On Markovian Cocycle Perturbations in Classical and Quantum Probability

We introduce Markovian cocycle perturbations of the groups of transformations associated with the classical and quantum stochastic processes with stationary increments, which are characterized by a localization of the perturbation to the algebra of events of the past. It is namely the definition one needs because the Markovian perturbations of the Kolmogorov flows associated with the classical and quantum noises result in the perturbed group of transformations which can be decomposed in the sum of a part associated with deterministic stochastic processes lying in the past and a part associated with the noise isomorphic to the initial one. This decomposition allows to obtain some analog of the Wold decomposition for classical stationary processes excluding a nondeterministic part of the process in the case of the stationary quantum stochastic processes on the von Neumann factors which are the Markovian perturbations of the quantum noises. For the classical stochastic process with noncorrelated increaments it is constructed the model of Markovian perturbations describing all Markovian cocycles up to a unitary equivalence of the perturbations. Using this model we construct Markovian cocyclies transformating the Gaussian state $\rho $ to the Gaussian states equivalent to $\rho $.


Introduction.
It is well known that every stochastic process with finite second moments and continuous in square mean can be considered as a continuous curve in the Hilbert space. In this framework, properties of the process, such as stationarity and noncorrelativity of the increments, appear as an invariance under the action of the group of unitaries in the Hilbert space and an orthogonality of the curve increments, respectively. Thus to investigate the stochastic process, one can use functional analysis techniques. This approach was introduced by Kolmogorov who considered in [26,27] a classification problem for the equivalence classes of continuous curves ξ = (ξ t ) t∈R in a Hilbert space H, which are invariant with respect to a strongly continuous one-parameter group of unitaries U = (U t ) t∈R (i.e., U t (ξ s − ξ r ) = ξ s+t − ξ s+r ) with the transformations of equivalence defined by the formulã ξ t = W ξ t + η, (1.1) where W and η are a unitary operator in H, and an element of H, correspondingly. Indeed, the continuous curveξ = (ξ t ) t∈R is invariant with respect to the group of unitariesŨ = (W t U t ) t∈R , where the one-parameter family of unitaries satisfies the condition of multiplicative U -cocycle, The multiplicative cocycle generated by a unitary operator W as in (1.2) is said to be U -coboundary. Notice that not every cocycle (1.3) is coboundary (1.2). Here we use ordinary definitions of the cohomologies of group theory for 1-cocycle and 1-coboundary associated with the standard bar-resolvent of the group R with values in the multiplicative group ᐁ(H) of all unitaries in the Hilbert space H with the module structure defined by the group action x → U t xU −t , x ∈ ᐁ(H), t ∈ R (see, e.g., [12,18]). Suppose that there exists a continuous curve ξ = (ξ t ) t∈R which is invariant with respect to the group U , and the increments of ξ are orthogonal such that ξ t 1 − ξ s 1 ⊥ ξ t 2 − ξ s 2 for all disjoint intervals (s 1 ,t 1 ) ∩ (s 2 ,t 2 ) = ∅. The curves of such type were called in [27] the Wiener spirals. Every multiplicative U -cocycle (W t ) t∈R , which is strongly continuous in t, defines a new strongly continuous one-parameter group of unitariesŨ = (W t U t ) t∈R . We will callŨ a cocycle perturbation of the group U . If (W t ) t∈R is a coboundary, then, for the cocycle perturbationŨ, there also exists an invariant curveξ which is the Wiener spiral. Indeed, this curve can be constructed by formula (1.1), where one must substitute for W the unitary operator generating the coboundary by means of (1.2). For arbitrary cocycle (W t ) t∈R which is not a coboundary, it is possible that there does not exist a Wiener spiral which is invariant with respect to the cocycle perturbationŨ. In this paper, we introduce a subset (2.2) of the set of all cocycles such that being given a cocycle from (2.2) generates the cocycle perturbation possessing the invariant Wiener spiral if the perturbing group satisfies this condition. The class of cocycle perturbations we propose is important because in applications (see [10,21,24,29,33]), the problem of constructing the dilation of a quantum positive evolution to a cocycle on the Fock space is often posed. The cocycles of this type are adapted with respect to a Fock filtration and can be constructed by the second quantization of cocycles from the subset (2.2) we introduce. Cocycles from (2.2) will be called Markovian. The usage of the term "Markovian cocycle" follows from [1,2]. The Markovian cocycles of [1,2] are connected with the classical Markov property in the sense that the perturbation of the Markovian stochastic process by the Markovian cocycle is also the Markovian stochastic process with the same algebras of the present, future, and past. Our definition is different from the definition in [1,2]. Nevertheless, under an appropriate interpretation, the perturbations we consider are Markovian in the sense of the definition in these cited papers, describing a wider class of perturbations. We consider this as a sufficient motivation for using this terminology.
The model example of the group of unitaries U possessing an invariant Wiener spiral is the group of shifts in the Hilbert space H = L 2 (R) defined by the formula (U t η)(x) = η(x + t), η ∈ H. In fact, the set of functions ξ t (x) = 1, x ∈ [−t, 0], ξ t (x) = 0, x ∉ [−t, 0], is the Wiener spiral which is invariant with respect to U (see [27]). We will call "a past" and "a future" of the system the subspaces H t] and H [t containing functions with the support belonging to [t, +∞) and (−∞,t], respectively. Notice that H t] = U t H 0] and H [t = U t H [0 , t ∈ R. In the case we consider, we call the cocycle W Markovian if the restriction of the unitary operator W t to the subspace of the future H [t is an identity transformation for every fixed t ≥ 0. In particular, this definition guarantees that the subspace of the past H 0] is invariant with respect to W −t for all t ≥ 0, which allows to consider the restriction of the cocycle perturbationŨ −t | H 0] , t ≥ 0. LetŨ be a cocycle perturbation of U by the cocycle which is Markovian in the sense of our definition. We will show that every such perturbation can be represented in the formŨ =Ũ (1) ⊕Ũ (2) , whereŨ (1) is an arbitrary group of unitaries in the subspace of the past H 0] andŨ (2) is unitarily equivalent to the initial group of shifts. This representation can be called the Wold decomposition of the cocycle perturbation ofŨ . The groupsŨ (1) andŨ (2) can be interpreted as associated with deterministic and nondeterministic parts of the process.
By a quantum stochastic process we mean (see [21,24,33]) a strongly continuous one-parameter family x = (x t ) t∈R of linear (unbounded in general) operators in a Hilbert space H. Usually one needs to require that all operators x t be closed and have the invariant common domain which is dense in H. Under this definition, stationary quantum stochastic process x = (x t ) t∈R can be defined by the condition x t = α t (x 0 ) + x 1 , t ∈ R, where α is a certain group of automorphisms and x 0 , x 1 are two fixed linear operators. In this way, a quantum stochastic process with stationary increments is a continuous operator-valued curve which is invariant with respect to a certain group of automorphisms α. In quantum probability theory, the role of the σ -algebras of events associated with the stochastic process is played by the von Neumann algebras generated by increments of the process, that is, the ultra-weak closed algebras of bounded operators in H obtained as the second commutant of the set of the increments x t − x s of the process x (see, e.g., [11,21,24]). Every classical stochastic process ξ consisting of the random variables ξ t ∈ L ∞ (Ω) can be considered as a quantum, where the operators x t = M ξ t forming the process are the operators of multiplications by the functions ξ t in the Hilbert space L 2 (Ω). Definition 1.1. The flow {T t , t ∈ R} on the probability space (Ω, ᏹ,µ) is said to be a Kolmogorov flow (see [28]) if there exists a σ -algebra of events In [15], a notion of the Kolmogorov flow was generalized into quantum probability. Here the σ -algebras of events ᏹ t] are replaced by the corresponding von Neumann algebras. Notice that in the well-known monograph [25], where the conditions on the spectral function of process, which result in the Kolmogorov flow were investigated, a term "completely nondeterministic process" is used for the stochastic process generating the Kolmogorov flow, while the term "Kolmogorov flow" is not used anywhere. Every classical or quantum process with independent (in the classical sense) increments results in the Kolmogorov flow. We define Markovian cocycle perturbations of the classical and quantum Kolmogorov flows such that the perturbed flows contain the parts which are isomorphic to the initial Kolmogorov flow. Notice that in the model situation of the Hilbert space L 2 (R) we considered above, the Kolmogorov flow can be associated with the flow of shifts. Thus, the possibility to exclude the part being the Kolmogorov flow in the perturbed dynamics can be considered as some analogue of the Wold decomposition allowing to exclude a nondeterministic part of the process.
For every classical or quantum stochastic process, a great role is played by the set of all (not necessarily linear) functionals of the process. In particular, for the Wiener process, we have the Wiener-Itô decomposition of the space of all L 2 -functionals into an orthogonal sum, which allows to solve effectively some stochastic differential equations (see [20]). Notice that from the viewpoint of the theory of the cohomologies of groups, the stochastic process with stationary increments determining the continuous curve being invariant with respect to the group of transformations U is an additive (1 − U)-cocycle. It is natural to consider the ring of cohomologies of all degrees generated by the (1−U )-cocycle of such type which can be interpreted as the space of all (nonlinear) functionals of the initial stochastic process. We show that the Markovian cocycle perturbations we introduced define homomorphisms of this ring of cohomologies.
This paper is organized as follows. In Section 2, for the group of unitaries U and the continuous curve ξ which is invariant with respect to U , we define a class of the Markovian U-cocycles. The group of unitariesŨ which is a perturbation of U by the cocycle of such type determines a continuous curveξ being invariant with respect toŨ and connected with the curve ξ by formula (1.1), where W is (nonunitary in general) isometrical operator satisfying the additional property of the localization of action to "the past." We call such isometrical operators Markovian and we prove that every Markovian operator is associated with a certain perturbation by a Markovian cocycle. The investigation we give in Section 3 shows in detail that the Markovian cocycle we introduced determines the Wold decomposition for the cocycle perturbation. In Section 4 we construct a model of the Markovian cocycle for the stochastic process with independent increments. The model we give allows to construct the Markovian cocycles with the property W t − I ∈ s 2 (the Hilbert-Schmidt class) which translate the fixed Gaussian measure to the equivalent Gaussian measures. In Section 5 we give the basic notion on the theory of Kolmogorov flows in classical and quantum probability. In Section 6, we define the rings of cohomologies generated by additive 1-cocycles and show in examples of the Wiener process and the quantum noise that the set of all functionals of the stochastic process with stationary increments can be considered as a ring of the cohomologies of the group which is composed by shifts of the increments in time. Moreover, we define a Markovian perturbation of the group resulting in a homomorphism of the ring of cohomologies of the group. Notice that for quantum stochastic processes x = (ξ t = x t ) t∈R , it is also possible to define transformations of the form (1.1), where one must take a morphism for W and a linear operator for η. In Section 7, we introduce the Markovian cocycle perturbations of the quantum noises being a generalization of the classical processes with independent stationary increments for the quantum case, which result in transformations of the quantum stochastic processes of the form (1.1). The Markovian perturbations we introduce determine homomorphisms of the ring of cohomologies associated with the stochastic process in the sense of Section 6. The techniques involved in Section 6 allow to obtain in Section 7 some analogue of the Wold decomposition for the classical stochastic processes in the quantum case, which permits to exclude a nondeterministic part of the process.

Stochastic processes with stationary increments as curves in a Hilbert
space. Let ξ = (ξ t ) t∈R be continuous in square mean stochastic process with stationary increments on the probability space (Ω, Σ,µ). Without loss of generality, we can suppose that the condition ξ(0) = 0 holds. Then, in the Hilbert space H ξ generated by the increments ξ t −ξ s , s, t ∈ R, one can define a strongly continuous group of unitaries U = (U t ) t∈R shifting the increments in time such that U t (ξ s − ξ r ) = ξ s+t − ξ r +t , s, t, r ∈ R, where ξ satisfies the condition of (additive) (1 − U )-cocycle, that is, ξ t+s = ξ t + U t ξ s , s, t ∈ R. Denote by H the Hilbert space with the inner product (ξ, η) = E(ξη) generated by the classes of equivalency of the random variables ξ, η in the space (Ω,µ), which possess finite second moments with respect to the Hilbert norm associated with the expectation E. We will identify the random variables with the elements of the Hilbert space H. A stochastic process ξ can be considered as a curve in the Hilbert space H ξ , which is invariant with respect to the action of the group U (see [26,27]). The space H ξ is a subspace of H but does not coincide with it in general. LetŨ be an arbitrary continuation of U to a strongly continuous group of unitaries in the Hilbert space H. Then ξ can also be considered as the curve in H, which is invariant with respect to the action of the groupŨ . Denote by Ᏽ(ξ) the set containing all possible strongly continuous groups of unitaries in H such that the stochastic process ξ is invariant with respect to them. One can define in the space H ξ an increasing family of subspaces H ξ t] generated by the increments ξ s − ξ r , s, r ≤ t, associated with "a past" before the moment t, and a decreasing family of subspaces H ξ [t generated by the increments ξ s − ξ r , s, r ≥ t, associated with "a future" after the moment t such that Notice that, for the processes with noncorrelated increments, the subspaces H ξ t] and H ξ [t are orthogonal. Fix the group U ∈ Ᏽ(ξ). The strongly continuous one-parameter family of unitaries W = (W t ) t∈R in H is said to be a (multiplicative) U -cocycle if the following condition holds: The cocycle W is called Markovian under the condition Property (2.1) exactly means that the strongly continuous one-parameter family of unitariesŨ = (W t U t ) t∈R forms a group. We consider Markovianity as a localization of the action of the cocycle W to the subspace of the past. Moreover, the Markovian property (2.2) preserves "a causality" such that "the future" of the system is not disturbed. Notice that our definition of a Markovian cocycle is based on the analogous definition introduced in [1,2] in a considerably more general case. We defer the examples of Markovian cocycles to Section 4, where a model of the Markovian cocycle is given for the group of shifts on the line, which describes all cocycles up to the unitary equivalence of perturbations.
We will call an isometrical operator R : H ξ (1) (1) [0 . (1) [−s , s, t ≥ 0, due to the Markovian property in the form (2.3). Hence, the limit exists for the set of elements f ∈ H ξ (1) [−s , s ≥ 0, which is dense in H ξ (1) . Thus, the strong limit exists by the Banach-Steinhaus theorem. The limiting operator W −∞ is Markovian because all operators W −t , t ≥ 0, satisfy this condition.

Proposition 2.3. In the space H, there exists a Markovian isometrical oper-
if and only if the stochastic processes ξ (1) and ξ (2) are connected by a Markovian cocycle W = (W t ) t∈R by the formula ξ Here we used the Markovian property for the operator R, which implies that ξ (1) t , t ≤ 0, and a cocycle W is Markovian. Then, due to Proposition 2.2, one can define a Markovian isometrical

Processes with noncorrelated increments: the Wold decomposition.
Given a stationary stochastic process ξ, there exists a group of unitaries U = (U t ) t∈R in the Hilbert space with the inner product defined by the formula It is evident that there exist processes which are neither nondeterministic nor deterministic. A stationary process ξ has the unique decomposition ξ = ξ (1) ⊕ξ (2) , where ξ (1) and ξ (2) are deterministic and nondeterministic processes, respectively, such that ξ (1) and ξ (2) have noncorrelated increments. In its turn, a nondetermnistic process ξ (2) is uniquely defined by the Wold decomposition where ζ(ds) is a noncorrelated measure such that E|ζ(ds)| 2 = ds and E(ζ(∆)ζ(∆ )) = 0 for all measurable disjoint sets ∆ and ∆ (see [35]). Thus, every stationary process ξ t = U t ξ 0 uniquely defines the process with noncorrelated stationary increments ζ t , which is an invariant curve with respect to the group U . This process can be called "the nondeterministic part" of ξ t . In the following, we will call the Wold decomposition the possibility to associate with the fixed stationary process the process with stationary noncorrelated increments, which is its nondeterministic part in the sense given above. Let V = (V t ) t∈R+ be a strongly continuous semigroup of nonunitary isometrical operators in a Hilbert space H. In functional analysis, the Wold decomposition is a decomposition of the form H = H (1) ⊕ H (2) , where subspaces H (1) and H (2) reduce the semigroup V to a semigroup of unitary operators and a semigroup of completely nonunitary isometrical operators, respectively. A completely nonunitary isometrical operator is characterized by the property that there is no subspace reducing it to a unitary operator. Every strongly continuous semigroup consisting of completely nonunitary isometrical operators is unitary equivalent to its model, which is the semigroup of right shifts S = (S t ) t∈R+ in the Hilbert space Recall that a deficiency index of the generator d = s − lim t→0 ((V t − I)/t) of the strongly continuous semigroup V is a number of linear independent solutions to the equation d * f = −f . The Hilbert space of values has the dimension equal to the deficiency index of the generator of V (see [32]). In the following, we will call the deficiency index of the generator an index of the semigroup. Notice that every semigroup of completely nonunitary isometrical operators V with the index n > 0 determines n noncorrelated processes ξ (i) , 1 ≤ i ≤ n, with noncorrelated increments such that ξ In the model case of the semigroup of right shifts S in L 2 (R + , ), the processes of such type can be constructed in the following way. Choose the orthonormal basis of the space consisting of the elements e i , 1 ≤ i ≤ n ≤ +∞, and put ξ (i) = e i ⊗χ [0,t] , where χ [0,t] is an indicator function of the interval [0,t]. We will investigate a behavior of the processes with noncorrelated increments with respect to perturbations by the Markovian cocycles we introduced in Section 2.
Proposition 3.1. Let ξ and W be the process with noncorrelated increments and the Markovian cocycle, respectively. Then, ξ t = W t ξ t , t ≤ 0 and ξ t = ξ t , t > 0, is a process with noncorrelated increments.
Proof. It follows from Proposition 2.3 that the Markovian isometrical operator W −∞ = s − lim t→+∞ W −t connects the perturbed process with the initial process by the formula Let the process with noncorrelated increments ξ = (ξ) t∈R be invariant with respect to the group of unitaries U = (U t ) t∈R . Proof. Every semigroup of completely nonunitary isometrical operators with unit index is unitarily equivalent to its model which is the semigroup of right shifts S = (S t ) t≥0 acting in the Hilbert space Linear combinations of the elements of the curve η form a dense set in the space L 2 (R + ), and η t+s = η t + S t η s , s, t ≥ 0. Notice that the stationarity and the orthogonality of the curve increments imply ξ t t≥0 be the semigroup of completely nonunitary isometrical operators in H ξ 0] defined in Proposition 3.2. (1) ⊕ H (2) associated with the semigroupṼ can be done by the condition H (2) Proof. It follows from Propositions 2.2 and 2.3 that there exists a limit (1) be a subspace of H ξ defined by the condition of orthogonality to all elements W −∞ ξ t , t ∈ R. This subspace is invariant with respect to the action of the group of unitaries W t U t , t ∈ R. On the other hand, by means of the Markovian property for W −∞ , the subspace H (1) is orthogonal to all elements W −∞ ξ t = ξ t , t ≥ 0, and, therefore, (1) , t ≥ 0, consists of unitary operators. The subspace H (2) ⊂ H ξ is determined by the condition of orthogonality to H (1) , which is H (2) = W −∞ H ξ by the definition of H (1) . In this way, the restriction V | H (2) is a semigroup of completely nonunitary isometrical operators by means of Proposition 3.2.

A model of the Markovian cocycle for the process with noncorrelated increments. Let S = (S t ) t∈R be a flow of shifts on the line acting by the formula
The group S is naturally associated with the stochastic process ξ t = χ [−t,0] with noncorrelated increments such that ξ t+s = ξ t + S t ξ s , s, t ∈ R. Let the subspace H t] consist of functions f with the support supp f ⊂ [−t, +∞). Then H t] is generated by the increments Here ᏸ is the Laplace transformation and Θ is a multiplication operator by Θ which is an inner function in the semiplane Re λ ≥ 0. Recall that a function Θ(λ) is said to be inner if it is analytical in the halfplane Re λ ≥ 0 and its modulus equals one on the imaginary axis (see [32]). Denote by P [0,t] , P [t,+∞) , P ᐂ , and P ᐂ ⊥ orthogonal projections on the subspaces of functions with the support belonging to the segment [0,t] and the interval [t, +∞), and on ᐂ and ᐂ ⊥ , respectively. Proposition 3.3 shows that arbitrary perturbation of the group of shifts associated with the process with noncorrelated increments by a Markovian cocycle W is completely described by a unitary part R =Ṽ | H (1) of the semigroup of isometriesṼ = (W −t V t ) t∈R+ . In the following theorem, we construct the Markovian cocycles resulting in the semigroup R which is unitarily equivalent to the one we set. In this way, we introduce a model describing all Markovian cocycles up to unitary equivalence of perturbations.
and extended for positive t by the formula Proof. Notice that the projection P ᐂ and the isometrical operator M Θ are commuting with the right shifts T t = S −t , t ≥ 0, and the projections P [t,+∞) . Hence, the family of unitary operators t≥0 is the semigroup of nonunitary isometrical operators with unit index and unitary part R.
Below, using the model of Markovian cocycle introduced in Theorem 4.1, we construct the Markovian cocycle satisfying the property W t − I ∈ s 2 , t ∈ R. Cocycles of such type can be named inner for further applications in quantum probability (see [3,5,6]). In fact, this condition appears, particularly, as the condition of innerness for the quasifree automorphism of the Fermion algebra, which is generated by the unitary operator W (see [3,5,31]). It is possible to explain why this condition appears in the following way. Attempts to define a measure in a Hilbert space H result in constructing the measure of white noise on the space E * involved in the triple E ⊂ H ⊂ E * , where we denote by E * the adjoint space of linear functionals on the space E which is dense in H (see [20]). This situation is realized, particularly, if ξ ∈ H are included in the parameter set of the generalized stochastic process. Given ξ ∈ E, x ∈ E * , denote by x, ξ the corresponding dual product. Notice that, if x ∈ H, then x, ξ = (x, ξ) coincides with the inner product in H. Fix a positive bounded operator R in the space H. Then there exists a space (Ω,µ) with the Gaussian measure µ such that E * e i x,ξ dµ(x) = e −(ξ,Rξ) , ξ ∈ E ⊂ H. Suppose that a unitary operator W in H does not map elements of H but the measure µ to certain other Gaussian measureμ such that E * e i<x,ξ> dμ(x) = e −(ξ,W * RW ξ) , ξ ∈ E ⊂ H. It is natural to ask: when are the Gaussian measures determined by the operators R and W * RW equivalent? The Feldman criterion (see [16,17]) gives the condition (ξ, Rξ)−(ξ, W * RW ξ) = (ξ, ∆ξ), where ∆ is a Hermitian operator of the Hilbert-Schmidt class. Thus, R −W * RW ∈ s 2 , which can be rewritten as W R −RW ∈ s 2 .
The condition of the type given above is satisfied for all positive operators R if W − I ∈ s 2 . Notice that the condition of the Feldman type appears in [7] as the condition of quasiequivalence for Gaussian states on the Boson algebra.
Let the inner function Θ involved in the condition of Theorem 4.1 be the Blaschke product (see [32]) constructed from the complex numbers λ k , 1 ≤ k ≤ N ≤ +∞, in the following way. By means of [3,4,5], take real parts such that Re λ k < 0, N k=1 | Re λ k | < +∞; imaginary parts can be chosen arbitrary, then Θ(λ) = N k=1 ((λ+λ k )/(λ−λ k )). The Blaschke product Θ(λ) is a regular analytical function in the semiplane Re λ > 0 and its module equals one on the imaginary axis. The corresponding subspace ᐂ of the Hilbert space H 0] = L 2 (R + ), which is invariant with respect to the semigroup of right shifts, is determined by the condition of orthogonality to all exponents e λ k x , 1 ≤ k ≤ N. Let the functions g k , 1 ≤ k ≤ N, be obtained by the successive orthogonalization of the system e λ k x . Then (g k ,g l ) = δ kl and g k , 1 ≤ k ≤ N, form an orthonormal basis of the space ᐂ ⊥ . Define a C 0 -semigroup of unitaries R = (R t ) t∈R by the formula R t g k = e i Im λ k t g k , 1 ≤ k ≤ N, t ∈ R + .

Proof of Theorem 4.2. Notice that
, t > 0. Thus, we need to prove a convergence of the series , t > 0, and prove a convergence of the series associated with these parts, that is, To check (4.2), it is sufficient to prove a convergence of the series N k=1 (R t − P ᐂ ⊥ S * t P ᐂ ⊥ )g k 2 , where the functions (g k ) N k=1 forming the orthonormal basis of the space ᐂ ⊥ are obtained by a successive orthogonalization of the exponents e λ k x . One can represent R t as a sum of R (1) t and R (2) t , where R (1) t g k = e λ k t g k , R (2) t g k = (e i Im λ k t − e λ k t )g k , 1 ≤ k ≤ N, t > 0. Then the series N k=1 (R (1) t − P ᐂ ⊥ S * t P ᐂ ⊥ )g k 2 converges by the theorem on the triangulation of the truncated shift (see [32]). Notice that N k=1 |e i Im λ k t − e λ k t | = N k=1 |1 − e Re λ k t | and the last series converges because N k=1 | Re λ k | < +∞. Therefore, R (2) t ∈ s 1 (the first Schatten class) such that k (x) = 0, x ∈ R − , and f (2) k (x) = 0, x < t. Using the Parseval equality for the Laplace transformation and taking into account that the Blaschke product Θ is an isometrical operator, we obtain and f (2) k . Applying the analytical functions techniques, we get Hence, Substituting in (4.5) the formulas (4.6) and (4.8), we obtain where C 1 , C 2 , and C 3 are some positive constants.

Processes with independent increments in classical and quantum probability: Kolmogorov flows. Denote by ᏸ s (H) and σ 1 (H) the sets of linear Hermitian and positive unit-trace operators in a Hilbert space H.
In the quantum probability theory, the elements x ∈ ᏸ s (H) and ρ ∈ σ 1 (H) are called random variables or observables and states of the system, respectively. Consider the spectral decomposition x = λdE λ of the random variable x ∈ ᏸ s (H), where E λ is a resolution of the identity in H. Then a probability distribution of x in the state ρ ∈ σ 1 (H) is defined by the formula P (x < λ) = Tr ρE λ . Thus, the expectation of x in the state ρ can be calculated as E(x) = Tr ρx (see [21,24]). Notice that the classical random variables from L ∞ (Ω) can be considered as linear operators of multiplication by the function in the Hilbert space H = L 2 (Ω), where Ω is some probability space. A quantum stochastic process (in the narrow sense of the word) is a strongly continuous family of operators (unbounded in general) x t ∈ ᏸ s (H), t ∈ R-"quantum observables." In applications (see [8,9,33]), one needs to require that the operators x t be closed and have the invariant common domain which is dense in H. We will call the quantum stochastic process a process with stationary increments if there exists an ultraweak continuous one-parameter group of * -automorphisms α t , t ∈ R, of the algebra of all bounded operators in H such that x t+s = x t + α t (x s ), s, t ∈ R. We do not suppose that the operators x t are bounded and assume that the action of α = (α t ) t∈R is correctly defined on x = (x t ) t∈R . The quantum stochastic process with stationary increments x t is called a stationary process if there exists x ∈ ᏸ s (H) such that x t = α t (x) − x, t ∈ R. As in the case of classical stochastic processes, the quantum stochastic process which is continuous in the square mean determines a continuous curve in the Hilbert space with the inner product defined by the expectation. This curve will be denoted by [x] = ([x t ]) t∈R . As we have identified continuous curves in a Hilbert space with classical stochastic processes, we obtain that a quantum stochastic process x can be associated with the classical stochastic process ξ = [x]. If x is a process with stationary increments or a stationary process and the state ρ ∈ σ 1 (H) is invariant with respect to the action of the group α, that is, Tr(ρα t (a)) = Tr(ρa), t ∈ R, where a is an arbitrary linear combination of the random variables x t , t ∈ R, then the classical stochastic process ξ = [x] is also stationary. The group of unitaries U shifting the increments ξ in time is defined by the formula (U t ξ s ,ξ r ) = Tr(ρα t (x s )x r ), s, t, r ∈ R. Because the notion of expectation in the quantum probability theory plays a major role, we will use it to define an independence in the classical probability theory and also, for convenience, to pass to quantum probability in the following. The classical stochastic process x = (x t ) t∈R is said to be a process with independent increments or a Lévy process if the identity holds for arbitrary choice of functions φ i ∈ L ∞ and disjoint intervals (s i ,t i ).
There are different approaches to a definition of the Lévy processes in quantum probability (see [21,24]). Anyway, besides that the condition (5.1) must be satisfied, they involve certain additional conditions concerning the algebraic structure of the process x. One of them is commutativity for increments, that is, for disjoint intervals (s i ,t i ). Condition (5.2) is associated with the bosonic independency and it is not a unique possibility (for the fermionic case, see [9]). Notice that, for the quantum processes with independent increments, it is possible to define the representation of the Lévy-Hinchin type (see [22,23]). For convenience, we recall Definition 1.1. In formula (1.6), we mean that the intersection of σ -algebras contains only two events which are the empty set and the whole space Ω. Let ᏹ t] be generated by the events associated with the stationary stochastic process ξ s , s < t. The investigation of the conditions which lead to the Kolmogorov flow generated by ξ is given in [25]. In particular, the flow of the Wiener process is a Kolmogorov flow (see [20]). Notice that to obtain the Kolmogorov flow from the stochastic process, it is not necessary to require the independence of the increments. Consider the quantum stochastic process with stationary increments x = (x t ) t∈R . In the quantum probability theory, a role of the σ -algebras of events is played by the von Neumann algebras generated by the quantum random variables. In this way, the conditional expectation is a completely positive projection on the von Neumann algebra (see [21,24] [15]). For classical random variables ξ with Eξ = 0, the condition Dξ = Eξ 2 = 0 means that ξ = 0 almost surely. The situation is different in the quantum case. For a quantum random variable x with Ex = 0, the equality Dx = Ex 2 = Tr ρx 2 = 0 does not imply that x = 0. The property of this type characterizes the expectation E = E ρ . If the situation appears, that is, Tr ρx = 0 implies x = 0 for all positive operators x belonging to a certain algebra ᏹ, then ρ is said to determine a faithful state E(·) = Tr ρ· on ᏹ.
In the following section, we will give an important example of the algebraic Kolmogorov flow, which complements the example of Proposition 5.1 in some sense.
6. Cohomology of groups as a language describing the perturbations of the space of all functionals from stochastic process. We recall some notion of the cohomology of groups (see, e.g., [12,18]). Define a certain action α of the real line R as an additive group on the algebra ᏹ. The element I ∈ Hom(R k , ᏹ) is said to be (additive) (k − α)-cocycle if the following identity is satisfied: α t 1 I t 2 ,t 3 ,...,t k+1 − I t 1 + t 2 ,t 3 ,...,t k+1 +··· + (−1) i I t 1 ,...,t i−1 ,t i + t i+1 ,t i+2 ,...,t k+1 +··· + (−1) k+1 I t 1 ,...,t k = 0, (6.1) t i ∈ R, 1 ≤ i ≤ k + 1. The (k − α)-cocycle I is said to be a coboundary if there exists an element J ∈ Hom(R k−1 , ᏹ) such that I t 1 ,...,t k = α t 1 J t 2 ,...,t k − J t 1 + t 2 ,...,t k +··· Denote by C k and B k the sets of all additive (k − α)-cocycles and (k−α)-coboundaries, correspondingly. Then H k = H k (α) = C k /B k is called a kth group of cohomologies of α with values in ᏹ. Define a cohomological multiplication ∪ : H k × H l → H k+l by the formula I k ∪ I l t 1 ,...,t k+l = I k t 1 ,...,t k α t 1 +···+t k I l t k+1 ,...,t k+l , (6.3) H i is a graded ring with respect to the multiplication ∪. Notice that a group of 0-cohomologies was omitted because we do not need it in the following. Now let α and α be two actions on the algebra ᏹ. We will say that the rings H(α) and H(α ) are isomorphic if there is a one-to-one correspondence w : H(α) → H(α ) mapping each H k (α) to H k (α ) and the action of α to the action of α .
The one-parameter family of automorphisms w = (w t ) t∈R of the algebra ᏹ is said to be a (multiplicative) α-cocycle if the following condition holds: w t+s = w t • α t • w s • α −t , s, t ∈ R, w 0 = Id. We will call a multiplicative αcocycle Markovian with respect to a certain set Ᏽ of additive (1 − α)-cocycles I(t) ∈ Ᏽ if the condition w t (I(t + s)− I(t)) = I(t +s)−I(t), t, s ≥ 0, is satisfied. The following proposition can be considered as some "abstract generalization" of the properties of the Markovian perturbations we have described in the previous sections. Proposition 6.1. Let the ring A ⊂ H(α) be generated by a set of additive (1 − α)-cocycles I(t) ∈ Ᏽ and the multiplication ∪. Then, a multiplicative αcocycle w which is Markovian with respect to Ᏽ determines the homomorphism of A into H(α ), where the group α is defined by the formula α t = w t • α t , t ∈ R. The image A of this homomorphism is a ring generated by the set of additive (1 − α )-cocycles I (t) = w t (I(t)), t ≤ 0, Proof. Check that I (t) = w t (I(t)) satisfies the condition for a (1 − α )cocycle. Fix s, t > 0 and notice that by means of the cocycle condition and the Markovian property for w. Therefore, Using the Markovian property of w, we obtain We give examples showing how the language of the theory of cohomology of groups can describe the set of all functionals from the classical and quantum stochastic processes with the stationary independent increments.

The Wiener process.
Consider the Wiener process {B(t), t ∈ R} implemented on the probability space (Ω,µ). Let S = (S t ) t∈R be the group of transformations shifting the increments of the process in time. Then the Wiener process satisfies the condition of the additive (1−S)-cocycle, that is, B(t +s) = B(t)+S t (B(s)), s, t ∈ R. Consider the ring of cohomologies H(S) for the group S with values in L ∞ (Ω). Then the Wiener process B(t) generates the subring A ⊂ H(S). Denote by Ᏼ the Hilbert space of all L 2 -functionals from the Wiener process. As it is known, one can define for Ᏼ the Wiener-Itô decomposition Ᏼ = ⊕ +∞ i=0 Ᏼ i in the orthogonal sum of spaces formed by polynomials of increasing degrees (see, e.g., [20]). Notice that the representation of the graded ring A as the sum of cohomologies of all degrees A = ⊕ +∞ i=1 A i is a cohomological analog of the Wiener-Itô decomposition. Take a function a(x) ∈ L 2 loc (R) and determine a one-parameter family of linear maps w = (w t ) t∈R acting on A by Every w t defines a unitary transformation in the space of L 2 -functionals of the white noise (see [20]) and satisfies the property for the Markovian cocycle by the definition. Thus the following proposition holds.
Proposition 6.2. The family w is a Markovian cocycle.

Quantum noises.
Let G be a certain Lie algebra with the involution. Then (see [36]) there exists a one-parameter family of * -homomorphisms j : x → j t (x) ∈ ᏸ(H) mapping each element x ∈ G to a strongly continuous one-parameter family of operators j t (x), t ∈ R, in the symmetric Fock space Ᏼ = Ᏺ(L 2 (R, )) over the one-particle Hilbert space L 2 (R, ) consisting of functions on the real line with values in the Hilbert space . Every homomorphism j t preserves the commutator such that [j t (x), j t (y)] = j t ([x, y]), x, y ∈ G, and satisfies the condition of additive (1 − α)-cocycle with respect to the group of automorphisms α generated by the group of shifts in the space L 2 (R, ). Then j can be extended to a * -homomorphism of the universal enveloping algebra ᐁ(G) determining the quantum Lévy process. Notice that the classification of possible Lévy processes over Lie algebras is given in [30,37]. The extension to a universal enveloping algebra is constructed in [19]. In the case if j t (x) ∈ ᏸ s (Ᏼ), t ∈ R, under fixed x ∈ G, the one-parameter family x t = j t (x), t ∈ R, is the quantum stochastic process in the sense of the definition given in Section 5. Moreover, x t = j t (x) is the process with independent increments, where the independence means that the conditions (5.1) and (5.2) of Section 5 are satisfied. Nevertheless, we will consider the curves x t = j t (x), t ∈ R, consisting of non-Hermitian operators as well. The scope of all curves x t = j t (x), x ∈ G, will be named a quantum noise because the increments of all curves x t are independent. Notice that the quantum stochastic processes j t (x) can play the roles of the Wiener and Poisson processes in the quantum case (see [21,24,33] for references and comments). Consider the ring of cohomologies A generated by additive (1 − α)cocycles j t (x), x ∈ G. Notice that the Fock space can be factorized such that Ᏺ(L 2 (R, )) = Ᏺ(L 2 (R + , )) ⊗ Ᏺ(L 2 (R − , )). In the following, we construct the example of a multiplicative α-cocycle generating the homomorphism of A which uses this factorization. Put w −t (·) = W t ⊗ I · W * t ⊗ I, t ≥ 0, where the family of unitary operators W t , t ≥ 0, in the Fock space Ᏺ(L 2 (R + , )) satisfies the quantum stochastic differential equation constructed in [33,Example 25.17,page 198], that is, where A + m and A m are the basic processes of creation and annihilation, respectively, generated by a function m ∈ , and Λ U −1 is the number of particles process generated by a unitary operator U (see [33]).
Let ᏹ t] and ᏹ [t be the von Neumann algebras generated by the increments of the quantum noise j t (x), x ∈ G, before the moment of time t and after the moment of time t, respectively. Then, for the von Neumann algebra ᏹ generated by all operators Proposition 6.3. Let the von Neumann algebra ᏹ generated by increments of the quantum noise j t (x), x ∈ G, be a factor. Then the group of automorphisms α is a Kolmogorov flow on ᏹ.
Proof. Take x ∈ ∩ t ᏹ t] ⊂ ᏹ. The increments of the quantum noise are independent quantum random variables by the condition of the proposition. Hence, the algebra ᏹ [t belongs to the commutant of the algebra ᏹ t] . Therefore, the operator x is commuting with all operators y ∈ ∪ t ᏹ [t = ᏹ. As the von Neumann algebra ᏹ is a factor, that is, ᏹ ∩ ᏹ = {C1}, we get x = const 1.
Notice that in the applications the algebra ᏹ is often the algebra of all bounded operators, that is, the factor of type I. In this case, the condition of Proposition 6.3 is satisfied. Take into account that the expectation in the quantum case is often determined by a pure state ρ such that E(x) = (Ω,xΩ).
Here Ω is some vector in the Hilbert space Ᏼ, where the operators x act. The pure state on the algebra of all bounded operators in Ᏼ cannot be faithful (see [11]). Hence, Proposition 6.3 complements Proposition 5.1.

Markovian perturbations of stationary quantum stochastic processes.
At first, we will generalize the definition of the quantum stochastic process given in Section 5. Following [2], a quantum stochastic process will refer to the one-parameter family of * -homomorphisms j = (j t ) t∈R from certain algebra with the involution Ꮽ to the algebra of linear (unbounded in general) operators ᏸ(H) in some Hilbert space Ᏼ. We will suppose that j 0 (x) = 0, x ∈ Ꮽ. For Ꮽ, one can, in particular, take a certain Lie algebra. Consider the minimal von Neumann algebra ᏹ generated by all operators j t (x), x ∈ Ꮽ, such that ᏹ={j t (x), x ∈ Ꮽ} , where denotes the commutant in the algebra Ꮾ(H) consisting of all bounded operators in Ᏼ. For fixed s, t ∈ R, we denote by ᏹ t] , ᏹ [t , and ᏹ [s,t] the von Neumann algebras generated by the operators {j t (x) − j r (x), r ≤ t}, {j r (x) − j t (x), r ≥ t}, and {j r (x) − j t (x), s ≤ r ≤ t}, respectively. Further, we will assume that the state ρ > 0, Tr ρ = 1, determining the expectation E(y) = Tr ρy for quantum random variables y ∈ ᏹ is fixed. Moreover, let ᏹ be in "the standard form" with respect to the state ρ, that is, the Hilbert space Ᏼ, where ᏹ acts, is defined by the map x → [x] from ᏹ to the dense set in Ᏼ such that the inner product is given by the formula ([x], [y]) = E(xy * ), x, y ∈ ᏹ. The quantum stochastic process j is said to be stationary if there exists a group of automorphisms α t ∈ Aut(ᏹ), t ∈ R, whose actions are correctly defined on the operators j s (x), Ꮽ such that α t (j s (x)) = j s+t (x), x ∈ Ꮽ, and the expectation E(·) is invariant with respect to the group α = (α t ) t∈R , that is, E(α t (y)) = E(y), y ∈ ᏹ. The stationary quantum stochastic process is a particular case of the quantum stochastic process with the stationary increments generated by * -homomorphisms j t which determine the curves j t (x) being additive (1 − α)-cocycles for every fixed x ∈ Ꮽ. For the example of the quantum stochastic process with stationary increments, the quantum noise can be chosen. We will suppose that α has ultraweak continuous orbits; that is, all functions η(α t (y)) are continuous in t for arbitrary η ∈ ᏹ * , y ∈ ᏹ. The one-parameter family of * -automorphisms w t , t ∈ R, of the algebra ᏹ is said to be a (multiplicative) Markovian α-cocycle if the following two conditions are satisfied: Notice that if the von Neumann algebra ᏹ [t is generated by the increments of the quantum noise {j t (x), x ∈ Ꮽ} the Markovian cocycle w in the sense of the definition given here is a cocycle, being Markovian with respect to the set {j t (x), x ∈ Ꮽ} in the sense of the definition of Section 6.
is the quantum noise isomorphic to the initial one. In particular, the processesj t (x), x ∈ Ꮽ, have independent increments.
Proof. The quantum noise j is a quantum stochastic process with independent increments. Denote by α the corresponding group of automorphisms shifting the increments in time. Then, due to Proposition 6.1,j t (x) is an additive (1 −α)-cocycle for each fixed x ∈ Ꮽ, where the groupα consists of automorphismsα t = w t • α t , t ∈ R. By means of property (5.2) guaranteeing the independence of increments of the quantum noise j, we obtain the commutator for all x 1 ,x 2 ∈ Ꮽ and disjoint intervals (s 1 ,t 1 ) and (s 2 ,t 2 ). We will prove that this property holds for the processj also. For s 1 , t 1 , s 2 , and t 2 , (7.1) takes place becausej r (x) = j r (x), x ∈ Ꮽ, r ≥ 0. The identity for the case where at least one of s 1 , t 1 , s 2 , and t 2 is less than zero can be obtained by applying the automorphismsα −r = w −r •α −r , r > 0, to formula (7.1). In fact, the automorphism α −r shifts the increments for r units backward, and formula (7.1), which is true for the processj with positive values s 1 , t 1 , s 2 , and t 2 , automatically appears to be true for negative values. Moreover, arguing in the way given above, it is easy to obtain that all algebraic properties satisfied for the operators j t (x), x ∈ Ꮽ, t ∈ R, are also satisfied for the operatorsj t (x), x ∈ Ꮽ, t ∈ R. It remains to check the condition of independence (5.1) for increments of the processj. But it holds because it is true for the process j and the expectation is invariant with respect to the action of the cocycle w by the condition. Thus, the processes j andj are isomorphic.
Consider the restriction β t = α −t , t ∈ R + , to the subalgebra ᏹ 0] which is invariant with respect to the action of α −t , t ∈ R + . The unital semigroup β = (β t ) t∈R+ consists of the endomorphisms of ᏹ 0] possessing the property β(ᏹ 0] ) ≠ ᏹ 0] and has the orbits continuous in the sense that η(β t (x)) is a continuous function for all η ∈ ᏹ 0] * , x ∈ ᏹ 0] . Hence, β is an E 0 -semigroup by the definition introduced in [34]. If α is a Kolmogorov flow, the E 0 -semigroup β is a semiflow of Powers shifts, that is, each β t , t > 0, is a Powers shift [34] such that ∩ n∈N β tn (ᏹ 0] ) = {C1}, t > 0 (see [14]). Notice that the Markovian cocycle w generates a new group of automorphismsα t = w t • α t , t ∈ R, on the von Neumann algebra ᏹ and a new E 0 -semigroupβ t = w −t • β t , t ∈ R + , on the von Neumann algebra ᏹ 0] . Recall that the conditional expectation of the unital algebra ᏹ onto the unital algebra ᏺ ⊂ ᏹ is a completely positive projection of ᏹ onto ᏺ (see, e.g., [21,24]). Using the techniques in [13], one can exclude the maximum subalgebra of the algebra ᏹ 0] such that the restriction of the semigroupβ to it is a semigroup of automorphisms and there exists a conditional expectation onto this algebra. The quantum random variables j t (x) = w t • j t (x), x ∈ Ꮽ, generate the von Neumann subalgebrasᏹ = {j t (x), x ∈ Ꮽ, t ∈ R} andᏹ + = {j −t (x), x ∈ Ꮽ, t ∈ R + } of the algebras ᏹ and ᏹ 0] , respectively. Suppose that ᏹ is a factor. Then, due to Proposition 6.3, α is a Kolmogorov flow on ᏹ, and by means of Proposition 7.1, the restrictioñ α|ᏹ is also a Kolmogorov flow. Thus,β|ᏹ + is a semiflow of Powers shifts. The Wold decomposition (3.1) allows to uniquely determine the stochastic process with noncorrelated increments associated with the stationary process. In the following theorem we establish the possibility to exclude a restriction of the group of automorphisms obtained by a perturbation of the Kolmogorov flow generated by the quantum noise, which is isomorphic to the initial Kolmogorov flow. In this way, our conjecture can be considered as some analogue of the Wold decomposition for the quantum case.
Theorem 7.2. Let the group of automorphismsα on the von Neumann factor ᏹ be obtained through a Markovian cocycle perturbation of the Kolmogorov flow generated by the quantum noise j with the expectation defining the probability distribution, which is invariant with respect to the Markovian cocycle w. Then there exists a subfactorᏹ ⊂ ᏹ such that the restrictionα|ᏹ is the Kolmogorov flow generated by the quantum noisej which is isomorphic to the initial one. The limit lim t→+∞ w −t = w −∞ correctly defines a normal *endomorphism w −∞ with the propertyᏹ = w −∞ (ᏹ),j t = w −∞ • j t , t ∈ R.
Proof. The first part of the theorem follows from Proposition 7.1. We will prove that there exists the limit lim t→+∞ w −t = w −∞ defining the normal *endomorphism on ᏹ with the properties we claimed. Notice that w −t−s (y) = w −t (y), y ∈ ᏹ t] . It follows that the limit exists on the dense set of elements. In the following, we will prove that it exists. The formulas (U t [x], [y]) = E(α t (x)y * ) and (W t [x], [y]) = E(w t (x)y * ), x, y ∈ ᏹ, define a strongly continuous group U = (U t ) t∈R of unitary operators in Ᏼ and a unitary U -cocycle W = (W t ) t∈R , respectively. Let Ᏼ [t be a subspace of Ᏼ generated by the elements [x], x ∈ ᏹ [t . Then W t ξ = ξ, ξ ∈ Ᏼ [t , t ≥ 0, by means of the Markovian property for w. Therefore, W is a Markovian U-cocycle in the sense of the definition of Section 2, and the limit s − lim t→+∞ W −t = W −∞ exists due to Proposition 2.2. Notice that w t (x) = W t xW * t , where x ∈ ᏹ and t ∈ R. It implies that w −t−s (x)ξ = W −t−s xW −t−s ξ = W −t−s xW −t ξ → W −∞ xW −t ξ, where x ∈ ᏹ, ξ ∈ Ᏼ [−t , and s → +∞. Hence, the limit lim t→+∞ w −t (x)ξ exists for a dense set of the elements ξ ∈ Ᏼ. Because w −t is an automorphism, we get w −t (x) = x . Therefore, the strong limit s − lim t→+∞ w −t (x) is defined by the Banach-Steinhaus theorem for all x ∈ ᏹ. Moreover, the limiting map w −∞ preserves the identity because all w −t satisfy this property and w −∞ = 1. In this way, the map w −∞ is positive (see [11]). On the other hand, w −∞ is a normal * -endomorphism for it is a limit of the series of normal * -automorphisms w −t . Thus, w −∞ is completely positive. Notice that w −∞ (ᏹ 0] ) =ᏹ 0] . The Markovian property gives w −t−s (y) = w −t (y), where y ∈ ᏹ [−t and s, t ≥ 0. Due to Proposition 7.1, For positive values of time,j t = j t = w −∞ • j t , t ≥ 0, by means of w −t (y) = w −∞ (y) = y, where t ≥ 0 and y ∈ ᏹ [0 .

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models. In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset. Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning • Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation