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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. $\int^t_0 \mathbf{1}_{F_i} \bigl( Z(s) \bigr) dY_i (s) = Y_i (t), \quad t \geq 0,\;i = 1, \dots, m.$

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