Stochastic integration

In order to develop good sampling paths for the Gibbs-Boltzmann distribution, we need to introduce a method of adding random perturbations into a differential equation. Let us first consider the definition of the integral with respect to Wiener measure. 

Recall that the Riemann integral of a continuous function g on [0, t] is the limit of Riemann sums constructed as follows partition the interval [0, t] into K subintervals each of width St = r/v, choose an arbitrary tk from the Mi subinterval, 1 A v, and evaluate the integral as a limit of the sum of products  

We know that this limit does not depend on where the 4 are chosen within the subintervals. For example we can take the left or right endpoint of each subinterval or the midpoint. 

By analogy, we would like to write the stochastic integral of a function g, where g may, itself, be defined in terms of a stochastic process, as the limit of a sum  

unlike in the case of the Riemann sums, it turns out that the result depends on where we place the points tk. Using the properties of the Wiener process we may easily show that the choice 4 = kSt (left endpoint) and the choice tk = (k+ l/2)St yield different formulas for the integral. The first choice is commonly referred to as ltd integration, and the second as Stratonovich integration. As an example, consider the case where g(t) = W(t). We suppose this to be approximated by a sum 

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The function a(x, t) appearing in the FPE is called the drift coefficient, which, due to Stratonovich s definition of stochastic integral, has the form [2]... 

The solution of a stochastic differential equation is expressed in terms of the integral with respect to a sample function W(t). The stochastic integral... 

The Ito integration formula, which is presented in the next subsection, is useful for the purpose of evaluating a stochastic integral. 

The following example shows the application of the Ito integration formula to evaluate a stochastic integral. 

For multiplicative noise the determination of these moments requires a more detailed consideration of the stochastic integral since white noise is too irregular for Riemann integrals to be applied. Application of Stieltjes integration yields a dependence of the moments on how the limit to white noise is taken. If t) is the limit of the Ornstein-Uhlenbeck -process with r —> 0 (Stratonovich sense) the coefficients read [50]... 

Protter P (1990) Stochastic Integration and and Differential Equations. Springer-Verlag, Berlin, Heidelberg, New York. 2nd Ed. 

The standard Wiener process is a close approximation of the behaviour of asset prices but does not account for some specific aspects of market behaviour. In the first instance, the prices of financial assets do not start at zero, and their price increments have positive mean. The variance of asset price moves is also not always unity. Therefore, the standard Wiener process is replaced by the generalised Wiener process, which describes a variable that may start at something other than zero, and also has incremental changes that have a mean other than zero as well as variances that are not unity. The mean and variance are still constant in a generalised process, which is the same as the standard process, and a different description must be used to describe processes that have variances that differ over time these are known as stochastic integrals (Figure 2.3). 

Harrison, J., Pliska, S., 1981. Martingales and stochastic integrals in the theory of continuous trading. Stochast. Proc. Appl. 11, 216-260. 

Rosenkrantz, W. A., Simha, R., Some theorems on conditional polygons, A stochastic integral approach. Operations Research Letters, 11(3), pp. 173-177 (1992). 

Plainly this is different than the Ito version of the integral, but in the cases we consider it is possible to reformulate Stratonovich integrals as Ito integrals. In the remainder of this book we will adopt the Ito form of stochastic integration. 

Proposition 6.3 Let g be a smooth deterministic function and W(f) a Wiener process. The stochastic integral Y(t) = fQg(s)dW(s) is, for all times t > 0, a normally distributed random variable such that... 

Finally, we observe that by calculating cr t ) and generating random increments Rn J (0, y(tn)) we have a discrete representation of the continuous time stochastic process Y(t), effectively an exact stroboscopic representation. Also it is natural to view the stochastic integral as the solution of the stochastic differential equation (SDE) with zero initial condition ... 

This notion of stochastic integration carries over to stochastic integration with respect to more general categories of stochastic processes. Importantly, with appropriate regularity we have the familiar relation from calculus ... 

In this section we briefly derive a more general stochastic differential equation as the limit of a biased random walk. The treatment here is intuitive, based on the concepts of limiting processes and stochastic integration introduced above. For a rigorous treatment of the material presented here, it is recommended to consult a reference such as the excellent book of Nelson [279] (which also contains a very lively historical discussion). 

Very important differences emerge if we attempt to use the stochastic integrators to compute dynamics, e.g. a time-correlation function. Velocity auto-correlation functions are shown in Fig. 8.4 for various choices of the parameters. 

Harrison, J., and S. Pliska. 1981. Martingales and Stochastic Integrals in the Theory of Continuous Trading. Stochastic Processes and Their Applications 11, 216-260. 

In the literature of application of stochastic processes there was extensive discussion of the interpretation of stochastic integrals. The key example is the particular case b(X,) = and so the integral IV dW has to be calculated. Using the Ito calculus (1951) we set... 

Finally, an entirely different approach to simulating gelation is the Dynamic Monte Carlo (DMC) method, in which chemical reactions are modeled by stochastic integration of phenomenological kinetic rate laws [23]. This has been used successfully to understand the onset of gel formation, first-shell substitution effects, and the influence of cyclization in silicon alkoxide systems [24—26]. However, this approach has not so far been extended to include the instantaneous positions and diffusion of each oUgomer, which would be necessary in order for the calculation to generate an actual model of an aerogel that could be used in subsequent simulations. 

DMC Dynamic Monte Carlo, a stochastic integration of the phenomenological kinetic equations for a chemical system DMDMS Dimethoxydimethylsilane... 

After the divisimi of the interval [0, t onto disjoint, contiguous subintervals, X(t) can be written down as the Riemann-Stieltjes sum. The limit, in the mean-square sense, of the sequence of such sums, is the mean-square Riemann-Stieltjes integral with respect to the counting process N t) or the stochastic integral ... 

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