tag:blogger.com,1999:blog-264226589944705290.post6115673277474006943..comments2022-08-07T01:05:01.413-07:00Comments on God Plays Dice: Solvable PDEs have measure zeroMichael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-264226589944705290.post-22195368824609008412008-11-03T12:56:00.000-08:002008-11-03T12:56:00.000-08:00> And options markets probably would have devel...> And options markets probably would have developed very differently if there hadn't been an exact solution, because solving such an equation numerically, while possible, is a lot slower.<BR/><BR/>I find it very difficult to believe. It's not the case that we have some particularly nice (and fast to compute) functions, that we use to construct closed-form solutions, it's rather a historic coincidence what we consider elementary.<BR/><BR/>For example, erf is not on the same footing as sin or log, but it's merely a matter of tradition. We "invented" erf to compute an important integral, and now we know enough about it to compute it very fast. If there was economic advantage in solving that PDE quickly, we would soon invent and research another "special function" or even a family of them (remember Bessel functions?;).Vekyhttps://www.blogger.com/profile/12207072339468136950noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-31646777399318400822008-10-20T19:36:00.000-07:002008-10-20T19:36:00.000-07:00More generally, mathematicians form a set of measu...More generally, mathematicians form a set of measure zero.CarlBrannenhttps://www.blogger.com/profile/17180079098492232258noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-71831299597785721162008-10-20T10:13:00.000-07:002008-10-20T10:13:00.000-07:00One might point out that knowing particular closed...One might point out that knowing particular closed-form solutions to PDE doesn't necessarily tell you much about the PDE itself.<BR/><BR/>If you have a uniqueness theorem, then it does--because all solutions are your closed-form one. But in the presence of nonuniquess, knowing the behaviour of one solution doesn't tell you much about the PDE itself.<BR/><BR/>That's why the prototypical PDE theorems are a priori estimates--they're theorems about the equation, not theorems about its solutions.Anonymousnoreply@blogger.com