Suppose $$L(v) = \dfrac{\partial v}{\partial t} + rS\dfrac{\partial v}{\partial S} + \dfrac{1}{2}\sigma^2S^2$\dfrac{\partial^2 v}{\partial S^2} -rv$$ is Black-Scholes operator.

First version is $$P = K - S, L(P)<0\quad 0\leq S< B(t).$$ $$P>K - S,\ L(P) =0\quad S> B(t)$$ $$P = \max\{K - S, 0 \},\ \dfrac{\partial P}{\partial S} = -1\quad S = B(t).$$

Second version: $$P = \max\{K - S\}\quad 0\leq S< B(t).$$ $$L(P) = 0,\quad S>B(t)$$ $$P(S,T) = \max(K-S,0),\quad \lim\limits_{S\rightarrow\infty}P = 0;$$ $$P = K-S,\ \dfrac{\partial P}{\partial S} = -1\quad S = B(t),\ B(T) = K$$

Could any one tell the lack or repeat conditions in each version? Give me a correct version.

By the way, do we need the condition $B(t) \leq K?$


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