2019/03/09 by Mihail Zervos, Néofytos Rodosthenous, Zervos, Mihail +5
Economics, Econometrics and Finance · Social Sciences · #60G40 #60J55 #60J60 #91G80 #Climate Change Policy and Economics #FOS: Mathematics #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1903.03834
openalex publication_date 2019/03/09 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
We consider the problem of optimally stopping a general one-dimensional\nstochastic differential equation (SDE) with generalised drift over an infinite\ntime horizon. First, we derive a complete characterisation of the solution to\nthis problem in terms of variational inequalities. In particular, we prove that\nthe problem's value function is the difference of two convex functions and\nsatisfies an appropriate variational inequality in the sense of distributions.\nWe also establish a verification theorem that is the strongest one possible\nbecause it involves only the optimal stopping problem's data. Next, we derive\nthe complete explicit solution to the problem that arises when the state\nprocess is a skew geometric Brownian motion and the reward function is the one\nof a financial call option. In this case, we show that the optimal stopping\nstrategy can take several qualitatively different forms, depending on parameter\nvalues. Furthermore, the explicit solution to this special case reveals that\nthe so-called "principle of smooth fit" does not hold in general for this type\nof optimal stopping problems in standard senses that this can be formulated.\n