Stochastic calculus

Using a Langevin dynamics approach, the stochastic LLG equation [Eq. (3.46)] can be integrated numerically, in the context of the Stratonovich stochastic calculus, by choosing an appropriate numerical integration scheme [51]. This method was first applied to the dynamics of noninteracting particles [51] and later also to interacting particle systems [13] (see Fig. 3.5). 

Karatzas, I. Brownian Motion and Stochastic Calculus. Springer-Verlag, Inc.. New York, NY, 1998. 

Vol. 579 D. Sondermann, Introduction to Stochastic Calculus for Finance. X, 136 pages, 2006. 

To carry out stochastic homogenisation, elements of local stochastic calculus are needed. For more details, the reader is referred to [4, 11, 13]. 

Equations for a2. P2 are obtained from (7a) and (7b) by exchanging the subscripts 1 and 2. Our derivation is based on the Ito stochastic calculus and R are Gaussian noise terms with zero means and the following nonzero correlators ... 

R. Durrett. Stochastic Calculus A practical introduction. CRC-Press, Cornell University, Ithaca, New York, USA, 1996. 

Vol, 1865 D. Applebaum, B.V. R. Bhat, J. Kustermans, J, M, Lindsay, Quantum Independent Increment Processes I, From Classical Probability to Quantum Stochastic Calculus. Editors M. Schtlrmann, U. Franz (2005) Vol, 1866 O.E. Barndorff-Nielsen, U. Franz, R. Gohm,... 

Vol. 1929 Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes (2008)... 

The next step in the analysis involves using stochastic calculus. Without going into this field here, we summarise from Baxter and Rennie (1996) and state that a stochastic process X will incorporate a Newtonian term that is based on df and a Brownian term based on the infinitesimal increment of W that is denoted by dWf The Brownian term has a noise factor of o,. The infinitesimal change of X at X is given by the differential equation... 

STOCHASTIC CALCULUS MODELS BROWNIAN MOTION AND ITO CALCULUS... 

Ito s theorem provides an analytical formula that simplifies the treatment of stochastic differential equations, which is why it is so valuable. It is an important rule in the application of stochastic calculus to the pricing of financial instruments. Here, we briefly describe the power of the theorem. 

Pliska, S., 1986. A stochastic calculus model of continuous trading optimal portfolios. Math. Oper. Res. 11,371-382. 

Applebaum, D. L vy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics, vol. 93. Cambridge University Press, Cambridge, UK (2004)... 

Beyond the cubic polynomial, there are two main approaches to fitting the term structure parametric and non-parametric curves. Parametric curves are based on term-structure models such as those discussed in chapter 4. As such, they need not be discussed here. Non-parametric curves, which are constructed employing spline-based methods, are not derived from any interest rate models. Instead, they are general approaches, described using sets of parameters. They are fitted using econometric principles rather than stochastic calculus, and are suitable for most purposes. 

Karatzas, L Shreve, S. Brownian Motion and Stochastic Calculus Springer-Verlag New York, 1991. 

In Section 2.4, we had raised the possibility that particle state could change in a random manner. Since the deliberations until this stage have taken a deterministic view of the rate of change of particle state, we shall address this issue at some length in the next section. For this section to be comprehensible, the reader must be familiar with Ito s stochastic calculus and elementary aspects of the theory of stochastic differential equations. 

We now consider probability theory, and its applications in stochastic simulation. First, we define some basic probabihstic concepts, and demonstrate how they may be used to model physical phenomena. Next, we derive some important probability distributions, in particular, the Gaussian (normal) and Poisson distributions. Following this is a treatment of stochastic calculus, with a particular focus upon Brownian dynamics. Monte Carlo methods are then presented, with apphcations in statistical physics, integration, and global minimization (simulated annealing). Finally, genetic optimization is discussed. This chapter serves as a prelude to the discussion of statistics and parameter estimation, in which the Monte Carlo method will prove highly usefiil in Bayesian analysis. 

We next consider an important application of probability theory to physical science, the theory of Brownian motion, and introduce the subject of stochastic calculus. Let us consider the jc-direction motion of a small spherical particle immersed in a Newtonian fluid. As observed by the botanist Robert Brown in the early 1800s, the motion of the particle is very irregular, and apparently random. Let Vx t) be the x-direction velocity as a function of time. For a particle of mass m and radius R in a fluid of viscosity /x, the equation of motion is... 

The lack of a proper definition for (7.149) means that we cannot apply the traditional rules of calculus to Brownian motion rather, we must use the special rules of stochastic calculus. Thus, integrals of the form of (7.137) and ODEs of the form of (7.132) are not to be defined using deterministic calculus as we have done above. Let us now write (7.132) in a form that is well defined by multiplying it by dt. 

While the explicit Euler method is simple, it is not very accurate. For a deterministic differential equation, we build higher-order methods through Taylor series expansions however, the rules of stochastic calculus are different. Consider the SDE... 

We use here ltd s stochastic calculus, in which we approximate the stochastic integral by quadrature using the values at the beginning of each subinterval ... 

Stochastic calculus is used heavily in quantitative finance, a significant employer of numerate engineers. In Problem 6.B.5, we solved the Black-Scholes equation for the fair value of an option. Here, we show how this equation is obtained, through stochastic calculus. 

Next follows a detailed discussion of probability theory, stochastic simulation, statistics, and parameter estimation. As engineering becomes more focused upon the molecular level, stochastic simulation techniques gain in importance. Particular attention is paid to Brownian dynamics, stochastic calculus, and Monte Carlo simulation. Statistics and parameter estimation are addressed from a Bayesian viewpoint, in which Monte Carlo simulation proves a powerful and general tool for making inferences and testing hypotheses from experimental data. 

---