Abstract. We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a three-parameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Lévy processes. 1.

This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a two-parameter extension of the Al-Salam–Chihara polynomials and a relation between these polynomials for different values of parameters. 1. Introduction. The

Abstract: We introduce a non-commutative extension of Tsirelson-Vershik’s noises [TV98, Tsi04], called (non-commutative) continuous Bernoulli shifts. These shifts encode stochastic independence in terms of commuting squares, as they are familiar in subfactor theory [Pop83, GHJ89]. Such shifts are, in particular, capable of producing Arveson’s product system of type I and type II [Arv03]. We investigate the structure of these shifts and prove that the von Neumann algebra of a (scalar-expected) continuous Bernoulli shift is either finite or of type III. The role of (‘classical’) G-stationary flows for Tsirelson-Vershik’s noises is now played by cocycles of continuous Bernoulli shifts. We show that these cocycles provide an operator algebraic notion for Lévy processes. They lead, in particular, to units and ‘logarithms ’ of units in Arveson’s product systems [Kös04a]. Furthermore, we introduce (non-commutative) white noises, which are operator algebraic versions of Tsirelson’s ‘classical ’ noises. We give examples coming from probability, quantum probability and from Voiculescu’s theory of free probability [VDN92]. Our main result is a bijective correspondence between additive and unital shift cocycles. For the proof of the correspondence we develop tools which are of interest on their own: non-commutative extensions of stochastic Itô integration, stochastic logarithms and exponentials.

Quadratic harness condition: = Qt,s,u(Xs, Xu), (3) E ( X 2 ∣ t Fs,u math.uc.edu/~brycw/preprint/5-param/q-harnesses.pdf where Qt,s,u(x, y) = At,s,ux 2 +Bt,s,uxy +Ct,s,uy 2 +Dt,s,ux+Et,s,uy +Ft,s,u (4)

Abstract. We present some properties of measures orthogonalizing set of q −Hermite polynomials so called q −Gaussian measures. We also present an algorithm simmulating i.i.d. sequencs of random variables having q−Gaussian distribution. 1.

Abstract. We present some properties of measures (q−Gaussian) that orthogonalize the set of q−Hermite polynomials. We also present an algorithm for simulating i.i.d. sequences of random variables having q−Gaussian distribution. 1.

Abstract. We present some properties of measures orthogonalizing set of q −Hermite polynomials so called q −Gaussian measures. We also present an algorithm simmulating i.i.d. sequencs of random variables having q−Gaussian distribution. 1.

Abstract. We present some properties of measures orthogonalizing set of q −Hermite polynomials so called q −Gaussian measures. We also present an algorithm simmulating i.i.d. sequencs of random variables having q−Gaussian distribution. 1.

Abstract. We present some properties of measures orthogonalizing set of q −Hermite polynomials so called q −Gaussian measures. We also present an algorithm simmulating i.i.d. sequencs of random variables having q−Gaussian distribution. 1.

class of non-commutative generalized stochastic processes with freely independent values