PARTIAL DIFFERENTIAL EQUATIONS

Finally, what is a partial differential equation? After all the buildup, the answer is quite anticlimactic. Very simply, PDEs are to ODEs as partial derivatives are to ordinary derivatives. PDEs are differential equations in which the answer, the function you are trying to find, depends on more than one variable (equation A.17):

I understand that the notation is inconsistent, but the differentials in PDEs are typically written just as they are in ODEs. Equation A.17 essentially states that our variable, r, is now a function of both the time, t, and a process, W. Since dW is a Weiner process and by its definition is a stochastic variable, equation A.17 is actually a special type of PDE. PDEs with stochastic elements are called, not surprisingly, stochastic differential equations (SDEs), which we will discuss next.