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<p(0)}, 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).

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