# Existence of a hedging portfolio and martingale property

Lets assume that the underlying follows a Brownian motion and the market has the standard properties of the Black Scholes setting.

Is there a way to find a hedging portfolio for every discounted price process X which is a supermartingale?

Edit: The price process X is a stochastic process such that $X_0\ge E[X_t]$ and $X_t \ge \max(V_t, 0)$ for all t. That is the process is at least self-financing an at every time it is possible to pay oft the value of the option if its exercised.

-
what do you mean by a super martingale price process? –  quasi May 22 at 17:59
ok, so the supermartingale is the value process corresponding to an american option. if i understand your question, then at a theoretical level, the answer is yes: doob-meyer decomposition to split the supermartingale into a martingale and decreasing process, and then martingale representation to get a hedging strategy for the martingale part. –  quasi May 22 at 20:11