Discrete approximation of optimal stopping time in the problem of irreversible investment (Q2740065)

The author considers the problem of finding the stopping time \(\tau^{*}\in [0,T]\) such that NEWLINE\[NEWLINEE[X_1(\tau^{*})-X_2(\tau^{*})]= \sup_{\tau}E[X_1(\tau)- X_2(\tau)],NEWLINE\]NEWLINE where \(X_{i}(t),\;i=1,2\), are solutions of the stochastic differential equations NEWLINE\[NEWLINEdX_{i}(t)= b_{i}X_{i}(t) dt+ X_{i}(t)[q_{i1}dB_1(t)+ q_{i2}dB_2(t)] , \quad X_{i}(0)=x_{i};NEWLINE\]NEWLINE \(B_{i}(t)\), \(i=1,2,\) are Brownian motions. The sequence of stopping times that is easy to calculate and which approximates an optimal stopping time is constructed.