Stochastic Differential Equations

It is well known that the straight-line segments are the backbone of the differential calculus. Moreover, differentiable functions are the basics of differential calculus. The behaviour may be locally approximated by straight-line segments. The same idea has been adapted in various methods, such as the Euler method for approximating differentiable functions defined by differential equations.

But in the case of the Brownian motion, the notion of a straight line produces another image of the Brownian motion. The mentioned self-similarity is ideal for an infinitesimal building block, and we can make the global Brownian motion out of lots of local Brownian motions. As such, one may build other stochastic processes out of the suitably scaled Brownian motion. Furthermore, if one may consider straight-line segments, then one can overlay the behaviour of differentiable functions onto the stochastic processes as well. Thus, straight-line segments and the local Brownian motion are the backbone of stochastic calculus.

With stochastic differential calculus, one can make new stochastic processes. We can make new stochastic processes by specifying the base deterministic function, the straight line, the base stochastic process and the standard Brownian motion. As such, the local change in the value of the stochastic process over a time interval of (infinitesimal) length dt as

In the stochastic differential equation, the initial point (t0, X0) is specified, possibly with a random variable X0 for a given distribution. A deterministic component at each point has a slope determined through a at that point. In addition, some random perturbation affects the evolution of the process. The random perturbation is normally distributed with mean zero. The variance of the random perturbation is (b(t, X))2 at (t, X(t)). This is a simple expression of a stochastic differential equation (SDE), which determines a stochastic process, just as an ordinary differential equation (ODE) determines a differentiable function. We extend this process with the incremental change information and repeat. This is an expression in words of the Euler-Maruyama method for numerically simulating the stochastic differential expression.

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