random increase the left point of the time interval dt_i. This is not by chance the condition, to make the process markov (use information at time t only to construct the path) and not anticipating (causal, no use of information from please). But it is possible to take another evaluation point, but then correlated parts of the path differential dW_t are added: While Ito chooses as differential dW_(t_i) = W_(t_{i+1})-W_{t_i} Stratanovich takes a symmetric differential (evaluation at midpoints) dW_{t_i} = 1/2 (W_{t_{i+1}-W_{t_{i-1}) This leads to addition of correlated parts of the white noise differential and in case of nonlinear functions one has quite different formulas for integrals and differentials. Consider the exponential Ito ODE with driving white noise dX_t = a X_t dW_t if you change the evalution point to the mid of the interval, you get a sum of squares dW_t^2/2. But here the independent Gaussian variable sum theorem gives sum dW_{t_i}^2 = sum dt_i G_{t_}^2 = sum dt_i = t with probablity one. The convergence: sum (dW_t)^2 = t means this is a Gaussian random variable with mean t and variance 0 (almost sure a linear function). The construction of the probabilty measure the paths are living on is a bit tricky: It is a cylinder measure generated by finite sets of gates the path passes through. General Ito stochastic differential euqations are modelelled like dX_t = v_t(X_t) dt + \sigma_t(X_t) dW_t and after carefull analysis of of expection of correlated terms one finds d f(t,X_t) = ( \partial_1 f + 1/2 \partial _2^2 f)(t,X_t) +(\partial_2 f)(t,X_t) dX_t Choosing mid evaluation (Stratanovich), the second derivative term arising from dW_t^2 = dt disappears. As a reference I would recommend Arnold, Ludwig: Stochastic Differential Equations (more about martingales, stopping times etc.) Gichman/Skohorod: Stochastic Differential Equations (more about diffusion processes and transition probabilities) Nicolas Roth wrote: > > Hi, > > I am looking for an intuitive approach of the stochastic integral, in an > Ito sense (by opposition with Stratonovicth). > > My main ref. book is OKSENDAL "Stochastic diff. equation" springer ed. > > If someone can explain me in a non formal way the concepts of the > construction of thge stochastic integral, it would be gladly > appreciated! > > Thanks a lot > > --- > Nicolas Roth > Econometrics Dept. > University of Geneva -- Roland Franzius +++ exactly <> lines of this message have value <> +++ ============================================================================== From: James Lawry Subject: Re: Stochastic integration Date: Fri, 2 Jun 2000 20:55:49 -0400 Newsgroups: sci.math.research sartorellig@gruppocredit.it says... > I think some non-specialistic book could be helpful for you. Most > financial mathematics books introduce stochastic calculus in an > non-formal way. > Try > T. Bjoerk - Arbitrage theory in continuous time > M. Baxter, A. Rennie - Financial Calculus Salih N. Neftci, "An Introduction to the Mathematics of Financial Derivatives" (Academic Press) has a good intelligible introduction to stochastic integration. James Lawry.