We review several problems on the quantile hedging given by [FL99].
We assume that the discounted price process of stock is given by a semimartingale 
on a probability space 
. Under self-financing, the wealth process 
with inital capital 
and a predictable strategy 
follows
![\displaystyle V_t^{v,\xi} = v + \int_0^t \xi_s d X_s, \ \forall t \in [0,T], \ \mathbb{P}-a.s.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+V_t%5E%7Bv%2C%5Cxi%7D+%3D+v+%2B+%5Cint_0%5Et+%5Cxi_s+d+X_s%2C+%5C+%5Cforall+t+%5Cin+%5B0%2CT%5D%2C+%5C+%5Cmathbb%7BP%7D-a.s.&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle V_t^{v,\xi} = v + \int_0^t \xi_s d X_s, \ \forall t \in [0,T], \ \mathbb{P}-a.s.")
belongs to a space of adimissible strategy given by 
,
![\displaystyle \mathcal{A}(v) = \{\xi: V_t^{v,\xi} \ge 0, \ \forall t\in [0,T], \ \mathbb{P}-a.s.\}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cmathcal%7BA%7D%28v%29+%3D+%5C%7B%5Cxi%3A+V_t%5E%7Bv%2C%5Cxi%7D+%5Cge+0%2C+%5C+%5Cforall+t%5Cin+%5B0%2CT%5D%2C+%5C+%5Cmathbb%7BP%7D-a.s.%5C%7D&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle \mathcal{A}(v) = \{\xi: V_t^{v,\xi} \ge 0, \ \forall t\in [0,T], \ \mathbb{P}-a.s.\}")
Let 
be a given claim.
Problem [SP]. Given a fixed constant 
, maximize success probability, i.e. find
Problem [SR] Given a fixed constant , maximize success ratio, i.e. find
![\displaystyle \sup_{v\le \hat v} \sup_{\xi\in \mathcal{A}(v)} \mathbb{E} \Big[ I_{\{H\le V_T^{v, \xi}\}} + \frac{V_T^{v,\xi}}{H} I_{\{H > V_T^{v, \xi}\}} \Big].](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csup_%7Bv%5Cle+%5Chat+v%7D+%5Csup_%7B%5Cxi%5Cin+%5Cmathcal%7BA%7D%28v%29%7D+%5Cmathbb%7BE%7D+%5CBig%5B+I_%7B%5C%7BH%5Cle+V_T%5E%7Bv%2C+%5Cxi%7D%5C%7D%7D+%2B+%5Cfrac%7BV_T%5E%7Bv%2C%5Cxi%7D%7D%7BH%7D+I_%7B%5C%7BH+%3E+V_T%5E%7Bv%2C+%5Cxi%7D%5C%7D%7D+%5CBig%5D.&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle \sup_{v\le \hat v} \sup_{\xi\in \mathcal{A}(v)} \mathbb{E} \Big[ I_{\{H\le V_T^{v, \xi}\}} + \frac{V_T^{v,\xi}}{H} I_{\{H > V_T^{v, \xi}\}} \Big].")
Problem [CP] Given a constant 
, minimize the cost to beat 
with probability 
, i.e. find
Problem [CR] Given a constant , minimize the cost to beat with ratio , i.e. find
![\displaystyle \inf_v\Big\{v: \mathbb{E} \Big[ I_{\{H\le V_T^{v, \xi}\}} + \frac{V_T^{v,\xi}}{H} I_{\{H > V_T^{v, \xi}\}} \Big] \ge 1-\varepsilon \hbox{ for some } \xi\in \mathcal{A}(v) \Big\}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cinf_v%5CBig%5C%7Bv%3A+%5Cmathbb%7BE%7D+%5CBig%5B+I_%7B%5C%7BH%5Cle+V_T%5E%7Bv%2C+%5Cxi%7D%5C%7D%7D+%2B+%5Cfrac%7BV_T%5E%7Bv%2C%5Cxi%7D%7D%7BH%7D+I_%7B%5C%7BH+%3E+V_T%5E%7Bv%2C+%5Cxi%7D%5C%7D%7D+%5CBig%5D+%5Cge+1-%5Cvarepsilon+%5Chbox%7B+for+some+%7D+%5Cxi%5Cin+%5Cmathcal%7BA%7D%28v%29+%5CBig%5C%7D.&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle \inf_v\Big\{v: \mathbb{E} \Big[ I_{\{H\le V_T^{v, \xi}\}} + \frac{V_T^{v,\xi}}{H} I_{\{H > V_T^{v, \xi}\}} \Big] \ge 1-\varepsilon \hbox{ for some } \xi\in \mathcal{A}(v) \Big\}.")
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