A generalized conditional Wiener integral with a drift and its applications

1 Introduction

Let C 0 ⁢ [ 0 , T ] denote the Wiener space, the space of continuous real-valued functions x on [ 0 , T ] with x ⁢ ( 0 ) = 0 . A time integral is simply the Riemann integral of a function of the continuous random variable X ⁢ ( x , t ) = x ⁢ ( t ) with respect to the parameter t for x ∈ C 0 ⁢ [ 0 , T ] . The Feynman–Kac functional on the Wiener space C 0 ⁢ [ 0 , T ] is given by exp ⁡ { - ∫ 0 T V ⁢ ( t , X ⁢ ( x , t ) ) ⁢ 𝑑 t } including the time integral, where V is a complex-valued potential. Calculations involving the conditional Wiener integrals of the Feynman–Kac functional are important in the study of the Feynman integral [9], and it can provide a solution of the integrals equation which is formally equivalent to the Schrödinger equation [6]. In particular, when 0 = t 0 < t 1 < t 2 < ⋯ < T and ξ j ∈ ℝ for j = 0 , 1 , 2 , … , the conditional Wiener integrals of functions involving the time integral, in which the paths pass through ξ j at each time t j , are very useful in describing the Brownian motion. As for one of simple formulas used to calculate the conditional Wiener integrals stated above, Park and Skoug [10] derived a simple formula for conditional Wiener integrals containing the time integral with the conditioning function ( ∫ 0 T e 1 ⁢ ( t ) ⁢ 𝑑 x ⁢ ( t ) , ∫ 0 T e 2 ⁢ ( t ) ⁢ 𝑑 x ⁢ ( t ) , … ) for x ∈ C 0 ⁢ [ 0 , T ] , where the e j are in L 2 ⁢ [ 0 , T ] . In their simple formula, they expressed the conditional Wiener integrals directly in terms of ordinary Wiener integrals, which generalizes the evaluations of conditional Wiener integrals with a finite dimensional conditioning function. We note that the Wiener measure used in [10] has no drifts with the variance function β ⁢ ( t ) = t for t ∈ [ 0 , T ] .

On the other hand, let C ⁢ [ 0 , T ] denote the space of continuous real-valued functions on [ 0 , T ] . Ryu [12, 13] introduced a finite positive measure w α , β ; φ on C ⁢ [ 0 , T ] , where α , β : [ 0 , T ] → ℝ are appropriate functions and φ is a finite positive measure on the Borel class ℬ ⁢ ( ℝ ) of ℝ . We note that w α , β ; φ is equivalent to the Wiener measure on C 0 ⁢ [ 0 , T ] if α ⁢ ( t ) = 0 , β ⁢ ( t ) = t and φ = δ 0 , which is the Dirac measure concentrated at 0. When e 1 , … , e n are of bounded variations on [ 0 , T ] and Z e → , n ⁢ ( x ) = ( x ⁢ ( t 0 ) , ∫ 0 T e 1 ⁢ ( t ) ⁢ 𝑑 x ⁢ ( t ) , … , ∫ 0 T e n ⁢ ( t ) ⁢ 𝑑 x ⁢ ( t ) ) for x ∈ C ⁢ [ 0 , T ] , the author of [4] derived a simple evaluation formula for generalized conditional Wiener integrals of functions on C ⁢ [ 0 , T ] with the conditioning function Z e → , n . As applications of the formula, he calculated the conditional Wiener integrals of the time integral, the cylinder functions and the functions in a Banach algebra which generalizes the Cameron–Storvick’s one [1]. We note that the value space of Z e → , n is finite dimensional.

Let Z e → , ∞ ⁢ ( x ) = ( x ⁢ ( 0 ) , ∫ 0 T e 1 ⁢ ( t ) ⁢ 𝑑 x ⁢ ( t ) , ∫ 0 T e 2 ⁢ ( t ) ⁢ 𝑑 x ⁢ ( t ) , … ) for x ∈ C ⁢ [ 0 , T ] , where the e j are orthonormal in the space of functions that are square integrable with respect to β. In this paper, we derive a simple evaluation formula for calculating Radon–Nikodym derivatives similar to the conditional Wiener integrals of functions on C ⁢ [ 0 , T ] with the conditioning function Z e → , ∞ . Regarding the applications of the formula, we evaluate the derivatives of various functions involving the time integral, such as cylinder-type functions and especially the function exp ⁡ { ∫ 0 T x ⁢ ( t ) ⁢ 𝑑 β ⁢ ( t ) } on C ⁢ [ 0 , T ] , which is a specific type of the Feynman–Kac functional and plays a significant role in Feynman integration theory. We note that the value space of Z e → , ∞ is infinite dimensional and Z e → , ∞ has a kind of drift with the more generalized variance function β. Furthermore, while every path in C 0 ⁢ [ 0 , T ] starts at the origin, the paths in our underlying space C ⁢ [ 0 , T ] may not. Our measure in this paper may not be a probability measure, so that the results of this work generalize those of [4, 10].

To summarize, with the conditioning function Z e → , ∞ on C ⁢ [ 0 , T ] , this paper extends the simple evaluation formula in [4] stated above. Using the extended formula, we will evaluate the Radon–Nikodym derivatives of various functions involving time integrals, particularly focusing on a specific type of Feynman–Kac functional exp ⁡ { ∫ 0 T x ⁢ ( t ) ⁢ 𝑑 β ⁢ ( t ) } .

2 A generalized analogue of Wiener space

Let C ⁢ [ 0 , T ] denote the space of continuous real-valued functions on the interval [ 0 , T ] . Let α be absolutely continuous on [ 0 , T ] and let β be continuous, strictly increasing on [ 0 , T ] . For t → k = ( t 0 , t 1 , … , t k ) with 0 = t 0 < t 1 < ⋯ < t k ≤ T , let J t → k : C ⁢ [ 0 , T ] → ℝ k + 1 be the function given by J t → k ⁢ ( x ) = ( x ⁢ ( t 0 ) , x ⁢ ( t 1 ) , … , x ⁢ ( t k ) ) . For any Borel set B j ( j = 0 , 1 , … , k ) in ℬ ⁢ ( ℝ ) , the subset J t → k - 1 ⁢ ( ∏ j = 0 k B j ) of C ⁢ [ 0 , T ] is called an interval I and let ℐ be the set of all such intervals I. Define a premeasure m α , β ; φ on ℐ by

where m L denotes the Lebesgue measure on ℬ ⁢ ( ℝ ) , and for u 0 ∈ ℝ , u → k = ( u 1 , … , u k ) ∈ ℝ k ,

The Borel σ-algebra ℬ ⁢ ( C ⁢ [ 0 , T ] ) of C ⁢ [ 0 , T ] with the supremum norm coincides with the smallest σ-algebra generated by ℐ , and there exists a unique positive finite measure w α , β ; φ on ℬ ⁢ ( C ⁢ [ 0 , T ] ) with w α , β ; φ ⁢ ( I ) = m α , β ; φ ⁢ ( I ) for all I ∈ ℐ . This measure w α , β ; φ is called a generalized analogue of Wiener measure on ( C ⁢ [ 0 , T ] , ℬ ⁢ ( C ⁢ [ 0 , T ] ) ) according to φ (see [12, 13]).

Let ν α , β denote the Lebesgue–Stieltjes measure defined by ν α , β ( E ) = ∫ E d ( | α | + β ) ( t ) for each Lebesgue measurable subset E of [ 0 , T ] , where | α | denotes the total variation of α. Define L α , β 2 ⁢ [ 0 , T ] to be the space of functions on [ 0 , T ] that are square-integrable with respect to ν α , β (see [11]); that is,

The space L α , β 2 ⁢ [ 0 , T ] is a (real) Hilbert space and has the inner product

We note that L α , β 2 ⁢ [ 0 , T ] ⊆ L 0 , β 2 ⁢ [ 0 , T ] , where L 0 , β 2 ⁢ [ 0 , T ] denotes the space L α , β 2 [ 0 , T ] with α ≡ 0 . Throughout the remainder of this paper, we give additional conditions for α and β. Assume that β ′ > 0 and | α | ′ β ′ is bounded. In this case, L α , β 2 ⁢ [ 0 , T ] = L 0 , β 2 ⁢ [ 0 , T ] , and the equality means that they are equal as vector spaces and the two norms on them are equivalent so that they have the same topology, but that they need not be equal isometrically.

Let S ⁢ [ 0 , T ] be the collection of step functions on [ 0 , T ] and let ∫ 0 T ϕ ⁢ ( t ) ⁢ 𝑑 x ⁢ ( t ) denote the Riemann–Stieltjes integral. For f ∈ L α , β 2 ⁢ [ 0 , T ] , let { ϕ n } be a sequence of step functions in S ⁢ [ 0 , T ] with lim n → ∞ ⁡ ∥ ϕ n - f ∥ α , β = 0 . Define I α , β ⁢ ( f ) by the L 2 ⁢ ( C ⁢ [ 0 , T ] ) -limit

for all x ∈ C ⁢ [ 0 , T ] for which this limit exists or I α , β ⁢ ( f ) ⁢ ( x ) = lim n → ∞ ⁡ ∫ 0 T ϕ n ⁢ ( t ) d ⁢ x ⁢ ( t ) pointwisely if exists. We note that for f ∈ L α , β 2 ⁢ [ 0 , T ] , I α , β ⁢ ( f ) ⁢ ( x ) exists for w α , β ; φ a.e. x ∈ C ⁢ [ 0 , T ] . Moreover, we have the following theorems [2].

Throughout this paper, for x ∈ C ⁢ [ 0 , T ] , we redefine I α , β ⁢ ( f ) ⁢ ( x ) = ∫ 0 T f ⁢ ( t ) d ⁢ x ⁢ ( t ) if ∫ 0 T f ⁢ ( t ) ⁢ 𝑑 x ⁢ ( t ) exists.

Let W be an ℝ ℵ 0 -valued Borel measurable function defined for w α , β ; φ a.e. on C ⁢ [ 0 , T ] , and let F : C ⁢ [ 0 , T ] → ℂ be integrable. Let m W be the image measure on the Borel class ℬ ⁢ ( ℝ ℵ 0 ) of ℝ ℵ 0 induced by W. By the Radon–Nikodym theorem, there exists an m W -integrable function ψ defined on ℝ ℵ 0 which is unique up to m W a.e. such that for every B ∈ ℬ ⁢ ( ℝ ℵ 0 ) ,

Define the function ψ as a generalized conditional Wiener integral of F for a given W and denote it by G E [ F | W ] . We note that G E [ F | W ] is a Radon–Nikodym derivative rather than a conditional expectation (or a conditional Wiener integral) since m W need not be a probability measure.

3 A simple formula for the generalized conditional Wiener integrals

In this section, we derive a simple evaluation formula for the generalized conditional Wiener integral as defined in the previous section.

Let { e j : j = 1 , 2 , … } be an orthonormal (preferably not complete) subset of L 0 , β 2 ⁢ [ 0 , T ] such that it is orthogonal in L α , β 2 ⁢ [ 0 , T ] . Such a set always exists for the following reason.

Let V be the closure of the subspace of L 0 , β 2 ⁢ [ 0 , T ] generated by { e 1 , e 2 , … } and let V ⊥ be the orthogonal complement of V. Let 𝒫 e → , ∞ , β : L 0 , β 2 ⁢ [ 0 , T ] → V and 𝒫 e → , ∞ , β ⊥ : L 0 , β 2 ⁢ [ 0 , T ] → V ⊥ be the orthogonal projections. By [7, Problem 8, p. 167], 𝒫 e → , ∞ , β and 𝒫 e → , ∞ , β ⊥ exist, and they can be expressed by

Since ∥ ⋅ ∥ 0 , β and ∥ ⋅ ∥ α , β are equivalent, V is also closed under the norm ∥ ⋅ ∥ α , β . Moreover, we have the following lemma.

For convenience, let φ 0 = 1 φ ⁢ ( ℝ ) ⁢ φ throughout this paper. We now provide the theorem which is needed in the sequel.

Let z 0 ⁢ ( x ) = x ⁢ ( 0 ) for x ∈ C ⁢ [ 0 , T ] . For j = 1 , 2 , … , define z j and Z e → , ∞ by z j ⁢ ( x ) = I α , β ⁢ ( e j ) ⁢ ( x ) and

for w α , β ; φ a.e. x ∈ C ⁢ [ 0 , T ] . For s ∈ [ 0 , T ] , w α , β ; φ a.e. x ∈ C ⁢ [ 0 , T ] and ξ → = ( ξ 0 , ξ 1 , ξ 2 , … ) ∈ ℝ ℵ 0 , let

and

We have

by Theorem 3.3. Moreover, since ξ → e → , ∞ , β ⁢ ( s ) is the evaluation of x e → , ∞ , β ⁢ ( s ) for z j ( x ) = ξ j ( j = 0 , 1 , … ) , ξ → e → , ∞ , β ⁢ ( s ) exists for m Z e → , ∞ a.e. ξ → ∈ ℝ ℵ 0 .

For the continuities of x e → , ∞ , β and ξ → e → , ∞ , β , assume that β ′ is bounded on a subinterval ( a , b ) of [ 0 , T ] throughout the remainder of this work.

We note that for w α , β ; φ a.e. x ∈ C ⁢ [ 0 , T ] and j = 1 , 2 , … ,

and

by Theorems 2.1 and 3.3 if each e j is of bounded variation on [ 0 , T ] . Moreover, we have the following properties:

1. For w α , β ; φ a.e. x ∈ C ⁢ [ 0 , T ] and s ∈ [ 0 , T ] , by Theorem 2.2, we have

   x ⁢ ( s ) - x e → , ∞ , β ⁢ ( s ) = x ⁢ ( s ) - x ⁢ ( 0 ) - I α , β ⁢ ( 𝒫 e → , ∞ , β ⁢ χ [ 0 , s ] ) ⁢ ( x ) = I α , β ⁢ ( χ [ 0 , s ] - 𝒫 e → , ∞ , β ⁢ χ [ 0 , s ] ) ⁢ ( x ) = I α , β ⁢ ( 𝒫 e → , ∞ , β ⊥ ⁢ χ [ 0 , s ] ) ⁢ ( x ) .

2. For 0 ≤ s 1 ≤ s 2 ≤ T ,

   〈 𝒫 e → , ∞ , β ⊥ ⁢ χ [ 0 , s 1 ] , 𝒫 e → , ∞ , β ⊥ ⁢ χ [ 0 , s 2 ] 〉 0 , β = β ⁢ ( s 1 ) - β ⁢ ( 0 ) - ∑ j = 1 ∞ c j ⁢ ( s 1 ) ⁢ c j ⁢ ( s 2 ) .