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Stochastic differential equations driven by \(G\)-Brownian motion with reflecting boundary conditions. (English) Zbl 1304.60068
The author considers a \(G\)-Brownian motion, stochastic integrals with respect to a \(G\)-Brownian motion and with respect to an increasing process, and a reflected \(G\)-Brownian motion. He proves the \(G\)-Itō formula.
These results are used to show the existence and uniqueness of the solution \((X,K)\) to the stochastic differential equation driven by the \(G\)-Brownian motion: \[ X_t= x+\int^t_0 f_s(X_s)\,ds+ \int^t_0 h_s(X_s)\,d\langle B\rangle_s+ \int^t_0 g_s(X_s)\,dB_s+ K_t,\tag{1} \] \(0\leq t\leq T\), where \(x\) is the initial condition, \(\langle B\rangle\) is the quadratic variation process of the \(G\)-Brownian motion \(B\), and \(K\) is an increasing process. The coefficients \(f, h, g\) satisfy a Lipschitz condition.
The author also establishes a comparison principle for solutions of (1).

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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