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Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping. (English) Zbl 1165.60020
In the paper a class of pathwise inequalities of the form $$H(B_t)\geq M_t+F(L_t)$$, where $$B_t$$ is Brownian motion, $$L_t$$ its local time at zero and $$M_t$$ a local martingale, is investigated. The concrete nature of the representation makes the inequality useful for a variety of applications. An example of such usage is constructions and optimality results of Vallois’ Skorokhod embeddings [see [P. Vallois, in: Semin. de probabilites XVII, Proc. 1981/82, Lect. Notes Math. 986, 227–239 (1983; Zbl 0512.60071) and Stochastic Processes Appl. 41, No. 1, 117–155 (1992; Zbl 0754.60089)]. The discussion of their financial interpretation in the context of robust pricing and hedging of options written on the local time is presented. Another application of the inequalities, included in the paper, is solution of a class of optimal stopping problems of the form sup$$_{\tau} \mathbb E[F(L_{\tau})-\int _0^{\tau} \beta(B_s)ds]$$. The solution is given via a minimal solution to a system of differential equations and thus resembles the maximality principle described by G. Peškir, [Ann. Probab. 26, No. 4, 1614–1640 (1998; Zbl 0935.60025)].

##### MSC:
 60G40 Stopping times; optimal stopping problems; gambling theory 60G44 Martingales with continuous parameter 91B28 Finance etc. (MSC2000)
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