RELATIONSHIPS OF CONVOLUTION PRODUCTS , GENERALIZED TRANSFORMS , AND THE FIRST VARIATION ON FUNCTION SPACES

We use a generalized Brownian motion process to define the generalized Fourier-Feynman transform, the convolution product, and the first variation. We then examine the various relationships that exist among the first variation, the generalized Fourier-Feynman transform, and the convolution product for functionals on function space that belong to a Banach algebra S(Lab[0,T ]). These results subsume similar known results obtained by Park, Skoug, and Storvick (1998) for the standard Wiener process.


Introduction.
The concept of L 1 analytic Fourier-Feynman transform (FFT) was introduced by Brue in [1].In [3], Cameron and Storvick introduced an L 2 analytic FFT.In [13], Johnson and Skoug developed an L p analytic FFT for 1 ≤ p ≤ 2, which extended the results in [1,3] and gave various relationships between the L 1 and L 2 theories.In [10], Huffman et al. defined a convolution product for functionals on Wiener space and obtained, in [11,12], various results involving the FFT and the convolution product.
Both the FFT and the convolution product are defined in terms of a Feynman integral.In this paper, we extend the ideas of [10,11,12] from the Wiener process to more general stochastic process.We note that the Wiener process is free of drift and is stationary in time.However, the stochastic process considered in this paper is subject to drift and nonstationary in time.
In Section 2, we consider the function space induced by a generalized Brownian motion process and define several concepts.In Section 3, we examine all relationships involving exactly two of three concepts of transform, convolution product and first variation of functionals in S(L ab [0,T ]).In Section 4, we examine all relationships involving all three of theses concepts where each concept is used exactly once.

Definitions and preliminaries.
Let D = [0,T ] and let (Ω, Ꮾ,P) be a probability measure space.A real-valued stochastic process Y on (Ω, Ꮾ,P) and D is called a generalized Brownian motion process if Y (0,ω) = 0 a.e.; and for 0 ≤ t 0 < t 1 < ••• < t n ≤ T , the n-dimensional random vector (Y (t 1 ,ω),...,Y (t n ,ω)) is normally distributed with the density function where η = (η 1 ,...,η n ), η 0 = 0, t = (t 1 ,...,t n ), t 0 = 0, and a(t) is a continuous realvalued function of bounded variation with a(0) = 0, and b(t) is a strictly increasing, continuous real-valued function with b(0) = 0.As explained in [19, pages 18-20], Y induces a probability measure µ on the measurable space (R D , Ꮾ D ) where R D is the space of all real-valued functions x(t), t ∈ D, and Ꮾ D is the smallest σ -algebra of subsets of R D with respect to which all the coordinate evaluation maps, e t (x) = x(t), defined on R D are measurable.The triple (R D , Ꮾ D ,µ) is a probability measure space.
We note that, the generalized Brownian motion process Y determined by a(•) and b(•) is a Gaussian process with mean function a(t) and covariance function r (s,t) = min{b(s), b(t)}.By [19, Theorem 14.2, page 187], the probability measure µ induced by Y , taking a separable version, is supported by C ab [0,T ] (which is equivalent to the Banach space of continuous functions x on [0,T ] with x(0) = 0 under the sup norm).Hence (C ab [0,T ],Ꮾ(C ab [0, T ]), µ) is the function space induced by Y where Ꮾ(C ab [0,T ]) is the Borel σ -algebra of C ab [0,T ].Note that we can also express x in the form where w(•) is the standard Brownian motion process [6,7].
Let L ab [0,T ] be the Hilbert space of functions on [0,T ] which are Lebesgue measurable and square integrable with respect to the Lebesgue Stieltjes measures on [0,T ] induced by a(•) and b(•), that is, where |a| denotes the total variation [6,7].
A subset B of C ab [0,T ] is said to be scale-invariant measurable [8,14] provided that ρB is Ꮾ(C ab [0,T ])-measurable for all ρ > 0, and a scale-invariant measurable set N is said to be scale-invariant null set provided that µ(ρN) = 0 for all ρ > 0. A property that holds except on a scale-invariant null set, is said to hold on scale-invariant almost everywhere (s-a.e.).If two functionals F and G defined on C ab [0,T ] are equal s-a.e., we write F ≈ G.
We denote the function space integral of a Ꮾ(C ab [0,T ])-measurable functional F by whenever the integral exists.
We are now ready to state the definitions of the generalized analytic Feynman integral.
exists for all λ > 0. If there exists a function J * (λ) analytic in C + such that J * (λ) = J(λ) for all λ > 0, then J * (λ) is defined to be the analytic function space integral of F over C ab [0,T ] with parameter λ, and for λ ∈ C + we write Let q ≠ 0 be a real number and let F be a functional such that E an λ [F ] exists for all λ ∈ C + .If the following limit exists, we call it the generalized analytic Feynman integral of F with parameter q and we write where λ approaches −iq through C + .
Next we state the definition of the generalized FFT (GFFT).
Definition 2.2.For λ > 0 and y ∈ C ab [0,T ], let (2.8) In the standard Fourier theory, the integrals involved are often interpreted in the mean; a similar concept is useful in the FFT theory [13,  and we call H the scale-invariant limit in the mean of order p .A similar definition is understood when n is replaced by the continuously varying parameter λ.Let real q ≠ 0 be given.For 1 < p ≤ 2, we define the L p analytic GFFT, if it exists.We define the L 1 analytic GFFT, T (1) q (F ) of F , by the formula (λ ∈ C + ) T (1)  q (F )(y) = lim if it exists.
We finish this section by giving the definition of the first variation δF of the functional F [2,5].
The following analytic Feynman integration formula is used throughout:

Relationships involving exactly two of three concepts of transform, convolution, and first variation. Let M(L ab [0,T ]) be the space of C-valued, countably additive finite Borel measures on L ab [0,T ]. The Banach algebra S(L ab [0,T ]) consists of functionals F on C ab [0,T ] expressible in the form
for s-a.e.x ∈ C ab [0,T ], where f is an element of M(L ab [0,T ]).Further works on S(L ab [0,T ]) show that it contains many functionals of interest in Feynman integration theory [3,4,6,7,13,14,15,16,17].Also, let Remark 3.1.Throughout, we choose the variance function b(•) which is strictly increasing such that the function p defined by This will insure that the first variation, δF (• | w) of F in S(L ab [0,T ]), that arises will exist for all w ∈ A (see [9]).Let G in S(L ab [0,T ]) be given by In our first lemma, we obtain a formula for the first variation of functionals in S(L ab [0,T ]).

Furthermore, as a function of y, δF (y | w) is an element of S(L ab [0,T ]).
Proof.By using the definition of the first variation, we see that where In our next theorem, we obtain the transform of functional in S(L ab [0,T ]).

Theorem 3.3. Let F ∈ S(L ab [0,T ]) be given by (3.1) and let p ∈ [1, 2] be given. Then the analytic generalized Fourier-Feynman transform T (p)
q (F ) exists for all real q ≠ 0 and is given by the formula for s-a.e.y ∈ C ab [0,T ].
Proof.By (2.8), the Fubini theorem, and (2.15), we have, for all λ > 0, for s-a.e.y ∈ C ab [0,T ].But the last equation above is analytic throughout C + and is continuous on C+ , since f is a finite Borel measure.Thus (3.9) is established.
In the following theorem, we obtain the convolution product of functionals in S(L ab [0,T ]).Theorem 3.4.Let F ∈ S(L ab [0,T ]) be given by (3.1), and let G ∈ S(L ab [0,T ]) be given by (3.5).Then their convolution product (F * G) q exists for all real q ≠ 0 and is given by the formula for s-a.e.y ∈ C ab [0,T ].
Proof.By using (2.13), the Fubini theorem, and (2.15), we have that for all λ > 0, for s-a.e.y ∈ C ab [0,T ].But the last equation above is analytic throughout C + , and is continuous on C+ .Thus we have the desired result.
Next, we obtain the transform of the convolution product.
Theorem 3.5.Let F and G be given as in Theorem 3.4 and p ∈ [1,2].Then for all real q ≠ 0, T (p) q ((F * G) q ) exists and is given by the formula for s-a.e.y ∈ C ab [0,T ], where F 1 and G 1 are given by (3.15) below.
In our next theorem, we have the convolution product of transforms of F and G in S(L ab [0,T ]).Theorem 3.6.Let F , G, and p be given as in Theorem 3.5.Then for all real q ≠ 0, (T (p) q (F ) * T (p) q (G)) −q (y) exists and is given by the formula for s-a.e.y ∈ C ab [0,T ] where F 2 and G 2 are given by (3.19) below.
In the next theorem, we obtain that the transform with respect to the first argument of the variation equals the variation of the transform.Theorem 3.7.Let F be given as in Lemma 3.2, p ∈ [1, 2], q ∈ R − {0}, and w ∈ A be given.Then for s-a.e.y ∈ C ab [0,T ].Also, both expressions in (3.21) are given by the expression Proof.By using (3.6), the Fubini theorem, (2.15), and (3.9), we have that for s-a.e.y ∈ C ab [0,T ] as desired.
In the next theorem, we obtain the transform with respect to the second argument of the variation.Theorem 3.8.Let F , p, q, and w be given as in Theorem 3.7.Then, for s-a.e.y ∈ C ab [0,T ], (u, a) exp i u, y df (u).(3.24) Proof.Using (2.11) and (3.6), we obtain (u, a) exp i u, y df (u) for s-a.e.y ∈ C ab [0,T ].In particular, if a ∈ A then (u, a) = u, a and so In our next theorem, we obtain the first variation of convolution product of functionals F and G in S(L ab [0,T ]).Theorem 3.9.Let F , p, q, and w be given as in Theorem 3.7 and let G be given by (3.5)
In our next theorem, we obtain the convolution product of the first variation with respect to the first argument.Theorem 3.10.Let F , G, p, q, and w be given as in Theorem 3.9.Then for s-a.e.
In our next theorem, we obtain the convolution product of the first variation with respect to the second argument.Theorem 3.11.Let F , G, p, q, and w be given as in Theorem 3.9.Then for s-a.e.y ∈ C ab [0,T ].
Proof.For each u, v ∈ L ab [0,T ], we have But, by using (2.13), (3.6), the Fubini theorem, and (3.31), we have (u, a) exp i u, y df (u) 2 and so we have (3.33) Also, by using (3.24), the alternative expression in (3.30) is given by (3.34) Thus we have the desired result.

Relationships involving three concepts.
In this section, we look at all the relationships involving the transform, the convolution, and the first variation where each operation is used exactly once.
In our next theorem, we obtain the formula for transform with respect to the first argument of the variation of the convolution product which equals the variation of the transform of the convolution product.Theorem 4.1.Let F , G, p, q, and w be given as in Theorem 3.9.Then for s-a.e.y ∈ C ab [0,T ], T (p) q (δ(F * G) q (• | w))(y) exists and is given by the formula where F 1 and G 1 are given by (3.15).
In our next theorem, we obtain the transform with respect to the second argument of the variation of the convolution product.Theorem 4.2.Let F , G, p, q, and w be given as in Theorem 4.1.Then for s-a.e.y ∈ C ab [0,T ], (4.4) Proof.By using (3.11), (3.24), and (3.27), we obtain (4.4) above.In particular, if a ∈ A, then (u + v, a) = u + v, a and hence we have Now we obtain formulas for the transforms of the convolution product with respect to the first argument of the variations.Theorem 4.3.Let F , G, p, q, and w be given as in Theorem 4.1.Then, for s-a.e.y ∈ C ab [0,T ], where F 1 and G 1 are given by (3.15) and Proof.By using (3.13) and (3.21), we obtain (4.6) above.To establish (4.7), we note that, by the use of (2.15), (3.24), (3.28), and (3.31), for s-a.e.y ∈ C ab [0,T ].
In our next theorem, we obtain the transforms of the convolution product with respect to the second argument of the variations.Theorem 4.4.Let F , G, p, q, and w be given as in Theorem 4.1.Then for s-a.e.y ∈ C ab [0,T ], (uv, b) (4.9) Proof.By using (2.11) and (3.30), we obtain (4.9).
Next, we obtain the variation of the convolution product of transforms.
Theorem 4.5.Let F , G, p, q, and w be given as in Theorem 4.1.Then for s-a.e.y ∈ C ab [0,T ] where F 2 and G 2 are given by (3.19).
Proof.By using (3.17) and the same calculation in the proof of Theorem 3.6, we obtain (4.10).Now, we obtain the formulas for convolution product of the variation of the transform.There are two cases; namely, we can take the convolution with respect to the first argument or the second argument of the variation.

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

Definition 2 . 3 .
Let F and G be functionals on C ab [0,T ].For λ ∈ C+ we define the convolution product (if it exists) by

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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