stochastic differential equation

Consider the ordinary differential equation, forexample, the population growth model

$$\frac{dX(t)}{dt}=a(t)X(t),X(0)=X_0,$$

where $a(t)$ is the relative rate of growth at time $t$, and $X(t)$is the solution-trajectoryof the system.

But we may want to take into account, in our model,the randomness or the uncertaintyof our knowledge of the data.In this case we may introduce the data $a(t)$as:

$$a(t)=r(t)+N(t),$$

where $N(t)$ is a noise term, represented by a random variablewith some postulated probability distribution.

In general, stochastic differential equationscan be posed in the case that the infinitesimal increment $dX(t)$is a Gaussian random variable. (Other types of random variables arealso possible, but require extensions of the basic theory.)A stochastic differential equation (SDE) is an equationof the form:

$$dX(t;\omega )=\mu (t;\omega )dt+\sigma (t;\omega )dW(t;\omega )$$

where $\omega$ lives in some probability space, and $W(t)$is a Wiener process on that probability space.The real-valued functions $\mu$ and $\sigma$ are to satisfy certain measurability requirements, and are usually assumed to be known, with the process $X(t)$ being sought.

The argument $\omega$ is usually suppressed in the notation:

$dX(t)=\mu (t)dt+\sigma (t)dW(t),$(1)

with the understanding that $X(t)$, $W(t)$, $\mu (t)$ and $\sigma (t)$ denoterandom variables for each time $t$.

The interpretation ofthe stochastic differential equation (1) is thata process $X(t)$ satisfies it if and only if it satisfiesthis relation amongst integrals:

$X(t_1)-X(t_0)={\displaystyle {\int }_{t_0}^{t_1}}\mu (t)𝑑t+{\displaystyle {\int }_{t_0}^{t_1}}\sigma (t)𝑑W(t)$(2)

for all times $t_0$ and $t_1$.The last integral is an Itô integral.

In many cases, the coefficients $\mu$ and $\sigma$depend on $X(t)$ itself:

$$dX(t)=\mu (t,X(t))dt+\sigma (t,X(t))dW(t).$$

In this case, equation (2) does not givean explicit solution for the stochastic differential equation.Nevertheless, there are theorems analogous to thoseof ordinary differential equations,that guarantee existence of solutions given certainbounds on the growth of the coefficients $\mu (t,x)$ and $\sigma (t,x)$.

In simpler cases, stochastic differential equations thatinvolve unknowns on the right-hand side may still be solvedexplicitly using changes of variables (often called Itô’s formulain this context).For example,

$$X(t)=X_0{e}^{-\kappa t}+\theta (1-e^{-\kappa t})+\sigma {\int }_0^t{e}^{-\kappa (t-s)}𝑑W(s)$$

(for any initial condition $X_0$) provides a solution to:

$$dX(t)=\kappa (\theta -X(t))dt+\sigma dW(t).$$