June 8th, 2020, 2:57 am

I second looking at Friedman, also his SDE book. Once you've got rigorous conditions for the validity of the Feynman-Kac type probabilistic solution,

[$]u(x,t) = E_{x,t}[g(X_T)][$], where [$]g(\cdot)[$] is a payoff function, then uniqueness becomes trivial. After all, if [$]u_1[$] and [$]u_2[$] are two solutions with the same payoff, then

[$](u_1 - u_2)(x,t) = E_{x,t}[ 0 ] = 0[$].

The general rule for uniqueness of PDE solutions associated to SDE's of the form [$]dS = a(S,t) S dW[$], is that you need [$]a(\cdot,t)[$] bounded on the positive S-axis. That's certainly true for the BS GBM process where [$]a = \sigma[$], a constant. You can violate that rule and still be ok if the payoff is bounded, but combining violations with call option payoffs will lead to (non-uniqueness) trouble. (Also, there will be big problems if S=0 is not either unreachable or absorbing).

Does it matter? This kind of thing is quite useful for knowing when generalized models are problematic. For example, the CEV process [$]dS = S^{\gamma} dW[$] with [$]\gamma > 1[$] has non-unique call option prices. Google `strict local martingale' for literature.