What is Black-Scholes-Barenblatt equation ? what is the use of the equation and what is the difference between it and Black-Scholes?When I read the article"Pricing and hedging derivative securities in markets with uncertain volatilities" written by M.Avellaneda, A.Levy and A.Paras. They talked about the equation ,worse case and best case. But I wonder how to get the worse case price and best case price. "maximise the worse case price and minimise the best case price" what does it mean?(You can find these words in Wilmott's book pp.307-311. )

BSB solves for the price of an option on an underlying that takes two different volatilities depending on the sign of gamma. It gives a minimum and maximum price, because you cannot hedge exactly. In some conditions, the minimum and maximum apply for any stochastic volatility specification, as long as the volatility is always within the band.From a hedging perspective, you can solve for a hedge that gives the highest value for the derivative in the worst case, or the one that gives the lowest value for the derivative in the best case. Any market price outside these limits would allow arbitrage.This is similar to trinomial pricing with only two hedge assets. You cannot deduce an exact price for a derivative, in general, but you can set a minimum and maximum.

G.I.Barenblatt(1978),a physicist, introduced a diffusion equation with similar non-linearilty to model flow in porous media.Arron, when we have a European call option with the volatility in a region: (sigma-,sigma+). Because the sign of gamma of European call option is positive, when we want to get the worse case V- (the best case V+), we can choose sigma+(sigma-).But if we put sigma+ and sigma- into BS equation, we also can get C+ and C-, the maximum price and the minmum price . What is the difference between C+(C-) and V-(V+)?

Perhaps you might have found info on the BSB already, but Vargiolu's paper related to it is quite good imo.found here (pdf)

Barenblatt is indeed the physicist, but neither he nor Fischer Black nor Myron Scholes ever derived this equation in finance!The first derivation of this non linear equation in a financial context predates Adobe pdf so I've attached a review article which refers to it.P