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Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedrons. (English. Russian original) Zbl 0854.60078
Theory Probab. Appl. 40, No. 1, 1-40 (1995); translation from Teor. Veroyatn. Primen. 40, No. 1, 3-53 (1995); correctional note Theory Probab. Appl. 50, No. 2, 346-347 (2006); translation from Teor. Veroyatn. Primen. 50, No. 2, 409 (2005).
Let \(\{n_i\), \(i = 1, \dots, m\}\) be a set of \(d\)-dimensional unit vectors, \(\{b_i\), \(i = 1, \dots, m\}\) a set of real numbers, \(S = \{x \in R^d : (n_i, x) \geq b_i\), \(i = 1, \dots, m\}\), assuming that for each \(i = 1, \dots, m\) the subsets \(F_i = \{x \in S : (n_i, x) = b_i\}\) are \((d - 1)\)-dimensional. Let \(\theta\) be a \(d\)-dimensional vector, \(\Gamma\) be a \((d \times d)\)-matrix and \(R\) be a \((d \times m)\)-matrix. General conditions are found for existence and weak uniqueness of the semimartingale reflecting Brownian motion \(Z\), corresponding to the data \((S, \theta, \Gamma, R)\) and starting from \(x \in S\), i.e. the continuous \(S\)-valued stochastic process such that \(Z(t) = X(t) + RY(t)\), \(t \geq 0\), where \(X\) is a Brownian motion starting from \(x\) with the drift vector \(\theta\) and the covariance matrix \(\Gamma\), and \(Y = (Y_1, \dots, Y_m)\) is an \(m\)-dimensional process with the nondecreasing continuous components \(Y_i\) starting from zero such that a.s. \[ \int^t_0 \mathbf{1}_{F_i} \bigl( Z(s) \bigr) dY_i (s) = Y_i (t), \quad t \geq 0,\;i = 1, \dots, m. \]

60J60 Diffusion processes
60G44 Martingales with continuous parameter
60J65 Brownian motion