1. On existence and uniqueness of solutionSince the Fokker-Planck equation describes the evolution of the probability density of a Brownian motion X_t, it makes sense that that the FP equation (with initial condition) has a unique solution.But does anyone know how to prove the existence and uniqueness of the solution analytically (i.e. by PDE or Functional Analysis theories or whatever analysis)?2. Between solving a stochastic equation and its corresponding Fokker-Planck equation, which one is easier to compute? Which are the widely used numerical methods?Thanks.

QuoteOriginally posted by: dongta1. On existence and uniqueness of solutionSince the Fokker-Planck equation describes the evolution of the probability density of a Brownian motion X_t, it makes sense that that the FP equation (with initial condition) has a unique solution.But does anyone know how to prove the existence and uniqueness of the solution analytically (i.e. by PDE or Functional Analysis theories or whatever analysis)?IMHO, the reference is Ladyzhenskaya and Ural'Tseva. Linear and Quasilinear elliptic equations. Academic Press, NY, 1968

QuoteOriginally posted by: dongta...2. Between solving a stochastic equation and its corresponding Fokker-Planck equation, which one is easier to compute? Which are the widely used numerical methods?Thanks.MC simulations in several dimensions are computational intensive and thus slow, so it's preferable to use the PDE approach in such cases. Also, some SDEs aren't analytical solvable so one is forced to use the PDE approach.

QuoteOriginally posted by: StaleQuoteOriginally posted by: dongta...2. Between solving a stochastic equation and its corresponding Fokker-Planck equation, which one is easier to compute? Which are the widely used numerical methods?Thanks.MC simulations in several dimensions are computational intensive and thus slow, so it's preferable to use the PDE approach in such cases. Also, some SDEs aren't analytical solvable so one is forced to use the PDE approach.This is not true. PDE methods are not used in higher dimensions because of the so-called "curse of dimensionality" -- the number of instructions is a higher than lineal polynomial in the number of dimensions. This is because of the link between PDE methods and matrix-type problems (inverting a matrix, finding eigenvalues, etc.). Monte Carlo, however, has the following scaling with dimension:ie, linear (for a given accuracy). While Monte Carlo methods might be slow in some "objective" measure (they do take some time), they are faster than PDE methods in high dimensions. Furthermore, any SDE can be discretized and simulated using Monte Carlo, not only the ones with analytical solutions. In fact, it is only a small subset of SDE's that have analytical solutions (log-normal, Hull White short rate etc.). Care must be taken when discretizing (for example CEV process crossing zero), but you will get an approximate answer. This is the same with any numerical method when the underlying equations have no exact solution, you'll inevitably end up with numerical error.

Two years ago,when I want to use FPE to simulate the pricing process,I have found so many problem,you will encounter with,such like how to get the probability at one time and the probability in the next time.If we cann't find what is the backbond of those probability,FPE is nothing!

Two years ago,when I want to use FPE to simulate the pricing process,I have found so many problem,you will encounter with,such like how to get the probability at one time and the probability in the next time.If we cann't find what is the backbone of those probability,FPE is nothing!

Some may find this interesting FD Rouah 'Heuristic Derivation of the Fokker-Planck Equation'http://www.frouah.com/finance%20notes/D ... uation.pdf