American type quantile hedging

We will formulate quantile hedging problem of American type. It seems no relevant solution so far in the literature

We fix a filtered probability space ${(\Omega,\;\mathcal {F}\;\mathbb{P},\;(\mathcal{F}_t)_{0\leq t\leq T})}$, let ${W_t}$ be a Brownian motion on the probability space, ${S_t}$ be the stock price given by

$\displaystyle dS_t=S_t(\lambda(S_t)dt+\sigma(S_t)dW_t).$

Suppose the wealth of the company is given by

$\displaystyle X_t^{x,\pi}=x+\int_0^t\pi_\nu dS_\nu,$

where ${x}$ is the initial capital, and ${\pi_t\in\mathscr{A}_x}$ is an admissible strategy. We can define the super-hedging price of an American option as

$\displaystyle p(0)=\inf\limits_{\pi_t\in\mathscr{A}_x,\tau\in\mathcal {T}_{0,T}}\{x: X_\tau^{x,\pi}\ge p(\tau)\;\;\;a.s.\},$

where ${p(t),\;t>0}$ is the price of an American option at time ${t}$, ${\mathcal {T}_{0,T}}$ is the set of stopping times in ${[0,T]}$. Given ${y, our objective is to choose an admissible control ${(\pi_t,\tau)}$ so as to maximize

$\displaystyle J(y,\pi,\tau):=\mathbb{P}\{X_\tau^{y,\pi}\ge p(\tau)\}.$

So we can define the value function as

$\displaystyle V(y)=\sup\limits_{\pi_t\in\mathscr{A}_x,\tau\in\mathcal {T}_{0,T}}J(y,\pi,\tau).$