Optimal stopping times for solutions of nonlinear stochastic differential equations and their applications to a problem of financial mathematics (Q2739631)

Let the capital \(X_t^0\) of an investor at the moment \(t\in[0,\tau]\) be described by the equation NEWLINE\[NEWLINE X_t^0=X_0+\int_0^ta_1(s,X_s^0) ds+ \int_0^tb(s,X_s^0) dw_s, \quad 0\leq t\leq T, NEWLINE\]NEWLINE and let the capital \(X_t^{\tau}\) of an investor at the moment \(t\in[\tau,T]\) be described by the equation NEWLINE\[NEWLINE X_t^{\tau}= e^{-\alpha\tau}X^0_{\tau}+ \int_{\tau}^ta_2(s,X_s^{\tau}) ds+ \int_{\tau}^tb(s,X_s^{\tau}) dw_s, \quad \tau\leq t\leq T, NEWLINE\]NEWLINE where \(w_s\) is a Wiener process, \(\tau\) is a switching time, \(e^{-\alpha\tau}\) is the discount factor. The problem is to find the optimal switching time between two alternative strategies at the financial market. The optimal switching time \(\tau\in[0,T]\) is found as the optimal stopping time for certain process \(Y(t)\) generated by the process \(X(t)\) in such a way that the average capital of an investor attains its maximum.