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Journal of Applied Mathematics and Stochastic Analysis
Volumeย 2009ย (2009), Article IDย 782572, 37 pages
http://dx.doi.org/10.1155/2009/782572
Research Article

Spectral Approximation of Infinite-Dimensional Black-Scholes Equations with Memory

1Mathematical Sciences Division, U. S. Army Research Office, P.O. Box 12211, RTP, NC 27709, USA
2Instrumental Sciences Inc., P.O. Box 4711, Huntsville, AL 35811, USA

Received 2 September 2009; Accepted 2 December 2009

Academic Editor: Kambizย Farahmand

Copyright ยฉ 2009 Mou-Hsiung Chang and Roger K. Youree. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper considers the pricing of a European option using a (๐ต,๐‘†)-market in which the stock price and the asset in the riskless bank account both have hereditary price structures described by the authors of this paper (1999). Under the smoothness assumption of the payoff function, it is shown that the infinite dimensional Black-Scholes equation possesses a unique classical solution. A spectral approximation scheme is developed using the Fourier series expansion in the space ๐ถ[โˆ’โ„Ž,0] for the Black-Scholes equation. It is also shown that the ๐‘›th approximant resembles the classical Black-Scholes equation in finite dimensions.

1. Introduction

The pricing of contingent claims in the continuous-time financial market that consists of a bank account and a stock account has been a subject of extensive research for the last decades. In the literature (e.g., [1โ€“5]), the equations that describe the bank account and the price of the stock are typically written, respectively, as๐‘‘๐ต(๐‘ก)=๐‘Ÿ๐ต(๐‘ก)๐‘‘๐‘ก,๐ต(0)=๐‘ฅ,๐‘‘๐‘†(๐‘ก)=๐›ผ๐‘†(๐‘ก)๐‘‘๐‘ก+๐œŽ๐‘†(๐‘ก)๐‘‘๐‘Š(๐‘ก),๐‘†(0)=๐‘ฆ,(1.1) where ๐‘Š={๐‘Š(๐‘ก),๐‘กโ‰ฅ0} is a one-dimensional standard Brownian motion defined on a complete filtered probability space (ฮฉ,๐น,๐;{๐น(๐‘ก),๐‘กโ‰ฅ0}) and ๐‘Ÿ, ๐›ผ, and ๐œŽ are positive constants that represent, respectively, the interest rate of the bank account, the stock appreciation rate, and the stock volatility rate. The financial market that consists of one bank account and one stock account will be referred to as a (๐ต,๐‘†)-market, where ๐ต stands for the bank account and ๐‘† stands for the stock.

A European option contract is a contract giving the buyer of the contract the right to buy (sell) a share of a particular stock at a predetermined price at a predetermined time in the future. The European option problem is, briefly, to determine the fee (called the rational price) that the writer of the contract should receive from the buyer for the rights of the contract and also to determine the trading strategy the writer should use to invest this fee in the (๐ต,๐‘†)-market in such a way as to ensure that the writer will be able to cover the option if it is exercised. The fee should be large enough that the writer can, with riskless investing, cover the option, but be small enough that the writer does not make an unfair (i.e., riskless) profit.

In [6], we noted reasons to include hereditary price structures to a (๐ต,๐‘†)-market model and then introduced such a model using a functional differential equation to describe the dynamics of the bank account and a stochastic functional differential equation to describe those of the stock account. The paper then obtained a solution to the option pricing problem in terms of conditional expectation with respect to a martingale measure. The importance of including hereditary price structure in the stock price dynamics was also recognized by other researchers in recent years (see, e.g., [7โ€“14]).

In particular, [6] was one of the firsts that took into consideration hereditary structure in studying the pricing problem of European option. There the authors obtained a solution to the option pricing problem in terms of conditional expectation with respect to a martingale measure. The two papers [7, 9] developed an explicit formula for pricing European options when the underlying stock price follows a nonlinear stochastic delay equation with fixed delays (resp., variable delays) in the drift and diffusion terms. The paper [8] computed the logarithmic utility of an insider when the financial market is modelled by a stochastic delay equation. There the author showed that, although the market does not allow free lunches and is complete, the insider can draw more from his wealth than the regular trader. The paper also offered an alternative to the anticipating delayed Black-Scholes formula, by proving stability of European call option proces when the delay coefficients approach the nondelayed ones. The paper [10] derived the infinite-dimensional Black-Scholes equation for the (๐ต,๐‘†)-market, where the bank account evolves according to a linear (deterministic) functional differential equation and the stock dynamics is described by a very general nonlinear stochastic functional differential equation. A power series solution is also developed for the equation. Following the same model studied in [10], the work in [11] shows that under very mild conditions the pricing function is the unique viscosity solution of the infinite-dimensional Black-Scholes equation. A finite difference approximation scheme for the solution of the equation is developed and convergence result is also obtained. We mention here that option pricing problems were also considered by [12โ€“14] for a financial market that is more restricted than those of [10, 11].

This paper considers the pricing of a European option using a (๐ต,๐‘†)-market, such as those in [6], in which the stock price and the asset in the riskless bank account both have hereditary price structures. Under the smoothness assumption of the payoff function, it is shown that the pricing function is the unique classical solution of the infinite-dimensional Black-Scholes equation. A spectral approximation scheme is developed using the Fourier series expansion in the space ๐ถ[โˆ’โ„Ž,0] for the Black-Scholes equation. It is also shown that the ๐‘›th approximant resembles the celebrated classical Black-Scholes equation in finite dimensions (see, e.g., [4, 5]).

This paper is organized as follows. Section 2 summarizes the definitions and key results of [6] that will be used throughout this paper. The concepts of Frรฉchet derivative and extended Frรฉchet derivative are introduced in Section 3, along with results needed to make use of these derivatives. In Section 4, the results regarding the infinite-dimensional Black-Scholes equation and its corollary are restated from [6, 10]. Section 5 details the spectral approximate solution scheme for this equation. Section 6 is the paper's conclusion, followed by an appendix with the proof of Proposition 3.2.

2. The European Option Problem with Hereditary Price Structures

To describe the financial model with hereditary price structures, we start by defining our probability space. Let 0<โ„Ž<โˆž be a fixed constant. This constant will be the length of the time window in which the hereditary information is contained. If ๐‘Ž,๐‘โˆˆโ„œ with ๐‘Ž<๐‘, denote the space of continuous functions ๐œ™โˆถ[๐‘Ž,๐‘]โ†’โ„œ by ๐ถ[๐‘Ž,๐‘]. Define ๐ถ+[][][]๐‘Ž,๐‘={๐œ™โˆˆ๐ถ๐‘Ž,๐‘โˆฃ๐œ™(๐œƒ)โ‰ฅ0โˆ€๐œƒโˆˆ๐‘Ž,๐‘}.(2.1) Note that ๐ถ[๐‘Ž,๐‘] is a real separable Banach space equipped with the uniform topology defined by the sup-norm โ€–๐œ™โ€–=sup๐‘กโˆˆ[๐‘Ž,๐‘]|๐œ™(๐‘ก)| and ๐ถ+[๐‘Ž,๐‘] is a closed subset of ๐ถ[๐‘Ž,๐‘]. Throughout the end of this paper, we let ๐‚=๐ถ[โˆ’โ„Ž,0] and ๐‚+[]}={๐œ™โˆˆ๐‚โˆฃ๐œ™(๐œƒ)โ‰ฅ0โˆ€๐œƒโˆˆโˆ’โ„Ž,0(2.2) for simplicity. If ๐œ“โˆˆ๐ถ[โˆ’โ„Ž,โˆž) and ๐‘กโˆˆ[0,โˆž), let ๐œ“๐‘กโˆˆ๐‚ be defined by ๐œ“๐‘ก(๐œƒ)=๐œ“(๐‘ก+๐œƒ), ๐œƒโˆˆ[โˆ’โ„Ž,0].

Let ฮฉ=๐ถ[โˆ’โ„Ž,โˆž), the space of continuous functions ๐œ”โˆถ[โˆ’โ„Ž,โˆž)โ†’โ„œ, and let ๐น=๐ต(๐ถ[โˆ’โ„Ž,โˆž)), the Borel ๐œŽ-algebra of subsets of ๐ถ[โˆ’โ„Ž,โˆž) under the topology defined by the metric ๐‘‘โˆถฮฉร—ฮฉโ†’โ„œ, where๐‘‘๎€ท๐œ”,๐œ”๎…ž๎€ธ=โˆž๎“๐‘›=112๐‘›supโˆ’โ„Žโ‰ค๐‘กโ‰ค๐‘›||๐œ”(๐‘ก)โˆ’๐œ”๎…ž||(๐‘ก)1+supโˆ’โ„Žโ‰ค๐‘กโ‰ค๐‘›||๐œ”(๐‘ก)โˆ’๐œ”๎…ž||.(๐‘ก)(2.3)

Let ๐ be the Wiener measure defined on (ฮฉ,๐น) with []๐{๐œ”โˆˆฮฉโˆฃ๐œ”(๐œƒ)=0โˆ€๐œƒโˆˆโˆ’โ„Ž,0}=1.(2.4) Note that the probability space (ฮฉ,๐น,๐) is the canonical Wiener space under which the coordinate maps ๐‘Š={๐‘Š(๐‘ก),๐‘กโ‰ฅ0}, ๐‘Š(๐‘ก)โˆถ๐ถ[โˆ’โ„Ž,โˆž)โ†’โ„œ, defined by ๐‘Š(๐‘ก)(๐œ”)=๐œ”(๐‘ก) for all ๐‘กโ‰ฅโˆ’โ„Ž and ๐œ”โˆˆฮฉ is a standard Brownian motion and ๐{๐‘Š0=0}=1. Let the filtration ๐น๐‘Š={๐น(๐‘ก),๐‘กโ‰ฅโˆ’โ„Ž} be the ๐-augmentation of the natural filtration of the Brownian motion ๐‘Š, defined by ๐น(๐‘ก)={โˆ…,ฮฉ} for all ๐‘กโˆˆ[โˆ’โ„Ž,0] and ๐น(๐‘ก)=๐œŽ(๐‘Š(๐‘ ),0โ‰ค๐‘ โ‰ค๐‘ก),๐‘กโ‰ฅ0.(2.5)

Equivalently, ๐น(๐‘ก) is the smallest sub-๐œŽ-algebra of subsets of ฮฉ with respect to which the mappings ๐‘Š(๐‘ )โˆถฮฉโ†’โ„œ are measurable for all 0โ‰ค๐‘ โ‰ค๐‘ก. It is clear that the filtration ๐น๐‘Š defined above is right continuous in the sense of [15].

Consider the ๐‚-valued process {๐‘Š๐‘ก,๐‘กโ‰ฅ0}, where ๐‘Š0=0 and ๐‘Š๐‘ก(๐œƒ)=๐‘Š(๐‘ก+๐œƒ), ๐œƒโˆˆ[โˆ’โ„Ž,0] for all ๐‘กโ‰ฅ0. That is, for each ๐‘กโ‰ฅ0, ๐‘Š๐‘ก(๐œ”)=๐œ”๐‘ก and ๐‘Š0=0. In [10], it is shown that ๐น0=๐น(๐‘ก) for ๐‘กโˆˆ[โˆ’โ„Ž,0] and ๐น(๐‘ก)=๐น๐‘ก for ๐‘กโ‰ฅ0, where ๐น๐‘ก๎€ท๐‘Š=๐œŽ๐‘ ๎€ธ,0โ‰ค๐‘ โ‰ค๐‘ก,๐‘กโ‰ฅ0.(2.6)

The new model for the (๐ต,๐‘†)-market introduced in [6] has a hereditary price structure in the sense that the rate of change of the unit price of the investor's assets in the bank account ๐ต and that of the stock account ๐‘† depend not only on the current unit price but also on their historical prices. Specifically, we assume that ๐ต and ๐‘† evolve according to the following two linear functional differential equations: ๎€ท๐ต๐‘‘๐ต(๐‘ก)=๐ฟ๐‘ก๎€ธ๎€ท๐‘†๐‘‘๐‘ก,๐‘กโ‰ฅ0,(2.7)๐‘‘๐‘†(๐‘ก)=๐‘€๐‘ก๎€ธ๎€ท๐‘†๐‘‘๐‘ก+๐‘๐‘ก๎€ธ๐‘‘๐‘Š(๐‘ก),๐‘กโ‰ฅ0,(2.8) with initial price functions ๐ต0=๐œ™ and ๐‘†0=๐œ“, where ๐œ™ and ๐œ“ are given functions in ๐‚+. In the model, ๐ฟ, ๐‘€, and ๐‘ are bounded linear functionals on the real Banach space ๐‚. The bounded linear functionals ๐ฟ,๐‘€,๐‘โˆถ๐‚โ†’โ„œ can be represented as (see [6]) ๎€œ๐ฟ(๐œ™)=0โˆ’โ„Ž๎€œ๐œ™(๐œƒ)๐‘‘๐œ‚(๐œƒ),๐‘€(๐œ™)=0โˆ’โ„Ž๐‘๎€œ๐œ™(๐œƒ)๐‘‘๐œ‰(๐œƒ),(2.9)(๐œ™)=0โˆ’โ„Ž๐œ™(๐œƒ)๐‘‘๐œ(๐œƒ),๐œ™โˆˆ๐‚,(2.10) where the above integrals are to be interpreted as Lebesgue-Stieltjes integrals and ๐œ‚, ๐œ‰, and ๐œ are functions that are assumed to satisfy the following conditions.

Assumption 2.1. The functions ๐œ‚,๐œ‰โˆถ[โˆ’โ„Ž,0]โ†’โ„œ, are nondecreasing functions on [โˆ’โ„Ž,0] such that ๐œ‚(0)โˆ’๐œ‚(โˆ’โ„Ž)>0 and ๐œ‰(0)โˆ’๐œ‰(โˆ’โ„Ž)>0, and the function ๐œโˆถ[โˆ’โ„Ž,0]โ†’โ„œ is a function of bounded variation on [โˆ’โ„Ž,0] such that โˆซ0โˆ’โ„Ž๐œ™(๐œƒ)๐‘‘๐œ(๐œƒ)โ‰ฅ๐œŽ>0 for every ๐œ™โˆˆ๐‚+.

We will, throughout the end, extend the domain of the above three functions to ๐‘… by defining ๐œ‚(๐œƒ)=๐œ‚(โˆ’โ„Ž) for ๐œƒโ‰คโˆ’โ„Ž and ๐œ‚(๐œƒ)=๐œ‚(0) for ๐œƒโ‰ฅ0, and so forth.

Proposition 2.3 in [6] provides an existence and uniqueness result under mild conditions, so the model makes sense mathematically to use. Note that the equations described by (2.7)-(2.8) include (1.1) as a special case. Therefore, the model considered in this paper is a generalization of that considered in most of the existing literature (see, e.g., [5]).

For the purpose of analyzing the discount rate for the bank account, let us assume that the solution process ๐ต(๐ฟ;๐œ™)={๐ต(๐‘ก),โˆ’โ„Žโ‰ค๐‘ก<โˆž} of (2.7) with the initial function ๐œ™โˆˆ๐‚+ takes the following form: ๐ต(๐‘ก)=๐œ™(0)๐‘’๐‘Ÿ๐‘ก,๐‘กโ‰ฅ0,(2.11) and ๐ต0=๐œ™โˆˆ๐‚+. Then the constant ๐‘Ÿ satisfies the following equation: ๎€œ๐‘Ÿ=0โˆ’โ„Ž๐‘’๐‘Ÿ๐œƒ๐‘‘๐œ‚(๐œƒ).(2.12) The existence and uniqueness of a positive number ๐‘Ÿ that satisfies the above equation is shown in [6].

Throughout the end, we will fix the initial unit price functions ๐œ™, and ๐œ“โˆˆ๐‚+, and the functional ๐‘โˆถ๐‚โ†’โ„œ for the stock price described in (2.8) and (2.10). For the purpose of making the distinction when we interchange the usage of ๐‘€โˆถ๐‚โ†’โ„œ and ๐ฟโˆถ๐‚โ†’โ„œ in (2.8), we write the stock price process ๐‘†(๐‘€,๐‘;๐œ“) as ๐‘†(๐‘€)={๐‘†(๐‘ก),๐‘กโ‰ฅโˆ’โ„Ž} for simplicity. And, when the functional ๐พโˆถ๐‚โ†’โ„œ, ๐พ(๐œ™๐‘ก)=๐‘Ÿ๐œ™(๐‘ก) is used in place of ๐‘€โˆถ๐‚โ†’โ„œ in (2.8), its solution process will be written as ๐‘†(๐พ)={๐‘†(๐‘ก),๐‘กโ‰ฅโˆ’โ„Ž}.

In [6], the basic theory of European option pricing using the (๐ต,๐‘†)-market model described in (2.7)-(2.8) is developed. We summarize the key definitions and results below.

A trading strategy in the (๐ต,๐‘†)-market is a progressively measurable vector process ๐œ‹={(๐œ‹1(๐‘ก),๐œ‹2(๐‘ก)),0โ‰ค๐‘ก<โˆž} defined on (ฮฉ,๐น,๐;๐น๐‘Š) such that for each ๐‘Ž>0, ๎€œ๐‘Ž0๐„๎€บ๐œ‹2๐‘–๎€ป(๐‘ก)๐‘‘๐‘ก<โˆž,๐‘–=1,2,(2.13) where ๐œ‹1(๐‘ก) and ๐œ‹2(๐‘ก) represent, respectively, the number of units of the bank account and the number of shares of the stock owned by the writer at time ๐‘กโ‰ฅ0, and ๐„ is the expectation with respect to ๐.

The writer's total asset is described by the wealth process ๐‘‹๐œ‹(๐‘€)={๐‘‹๐œ‹(๐‘ก),0โ‰ค๐‘ก<โˆž} defined by ๐‘‹๐œ‹(๐‘ก)=๐œ‹1(๐‘ก)๐ต(๐‘ก)+๐œ‹2(๐‘ก)๐‘†(๐‘ก),0โ‰ค๐‘ก<โˆž,(2.14) where again ๐ต(๐ฟ;๐œ™) and ๐‘†(๐‘€,๐‘;๐œ“) are, respectively, the unit price of the bank account and the stock described in (2.7) and (2.8). This wealth process can clearly take both positive and negative values, since it is permissible that (๐œ‹1(๐‘ก),๐œ‹2(๐‘ก))โˆˆโ„œ2.

We will make the following basic assumption throughout this paper.

Assumption 2.2 (self-financing condition). In the (๐ต,๐‘†)-market, it is assumed that all trading strategies ๐œ‹ satisfy the following self-financing condition: ๐‘‹๐œ‹(๐‘ก)=๐‘‹๐œ‹(๎€œ0)+๐‘ก0๐œ‹1(๎€œ๐‘ )๐‘‘๐ต(๐‘ )+๐‘ก0๐œ‹2(๐‘ )๐‘‘๐‘†(๐‘ ),0โ‰ค๐‘ก<โˆž,a.s.(2.15) or equivalently, ๐‘‘๐‘‹๐œ‹(๐‘ก)=๐œ‹1(๐‘ก)๐‘‘๐ต(๐‘ก)+๐œ‹2(๐‘ก)๐‘‘๐‘†(๐‘ก),0โ‰ค๐‘ก<โˆž.(2.16)

Using the same notation as in [6] (see also [10]) the set of all self-financing trading strategies ๐œ‹ will be denoted by SF(๐ฟ,๐‘€,๐‘;๐œ™,๐œ“) or simply SF if there is no danger of ambiguity.

For the unit price of the bank account ๐ต(๐ฟ;๐œ™)={๐ต(๐‘ก),๐‘กโ‰ฅ0} and the stock ๐‘†(๐‘€,๐‘;๐œ“)={๐‘†(๐‘ก),๐‘กโ‰ฅ0} described in (2.7) and (2.8), define ๎‚‹๎€œ๐‘Š(๐‘ก)=๐‘Š(๐‘ก)+๐‘ก0๐›พ๎€ท๐ต๐‘ ,๐‘†๐‘ ๎€ธ๐‘‘๐‘ ,๐‘กโ‰ฅ0,(2.17) where ๐›พโˆถ๐‚+ร—๐‚+โ†’โ„œ is defined by ๐›พ(๐œ™,๐œ“)=๐œ™(0)๐‘€(๐œ“)โˆ’๐œ“(0)๐ฟ(๐œ™).๐œ™(0)๐‘(๐œ“)(2.18)

Define the process ๐‘(๐ฟ,๐‘€,๐‘;๐œ™,๐œ“)={๐‘(๐‘ก),๐‘กโ‰ฅ0} by ๎‚ป๎€œ๐‘(๐‘ก)=exp๐‘ก0๐›พ๎€ท๐ต๐‘ ,๐‘†๐‘ ๎€ธ1๐‘‘๐‘Š(๐‘ )โˆ’2๎€œ๐‘ก0||๐›พ๎€ท๐ต๐‘ ,๐‘†๐‘ ๎€ธ||2๎‚ผ๐‘‘๐‘ ,๐‘กโ‰ฅ0.(2.19) The following results are proven in [6].

Lemma 2.3. The process ๐‘(๐ฟ,๐‘€,๐‘;๐œ™,๐œ“)={๐‘(๐‘ก),๐‘กโ‰ฅ0} defined by (2.19) is a martingale defined on (ฮฉ,๐น,๐;๐น๐‘Š).

Lemma 2.4. There exists a unique probability measure ๎‚๐ defined on the canonical measurable space (ฮฉ,๐น) such that ๎‚๎€บ๐Ÿ๐(๐ด)=๐„๐ด๎€ป๐‘(๐‘‡)โˆ€๐ดโˆˆ๐น๐‘‡,0<๐‘‡<โˆž,(2.20) where ๐Ÿ๐ด is the indicator function of ๐ดโˆˆ๐น๐‘‡.

Lemma 2.5. The process ๎‚‹๐‘Š defined by (2.17) is a standard Brownian motion defined on the filtered probability space ๎‚(ฮฉ,๐น,๐;๐น๐‘Š).

From the above, it has been shown (see [6, equation (14)]) that ๎€ท๐‘†๐‘‘๐‘†(๐‘ก)=๐‘Ÿ๐‘†(๐‘ก)๐‘‘๐‘ก+๐‘๐‘ก๎€ธ๐‘‘๎‚‹๐‘Š(๐‘ก),(2.21) with ๐‘†0=๐œ“โˆˆ๐‚+. It is also clear that the probabilistic behavior of ๐‘†(๐‘€) under the probability measure ๐ is the same as that of ๐‘†(๐พ) under the probability measure ๎‚๐; that is, they have the same distribution.

Define the process ๐‘Œ๐œ‹(๐ฟ,๐‘€,๐‘;๐œ™,๐œ“)={๐‘Œ๐œ‹(๐‘ก),๐‘กโ‰ฅ0}, called the discounted wealth process, by ๐‘Œ๐œ‹๐‘‹(๐‘ก)=๐œ‹(๐‘ก)๐ต(๐‘ก),๐‘กโ‰ฅ0.(2.22) We say that a trading strategy ๐œ‹ from SF(๐ฟ,๐‘€,๐‘;๐œ™,๐œ“) belongs to a subclass SF๐œโŠ‚SF if ๎‚๐ a.s. ๐‘Œ๐œ‹๎‚๐„๎€บ(๐‘ก)โ‰ฅโˆ’๐œโˆฃ๐น๐‘ก๎€ป,๐‘กโ‰ฅ0,(2.23) where ๎‚๐„ is the expectation with respect to ๎‚๐, ๐œ is a nonnegative ๐น-measurable random variable such that ๎‚๐„[๐œ]<โˆž. We say that ๐œ‹ belongs to SF+โŠ‚SF if ๐œโ‰ฅ0.

In [6, 10], it is shown that ๐‘Œ๐œ‹ is a local martingale; for ๐œ‹โˆˆSF๐œ, ๐‘Œ๐œ‹ is a supermartingale, and is a nonnegative supermartingale if ๐œ‹โˆˆSF+.

Throughout, we assume the reward function ฮ› is an ๐น๐‘‡-measurable nonnegative random variable satisfying the following condition: ๐„๎€บฮ›1+๐œ–๎€ป<โˆž,(2.24) for some ๐œ–>0. Here, ๐‘‡>0 is the expiration time. (Note that the above condition on ฮ› implies that ๎‚๐„[ฮ›]<โˆž.)

Let ฮ› be a nonnegative ๐น๐‘‡-measurable random variable satisfying (2.24). A trading strategy ๐œ‹โˆˆSF is a (๐‘€;ฮ›,๐‘ฅ)-hedge of European type if ๐‘‹๐œ‹(0)=๐œ‹1(0)๐œ™(0)+๐œ‹2(0)๐œ“(0)=๐‘ฅ(2.25) and ๎‚๐ a.s. ๐‘‹๐œ‹(๐‘‡)โ‰ฅฮ›.(2.26) We say that a (๐‘€;ฮ›,๐‘ฅ)-hedge trading strategy ๐œ‹โˆ—โˆˆSF(๐‘€) is minimal if ๐‘‹๐œ‹(๐‘‡)โ‰ฅ๐‘‹๐œ‹โˆ—(๐‘‡)(2.27) for any (๐‘€;ฮ›,๐‘ฅ)-hedge strategy ๐œ‹โˆˆSF(๐‘€).

Let ฮ (๐‘€;ฮ›,๐‘ฅ) be the set of (๐‘€;ฮ›,๐‘ฅ)-hedge strategies from SF+(๐‘€). Define ๐ถ(๐‘€;ฮ›)=inf{๐‘ฅโ‰ฅ0โˆถฮ (๐‘€;ฮ›,๐‘ฅ)โ‰ โˆ…}.(2.28) The value ๐ถ(๐‘€;ฮ›) defined above is called the rational price of the contingent claim of European type. If the infimum in (2.28) is achieved, then ๐ถ(๐‘€;ฮ›) is the minimal possible initial capital for which there exists a trading strategy ๐œ‹โˆˆSF+(๐‘€) possessing the property that ๎‚๐ a.s. ๐‘‹๐œ‹(๐‘‡)โ‰ฅฮ›.

Let ๐‘Œ(๐‘€)={๐‘Œ(๐‘ก),0โ‰ค๐‘กโ‰ค๐‘‡} be defined by ๎‚๐„๎‚ธฮ›๐‘Œ(๐‘ก)=โˆฃ๎‚๐น๐ต(๐‘‡)๐‘ก๎‚น,0โ‰ค๐‘กโ‰ค๐‘‡,(2.29) where ๎‚๐น๐‘ก๎‚‹๐‘Š=๐œŽ(๐‘ ,0โ‰ค๐‘ โ‰ค๐‘ก). In [10], it is shown that the process ๐‘Œ(๐‘€) is a martingale defined on ๎‚๎‚‹๐‘Š)(ฮฉ,๐น,๐;๐น and can be represented by ๎€œ๐‘Œ(๐‘ก)=๐‘Œ(0)+๐‘ก0๎‚‹๐›ฝ(๐‘ )๐‘‘๐‘Š(๐‘ ),(2.30) where ๐›ฝ={๐›ฝ(๐‘ก),0โ‰ค๐‘กโ‰ค๐‘‡} that is ๐น๎‚‹๐‘Š-adapted and โˆซ๐‘‡0๐›ฝ2(๐‘ก)๐‘‘๐‘ก<โˆž (๐ a.s.).

The following lemma and theorem provide the main results of [6, 10]. Let ๐œ‹โˆ—={(๐œ‹โˆ—1(๐‘ก),๐œ‹โˆ—2(๐‘ก)),0โ‰ค๐‘กโ‰ค๐‘‡} be a trading strategy, where ๐œ‹โˆ—2(๐‘ก)=๐›ฝ(๐‘ก)๐ต(๐‘ก)๐‘๎€ท๐‘†๐‘ก๎€ธ,๐œ‹โˆ—1๐‘†(๐‘ก)=๐‘Œ(๐‘ก)โˆ’(๐‘ก)๐œ‹๐ต(๐‘ก)โˆ—2[].(๐‘ก),๐‘กโˆˆ0,๐‘‡(2.31)

Lemma 2.6. ๐œ‹โˆ—โˆˆSF(๐‘€) and for each ๐‘กโˆˆ[0,๐‘‡], ๐‘Œ(๐‘ก)=๐‘Œ๐œ‹โˆ—(๐‘ก) for each ๐‘กโˆˆ[0,๐‘‡] where again ๐‘Œ๐œ‹โˆ— is the process defined in (2.22) with the minimal strategy ๐œ‹โˆ— defined in (2.31).

Theorem 2.7. Let ฮ› be an ๐น๐‘‡-measurable random variable defined on the filtered probability space (ฮฉ,๐น,๐;๐น๐‘Š) that satisfies (2.24). Then the rational price ๐ถ(๐‘€;ฮ›) defined in (2.28) is given by ๎‚๐„๎€บ๐‘’๐ถ(๐‘€;ฮ›)=โˆ’๐‘Ÿ๐‘‡ฮ›๎€ป,(2.32) where ๐‘Ÿ is the positive constant that satisfies (2.12). Furthermore, there exists a minimal hedge ๐œ‹โˆ—={(๐œ‹โˆ—1(๐‘ก),๐œ‹โˆ—2(๐‘ก)),0โ‰ค๐‘กโ‰ค๐‘‡}, where ๐œ‹โˆ—2(๐‘ก)=๐›ฝ(๐‘ก)๐ต(๐‘ก)๐‘๎€ท๐‘†๐‘ก๎€ธ,๐œ‹โˆ—1(๐‘ก)=๐‘Œ๐œ‹โˆ—(๐‘ก)โˆ’๐œ‹โˆ—2๐‘†(๐‘ก)(๐‘ก),๐ต(๐‘ก)(2.33) and the process ๐›ฝ={๐›ฝ(๐‘ก),0โ‰ค๐‘กโ‰ค๐‘‡} is given by (2.30).
If in addition, the reward ฮ› is intrinsic, that is, ฮ›=ฮ“(๐‘†(๐‘€)) for some measurable function ฮ“โˆถ๐‚+โ†’โ„œ, then the rational price ๐ถ(๐‘€;ฮ›) does not depend on the mean growth rate ๐‘€ of the stock and ๎‚๐„๎€บ๐‘’๐ถ(ฮ›)=โˆ’๐‘Ÿ๐‘‡ฮ›๎€ป.(2.34)

3. Frรฉchet and Extended Frรฉchet Derivatives

In this section, results are proven that allow the use of a Dynkins formula for stochastic functional differential equation as found in [16, 17]. We assume contingent claims of European type in which the ๐น๐‘‡-measurable reward function ฮ› has the explicit expression ฮ›=๐‘“(๐‘†๐‘‡), where again ๐‘†๐‘‡(๐œƒ)=๐‘†(๐‘‡+๐œƒ), ๐œƒโˆˆ[โˆ’โ„Ž,0] and ๐‘†(๐พ)={๐‘†(๐‘ก),๐‘กโ‰ฅ0} is the unit price of the stock described by the following equation: ๎€ท๐‘†๐‘‘๐‘†(๐‘ก)=๐‘Ÿ๐‘†(๐‘ก)๐‘‘๐‘ก+๐‘๐‘ก๎€ธ๐‘‘๎‚‹๐‘Š(๐‘ก),๐‘กโ‰ฅ0,(3.1) where ๐‘†0=๐œ“โˆˆ๐‚+. Throughout this section, we assume that ๐‘†(๐‘ก), and therefore ๐‘(๐‘†๐‘ก), are uniformly bounded almost surely. This assumption is realistic for the price of a stock during time interval [0,๐‘‡] in a financial system with finite total wealth.

The remaining sections make extensive use of Frรฉchet derivatives. Let ๐‚โˆ— be the space of bounded linear functionals ฮฆโˆถ๐‚โ†’โ„œ. ๐‚โˆ— is a real separable Banach space under the supremum operator norm โ€–ฮฆโ€–=sup๐œ™โ‰ 0||||ฮฆ(๐œ™).โ€–๐œ™โ€–(3.2) For ฮจโˆถ[0,๐‘‡]ร—๐‚โ†’โ„œ, we denote the Frรฉchet derivative of ฮจ at ๐œ™โˆˆ๐‚ by ๐ทฮจ(๐‘ก,๐œ™). The second Frรฉchet derivative at ๐œ™ is denoted as ๐ท2ฮจ(๐‘ก,๐œ™).

Let ฮ“ be the vector space of all simple functions of the form ๐‘ฃ๐Ÿ{0}, where ๐‘ฃโˆˆ๐‘… and ๐Ÿ{0}โˆถ[โˆ’โ„Ž,0]โ†’โ„œ is defined by ๐Ÿ{0}๎ƒฏ[(๐œƒ)=0,for๐œƒโˆˆโˆ’โ„Ž,0),1,for๐œƒ=0.(3.3) Form the direct sum ๐‚โŠ•ฮ“ and equip it with the complete norm โ€–โ€–๐œ™+๐‘ฃ๐Ÿ{0}โ€–โ€–=sup๐œƒโˆˆ[โˆ’โ„Ž,0]||||๐œ™(๐œƒ)+|๐‘ฃ|,๐œ™โˆˆ๐‚,๐‘ฃโˆˆโ„œ.(3.4) Then ๐ทฮจ(๐‘ก,๐œ™) has a unique continuous linear extension from ๐‚โŠ•ฮ“ to โ„œ which we will denote by ๐ทฮจ(๐‘ก,๐œ™), and similarly for ๐ท2ฮจ(๐‘ก,๐œ™); see [16] or [17] for more details.

Finally, we define ๎€ท๐†(ฮจ)๐‘ก,๎‚๐œ“๐‘ก๎€ธ=lim๐‘ขโ†’0+1๐‘ข๎€บฮจ๎€ท๐‘ก,๎‚๐œ“๐‘ก+๐‘ข๎€ธ๎€ทโˆ’ฮจ๐‘ก,๎‚๐œ“๐‘ก๎€ธ๎€ป(3.5) for all ๐‘กโˆˆ[0,โˆž) and ๐œ“โˆˆ๐‚+, where ๎‚๐œ“โˆถ[โˆ’โ„Ž,โˆž)โ†’โ„œ is defined by ๎ƒฏ[๎‚๐œ“(๐‘ก)=๐œ“(๐‘ก)if๐‘กโˆˆโˆ’โ„Ž,0)๐œ“(0)if๐‘กโ‰ฅ0.(3.6) Let ๐‘“โˆถ๐‚โ†’โ„œ. We say that ๐‘“โˆˆ๐ถ1(๐‚) if ๐‘“ has a continuous Frรฉchet derivative. Similarly, ๐‘“โˆˆ๐ถ๐‘›(๐‚) if ๐‘“ has a continuous ๐‘›th Frรฉchet derivative. For ๐‘“โˆถ๐‘…+ร—๐‚โ†’โ„œ, we say that ๐‘“โˆˆ๐ถโˆž,๐‘›([0,โˆž)ร—๐‚) if ๐‘“ is infinitely differentiable in its first variable and has a continuous ๐‘›th partial derivative in its second variable.

Proposition 3.1. Let ๐œ‘โˆˆ๐‚ and ๐‘“โˆถ๐‚โ†’โ„œ with ๐‘“โˆˆ๐ถ2(๐‚). Define ฮจโˆถ[0,๐‘‡]ร—๐‚โ†’โ„œ by ฮจ(๐‘ก,๐œ‘)=๐‘’โˆ’๐‘Ÿ(๐‘‡โˆ’๐‘ก)๎‚๐„๎€บ๐‘“๎€ท๐‘†๐‘‡๎€ธโˆฃ๐‘†๐‘ก๎€ป.=๐œ‘(3.7) Then ฮจโˆˆ๐ถโˆž,2([0,โˆž)ร—๐‚).

Proof. That ๐‘’โˆ’๐‘Ÿ(๐‘‡โˆ’๐‘ก) is ๐ถโˆž[0,โˆž) is clear, so we have only to show that ฮฅโˆˆ๐ถ2(๐‚), where ๎‚ฮฅ(๐œ‘)=๐„[๐‘“(๐‘†๐‘‡)โˆฃ๐‘†๐‘ก=๐œ‘] given that ๐‘“โˆˆ๐ถ2(๐‚).
We have that ๎€ท๐‘†๐‘‘๐‘†(๐‘ก)=๐‘Ÿ๐‘†(๐‘ก)๐‘‘๐‘ก+๐‘๐‘ก๎€ธ๐‘‘๎‚‹๐‘Š(๐‘ก),๐‘กโ‰ฅ0,(3.8) with ๐‘†0=๐œ“โˆˆ๐‚+. Under Assumption 2.1 on ๐‘โˆถ๐‚โ†’โ„œ and the properties of ฮฅ, it can be shown that there exists ๐ปโˆถโ„œร—โ„œร—๐‚โ†’๐‚ such that ๐‘†๐‘ก๎‚‹=๐ป(๐‘ก,๐‘Š(๐‘ก),๐œ“). Therefore, ๎‚๐„๎€บ๐‘“๎€ท๐‘†ฮฅ(๐œ‘)=๐‘‡๎€ธโˆฃ๐‘†๐‘ก๎€ป=1=๐œ‘โˆš๎€œ2๐œ‹โˆžโˆ’โˆž๐‘“(๐ป(๐‘‡โˆ’๐‘ก,๐‘ฆ,๐œ‘))๐‘’โˆ’๐‘ฆ2/2๐‘‘๐‘ฆ.(3.9) By Theorem 3.2, Chapter 2 of [16], ๐ป(๐‘ก,๐‘ฆ,โ‹…)โˆˆ๐ถ1(๐‚) as a function of ๐œ“. By a second application of the same theorem (since ๐‘“โˆˆ๐ถ2(๐‚)), we have that ๐ป(๐‘ก,๐‘ฆ,โ‹…)โˆˆ๐ถ2(๐‚) as a function of ๐œ“. Define ๐‘”โˆถโ„œร—โ„œร—๐‚โ†’โ„œ by ๐‘”=๐‘“โˆ˜๐ป. Since ๐‘“โˆˆ๐ถ2(๐‚) and ๐ป(๐‘ก,๐‘ฆ,โ‹…)โˆˆ๐ถ2(๐‚) in its third variable, ๐‘”(๐‘ก,๐‘ฆ,โ‹…)โˆˆ๐ถ2(๐‚). Hence, for ๐œ‘,๐œ™โˆˆ๐‚, ๎‚๐„๎€บ๐‘“๎€ท๐‘†๐‘‡๎€ธโˆฃ๐‘†๐‘ก๎€ปโˆ’๎‚๐„๎€บ๐‘“๎€ท๐‘†=๐œ‘+๐œ™๐‘‡๎€ธโˆฃ๐‘†๐‘ก๎€ป=1=๐œ‘โˆš๎€œ2๐œ‹โˆžโˆ’โˆž[]๐‘’๐‘“(๐ป(๐‘‡โˆ’๐‘ก,๐‘ฆ,๐œ‘+๐œ™))โˆ’๐‘“(๐ป(๐‘‡โˆ’๐‘ก,๐‘ฆ,๐œ‘))โˆ’๐‘ฆ2/2=1๐‘‘๐‘ฆโˆš๎€œ2๐œ‹โˆžโˆ’โˆž๐ท๐‘”(๐‘‡โˆ’๐‘ก,๐‘ฆ,๐œ‘)(๐œ™)๐‘’โˆ’๐‘ฆ2/21๐‘‘๐‘ฆ+โˆš๎€œ2๐œ‹โˆžโˆ’โˆž๐‘œ(๐œ™)๐‘’โˆ’๐‘ฆ2/2๐‘‘๐‘ฆ,(3.10) where ๐‘œ(๐œ™) is a function mapping continuous functions into the reals such that ๐‘œ(๐œ™)โ€–๐œ™โ€–โŸถ0asโ€–๐œ™โ€–โŸถ0.(3.11) The last integral is clearly ๐‘œ(๐œ™) and 1โˆš๎€œ2๐œ‹โˆžโˆ’โˆž๐ท๐‘”(๐‘‡โˆ’๐‘ก,๐‘ฆ,๐œ‘)(๐œ™)๐‘’โˆ’๐‘ฆ2/2๐‘‘๐‘ฆ(3.12) is bounded and linear in ๐œ™, so this integral is the first Frรฉchet derivative with respect to ๐œ‘. Since ๐‘”(๐‘ก,๐‘ฆ,โ‹…)โˆˆ๐ถ2(๐‚), the process can be repeated, giving a second Frรฉchet derivative with respect to ๐œ‘ and so ฮฅโˆˆ๐ถ2(๐‚).

Proposition 3.2. Let ๐œ‘โˆˆ๐‚ and ๐‘“โˆถ๐‚โ†’โ„œ. Further assume ๐‘“โˆˆ๐ถ2(๐‚) and let ฮจโˆถ[0,๐‘‡]ร—๐‚โ†’โ„œ be defined by ฮจ(๐‘ก,๐œ‘)=๐‘’โˆ’๐‘Ÿ(๐‘‡โˆ’๐‘ก)๎‚๐ธ๎€บ๐‘“๎€ท๐‘†๐‘‡๎€ธโˆฃ๐‘†๐‘ก๎€ป.=๐œ‘(3.13) Then if ๐ท๐‘“ and ๐ท2๐‘“ are globally Lipschitz, then so is ๐ท2ฮจ.

Recall from Proposition 3.1 that ๐‘”โˆถ๐‘…ร—๐‘…ร—๐‚โ†’๐‘… is ๐‘“โˆ˜๐ป where ๐‘†๐‘ก๎‚‹=๐ป(๐‘ก,๐‘Š(๐‘ก),๐œ“) with ๐‘†0=๐œ“โˆˆ๐‚+.

Proposition 3.3. Let ๐œ‘โˆˆ๐‚ and ๐‘“โˆถ๐‚โ†’โ„œ. Further assume ๐‘“โˆˆ๐ถ2(๐‚) and let ฮจโˆถ[0,๐‘‡]ร—๐‚โ†’โ„œ be defined by ฮจ(๐‘ก,๐œ‘)=๐‘’โˆ’๐‘Ÿ(๐‘‡โˆ’๐‘ก)๎‚๐ธ๎€บ๐‘“๎€ท๐‘†๐‘‡๎€ธโˆฃ๐‘†๐‘ก๎€ป.=๐œ‘(3.14) Then if ๐‘“ and ๐†(๐‘”)(๐‘‡โˆ’๐‘ก,๐‘ฆ,๎‚๐œ“๐‘ก) are globally bounded, then so is ๐†(ฮจ)(๐‘ก,๎‚๐œ“๐‘ก).

Proof. We have that ๎€ท๐†(ฮจ)๐‘ก,๎‚๐œ“๐‘ก๎€ธ=lim๐‘ขโ†’0+1๐‘ข๎€บฮจ๎€ท๐‘ก,๎‚๐œ“๐‘ก+๐‘ข๎€ธ๎€ทโˆ’ฮจ๐‘ก,๎‚๐œ“๐‘ก๎€ธ๎€ป=lim๐‘ขโ†’0+1๐‘ข1โˆš๎€œ2๐œ‹โˆžโˆ’โˆž๎€บ๐‘”๎€ท๐‘‡โˆ’๐‘ก,๐‘ฆ,๎‚๐œ“๐‘ก+๐‘ข๎€ธ๎€ทโˆ’๐‘”๐‘‡โˆ’๐‘ก,๐‘ฆ,๎‚๐œ“๐‘ก๐‘’๎€ธ๎€ปโˆ’๐‘ฆ2/2=1๐‘‘๐‘ฆโˆš๎€œ2๐œ‹โˆžโˆ’โˆžlim๐‘ขโ†’0+1๐‘ข๎€บ๐‘”๎€ท๐‘‡โˆ’๐‘ก,๐‘ฆ,๎‚๐œ“๐‘ก+๐‘ข๎€ธ๎€ทโˆ’๐‘”๐‘‡โˆ’๐‘ก,๐‘ฆ,๎‚๐œ“๐‘ก๐‘’๎€ธ๎€ปโˆ’๐‘ฆ2/2โ‰ค1๐‘‘๐‘ฆโˆš๎€œ2๐œ‹โˆžโˆ’โˆž๐‘€๐‘’โˆ’๐‘ฆ2/2๐‘‘๐‘ฆ=๐‘€<โˆž,(3.15) where we used the assumption that ๐‘“ and hence ๐‘” are globally bounded to move the limit inside the integral and ๐†(๐‘”)(๐‘‡โˆ’๐‘ก,๐‘ฆ,๎‚๐œ“๐‘ก)โ‰ค๐‘€<โˆž.

Remark 3.4. Note that since ๐†(ฮจ)(๎‚๐œ“๐‘ ) is bounded for all ๐‘ โˆˆ[0,๐‘‡], โˆซ๐‘ก0๐ทฮจ(๐‘ ,๎‚๐œ“๐‘ )(๐‘‘๎‚๐œ“๐‘ ) exits. Also, if ๐ท2๐‘“ is bounded, โˆซ๐‘ก0๐ท2ฮจ(๐‘ ,๎‚๐œ“๐‘ )(๐‘‘๎‚๐œ“๐‘ ,๐‘‘๎‚๐œ“๐‘ ) exits (see [18]).

4. The Infinite-Dimensional Black-Scholes Equation

It is known (e.g., [4, 5]) that the classical Black-Scholes equation is a deterministic parabolic partial differential equation (with a suitable auxiliary condition) the solution of which gives the value of the European option contract at a given time. Propositions 3.1 through 3.3 allow us to use the Dynkin formula in [16]. With it, a generalized version of the classical Black-Scholes equation can be derived for when the (๐ต,๐‘†)-market model is (2.7) and (2.8). The following theorem is a restatement of Theorem 3.1 in [10].

Theorem 4.1. Let ฮจ(๐‘ก,๐œ‘)=๐‘’โˆ’๐‘Ÿ(๐‘‡โˆ’๐‘ก)๎‚๐„[๐‘“(๐‘†๐‘‡)โˆฃ๐‘†๐‘ก=๐œ‘], where ๐‘†0=๐œ“โˆˆ๐‚+ and ๐‘กโˆˆ[0,๐‘‡]. Let ๐‘“ be a ๐ถ2(๐‚) function with ๐ท๐‘“ and ๐ท2๐‘“ globally Lipschitz and let ฮ›=๐‘“(๐‘†๐‘‡) and ๐‘ฅ=๐‘‹๐œ‹โˆ—(0). Finally, let ๐‘“ and ๐†(๐‘”)(๐‘‡โˆ’๐‘ก,๐‘ฆ,๎‚๐œ“๐‘ก) be globally bounded. Then if ๐‘‹๐œ‹โˆ—(๐‘ก)=ฮจ(๐‘ก,๐‘†๐‘ก) is the wealth process for the minimal (ฮ›,๐‘ฅ)-hedge, one has ๐œ•๐‘Ÿฮจ(๐‘ก,๐œ‘)=๎€ท๐œ•๐‘กฮจ(๐‘ก,๐œ‘)+๐†(ฮจ)๐‘ก,๎‚๐œ‘๐‘ก๎€ธ+๎€ท๐ทฮจ(๐‘ก,๐œ‘)๐‘Ÿ๐œ‘(0)๐Ÿ{0}๎€ธ+12๐ท2๎€ทฮจ(๐‘ก,๐œ‘)๐‘(๐œ‘)๐Ÿ{0},๐‘(๐œ‘)๐Ÿ{0}๎€ธ[,a.s.โˆ€(๐‘ก,๐œ‘)โˆˆ0,๐‘‡)ร—๐‚+,(4.1) where ฮจ(๐‘‡,๐œ‘)=๐‘“(๐œ‘)โˆ€๐œ‘โˆˆ๐‚+,(4.2) and the trading strategy (๐œ‹โˆ—1(๐‘ก),๐œ‹โˆ—2(๐‘ก)) is defined by ๐œ‹โˆ—2(๐‘ก)=๎€ท๐Ÿ๐ทฮจ(๐‘ก,๐œ‘){0}๎€ธ๐œ‹a.s.,โˆ—11(๐‘ก)=๎€บ๐‘‹๐ต(๐‘ก)๐œ‹โˆ—(๐‘ก)โˆ’๐œ‘(0)๐œ‹โˆ—2๎€ป.(๐‘ก)(4.3) Furthermore, if (4.1) and (4.2) hold, then ฮจ(๐‘ก,๐‘†๐‘ก) is the wealth process for the (ฮ›,๐‘ฅ)-hedge with ๐œ‹โˆ—2(๐‘ก)=๐ทฮจ(๐‘ก,๐‘†๐‘ก)(๐Ÿ{0}) and ๐œ‹โˆ—1(๐‘ก)=(1/๐ต(๐‘ก))[๐‘‹๐œ‹โˆ—(๐‘ก)โˆ’๐‘†(๐‘ก)๐œ‹โˆ—2(๐‘ก)].

Proof. The theorem is a restatement of Theorem 3.1 in [10] and is therefore omitted.

Note
Equations (4.1) and (4.2) are the generalized Black-Scholes equation for the (๐ต,๐‘†)-market with hereditary price structure as described by (2.7) and (2.8).

5. Approximation of Solutions

In this section, we will solve the generalized Black-Scholes equation (4.1)-(4.2) by considering a sequence of approximations of its solution. By a (classical) solution to (4.1)-(4.2), we mean ฮจโˆถ[0,๐‘‡]ร—๐‚โ†’โ„œ satisfying the following conditions:

(i)ฮจโˆˆ๐ถ1,2([0,๐‘‡]ร—๐‚),(ii)ฮจ(๐‘‡,๐œ‘)=๐‘“(๐œ‘) for all ๐œ‘โˆˆ๐‚,(iii)ฮจ satisfies (4.1).

The sequence of approximate solutions is constructed by looking at finite-dimensional subspaces of ๐‚, solving (4.1)-(4.2) on these subspaces, and then showing that as the dimension of the subspaces goes to infinity, the finite-dimensional solutions converge to a solution of the original problem. Theorem 5.2, Remark 5.3, and Corollary 5.4 show that the generalized Black-Scholes equation can be solved by solving two simpler equations. The first of these, a first-order partial differential equation, can be handled by traditional techniques once the second equation is solved. Theorem 5.5 provides a solution to the second. Proposition 5.7, which uses Lemma 5.6, gives a generalized Black-Scholes formula for the standard European call option when used in conjunction with Theorem 5.2.

We start by noting that ๐‚โŠ‚๐ฟ2[โˆ’โ„Ž,0] where ๐ฟ2[โˆ’โ„Ž,0] is the space of all square-integrable functions on the interval [โˆ’โ„Ž,0]. Furthermore, ๐‚ is dense in ๐ฟ2[โˆ’โ„Ž,0]. It is well known (e.g., [19]) that even extensions of a function ๐œ‘ in ๐ฟ2[โˆ’โ„Ž,0] may be represented by a cosine Fourier series where โ€–โ€–โ€–โ€–๐œ‘โˆ’๐‘๎“๐‘–=0๐‘Ž๐‘–๎‚€cos2๐œ‹๐‘–โ‹…โ„Ž๎‚โ€–โ€–โ€–โ€–2โŸถ0(5.1) as ๐‘โ†’โˆž where ๐‘Ž0=1โ„Ž๎€œ0โˆ’โ„Ž๐‘Ž๐œ‘(๐œƒ)๐‘‘๐œƒ,๐‘–=2โ„Ž๎€œ0โˆ’โ„Ž๎‚€๐œ‘(๐œƒ)cos2๐œ‹๐‘–๐œƒโ„Ž๎‚๐‘‘๐œƒ,๐‘–=1,2,3,โ€ฆ.(5.2) Here, โ€–๐‘“โ€–22=๎€œ0โˆ’โ„Ž๐‘“2(๐œƒ)๐‘‘๐œƒ(5.3) for ๐‘“โˆˆ๐ฟ2[โˆ’โ„Ž,0]. If ๐œ‘ is Hรถlder-continuous, then the convergence is also point wise (see, e.g., [20]).

Throughout this section, we let ๐ฟ2๐‘›[โˆ’โ„Ž,0] be the subspace of ๐ฟ2[โˆ’โ„Ž,0] consisting of functions that can be represented as a finite Fourier series, that is, ๐œ‘(๐‘›)โˆˆ๐ฟ2๐‘›[โˆ’โ„Ž,0] if ๐œ‘(๐‘›)(๐œƒ)=๐‘›๎“๐‘–=0๐‘Ž๐‘–๎‚€cos2๐œ‹๐‘–๐œƒโ„Ž๎‚(5.4) for all ๐œƒโˆˆ[โˆ’โ„Ž,0].

We will see that it is convenient having a spanning set {๐‘“๐‘–}โˆž๐‘–=0 for ๐ฟ2๐‘›[โˆ’โ„Ž,0] where ๐‘“๐‘–โˆถ[โˆ’โ„Ž,0]โ†’โ„œ for ๐‘–=0,1,โ€ฆ such that ๐‘“๐‘–(0)=1 for all ๐‘– and ๐‘(๐‘“๐‘–โˆซ)=0โˆ’โ„Ž๐‘“๐‘–(๐œƒ)๐‘‘๐œ(๐œƒ)=๐›ฟ for all ๐‘–. Here, โˆซ๐›ฟ=0โˆ’โ„Ž๐‘‘๐œ(๐œƒ)โˆˆโ„œ. Let ๐‘žโˆถ[โˆ’โ„Ž,0]โ†’โ„œ be any function such that ๐‘(๐‘ž)=๐‘ž(0)โ‰ 0. For example, let ๐‘ž(๐œƒ)=1+1โˆ’๐›ฟโˆ’๐‘‘2๐‘‘1๐œƒ+๐œƒ2,(5.5) where ๐‘‘1=โˆซ0โˆ’โ„Ž๐œƒ๐‘‘๐œ(๐œƒ) and ๐‘‘2=โˆซ0โˆ’โ„Ž๐œƒ2๐‘‘๐œ(๐œƒ). To this end, we define the following functions. Let ๐‘“0[],๐‘“(๐œƒ)=1โˆ€๐œƒโˆˆโˆ’โ„Ž,01(๐œƒ)=๐›ผ1,1+๐›ผ1,2๐‘ž[],๐‘“(๐œƒ)โˆ€๐œƒโˆˆโˆ’โ„Ž,02(๐œƒ)=๐›ผ2,1๐‘ž(๐œƒ)+๐›ผ2,2๎‚€cos2๐œ‹๐œƒโ„Ž๎‚[],โˆ€๐œƒโˆˆโˆ’โ„Ž,0(5.6) and for ๐‘–=3,4,โ€ฆ, ๐‘“๐‘–(๐œƒ)=๐›ผ๐‘–,1๎‚ตcos2๐œ‹(๐‘–โˆ’2)๐œƒโ„Ž๎‚ถ+๐›ผ๐‘–,2๎‚ตcos2๐œ‹(๐‘–โˆ’1)๐œƒโ„Ž๎‚ถ[]โˆ€๐œƒโˆˆโˆ’โ„Ž,0.(5.7) Recall that ๐‘โˆถ๐ฟ2[โˆ’โ„Ž,0]โ†’โ„œ is defined by ๎€œ๐‘(๐œ‘)=0โˆ’โ„Ž๐œ‘(๐œƒ)๐‘‘๐œ(๐œƒ),(5.8) and let ๐‘๐‘–๎‚€๎‚€=๐‘cos2๐œ‹๐‘–โ‹…โ„Ž=๎€œ๎‚๎‚0โˆ’โ„Ž๎‚€cos2๐œ‹๐‘–๐œƒโ„Ž๎‚๐‘‘๐œ(๐œƒ).(5.9) Here again ๐‘žโˆถ[โˆ’โ„Ž,0]โ†’โ„œ is any function such that ๐‘(๐‘ž)=๐‘ž(0)โ‰ 0. For example, ๐‘ž can be chosen as in (5.5). In this case, the constant ๐›ผ1,2 is nonzero but otherwise arbitrary, ๐›ผ1,1=1โˆ’๐›ผ1,2๐›ผ๐‘ž(0),2,1=๐›ฟโˆ’๐‘1๎€ท๐‘ž(0)1โˆ’๐‘1๎€ธ,๐›ผ2,2=1โˆ’๐›ผ2,1๐‘ž(0),(5.10) and so on with ๐›ผ๐‘–,2=๐›ฟโˆ’๐‘๐‘–โˆ’2๐‘๐‘–โˆ’1โˆ’๐‘๐‘–โˆ’2,๐›ผ๐‘–,1=1โˆ’๐›ผ๐‘–,2(5.11) for ๐‘–โ‰ฅ3.

Lemma 5.1. The set {๐‘“๐‘–}โˆž๐‘–=0 defined in (5.6) and (5.7) forms a spanning set for ๐ฟ2[โˆ’โ„Ž,0] in the sense that โ€–โ€–โ€–โ€–๐œ‘โˆ’๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๐‘“๐‘–โ€–โ€–โ€–โ€–2โŸถ0(5.12) as ๐‘›โ†’โˆž, where the ๐‘ฅ๐‘– are defined by ๐‘ฅ๐‘›+1=๐‘Ž๐‘›๐›ผ๐‘›+1,2,๐‘ฅ๐‘›=๐‘Ž๐‘›โˆ’1โˆ’๐‘ฅ๐‘›+1๐›ผ๐‘›+1,1๐›ผ๐‘›,2,(5.13) and continuing using ๐‘ฅ๐‘–=๐‘Ž๐‘–โˆ’1โˆ’๐‘ฅ๐‘–+1๐›ผ๐‘–+1,1๐›ผ๐‘–,2(5.14) until ๐‘ฅ1๐‘ฅ=โˆ’2๐›ผ2,1๐›ผ1,2,๐‘ฅ0=๐‘Ž0โˆ’๐‘ฅ1๐›ผ1,1.(5.15) This set of functions has the properties that ๐‘“๐‘–(0)=1 and ๐‘(๐‘“๐‘–)=๐›ฟ for all ๐‘–=0,1,โ€ฆ.

Proof. For any ๐œ‘โˆˆ๐ฟ2[โˆ’โ„Ž,0], we can construct an even extension ๐œ™โˆˆ๐ฟ2[โˆ’โ„Ž,โ„Ž] where ๐œ™(๐œƒ)=๐œ‘(๐œƒ) for all ๐œƒโˆˆ[โˆ’โ„Ž,0] and ๐œ™(๐œƒ)=๐œ‘(โˆ’๐œƒ) for all ๐œƒโˆˆ[0,โ„Ž]. The function ๐œ™ may be represented by a Fourier series of cosine functions ๐œ™(๐œƒ)โˆผ๐‘๎“๐‘–=0๐‘Ž๐‘–๎‚€cos2๐œ‹๐‘–๐œƒโ„Ž๎‚,(5.16) where the โ€œโˆผโ€™โ€™ is used to indicate that โ€–โ€–โ€–โ€–๐œ™โˆ’๐‘๎“๐‘–=0๐‘Ž๐‘–๎‚€cos2๐œ‹๐‘–โ‹…โ„Ž๎‚โ€–โ€–โ€–โ€–2โŸถ0(5.17) as ๐‘โ†’โˆž. In what mentioned before, ๐‘Ž0=1โ„Ž๎€œ0โˆ’โ„Ž๐‘Ž๐œ™(๐œƒ)๐‘‘๐œƒ,๐‘–=2โ„Ž๎€œ0โˆ’โ„Ž๎‚€๐œ™(๐œƒ)cos2๐œ‹๐‘–๐œƒโ„Ž๎‚๐‘‘๐œƒ(5.18) for all ๐‘–=1,2,โ€ฆ. For simplicity, we will replace the โ€œโˆผโ€™โ€™ with an equality sign knowing that mean-square convergence is implied.
For the Fourier series, the basis is ๎‚†๎‚€cos2๐œ‹๐‘–๐œƒโ„Ž๎‚๎‚‡โˆž๐‘–=0,(5.19) so the first term of this basis and {๐‘“๐‘–}โˆž๐‘–=0 are the same, namely, the constant โ€œ1.โ€™โ€™ Clearly ๐‘“0(0)=1 and ๐‘(๐‘“0)=๐›ฟ. The first part of this proof is to show that for all ๐‘–=0,1,โ€ฆ, ๐‘“๐‘–(0)=1 and ๐‘(๐‘“๐‘–)=๐›ฟ.
For ๐‘“1, we have that ๐‘“1(0)=๐›ผ1,1+๐›ผ1,2๐‘ž(0)=1 which implies that ๐›ผ1,1=1โˆ’๐›ผ1,2๐‘ž(0).(5.20) Also, ๐‘(๐‘“1)=๐›ผ1,1๐›ฟ+๐›ผ1,2๐‘(๐‘ž)=๐›ฟ. Since we do not want ๐›ผ1,2=0, we require that ๐‘(๐‘ž)=๐‘ž(0).(5.21) There are no restrictions on ๐›ผ1,2 other than ๐›ผ1,2โ‰ 0.
For ๐‘“2, ๐›ผ2,1๐‘ž(0)+๐›ผ2,2=1 requires that ๐›ผ2,2=1โˆ’๐›ผ2,1๐‘ž(0).(5.22) Since we want ๐›ผ2,1๐‘(๐‘ž)+๐›ผ2,2๐‘1=๐›ฟ, then ๐›ผ2,1=๐›ฟโˆ’๐‘1๐‘(๐‘ž)โˆ’๐‘ž(0)๐‘1=๐›ฟโˆ’๐‘1๎€ท๐‘ž(0)1โˆ’๐‘1๎€ธ.(5.23)
The rest of the ๐‘“๐‘–, that is, where ๐‘–โ‰ฅ3, are handled alike. In order that ๐‘“๐‘–(0)=1, we require that ๐›ผ๐‘–,1=1โˆ’๐›ผ๐‘–,2. To ensure that ๐‘(๐‘“๐‘–)=๐›ฟ, ๐›ผ๐‘–,2=๐›ฟโˆ’๐‘๐‘–โˆ’2๐‘๐‘–โˆ’1โˆ’๐‘๐‘–โˆ’2.(5.24) We have now shown that the sequence of functions {๐‘“๐‘–}โˆž๐‘–=0 is such that ๐‘“๐‘–(0)=1 and ๐‘(๐‘“๐‘–)=๐›ฟ for all ๐‘–=0,1,โ€ฆ. Now it must be shown that this sequence is a spanning set for ๐ฟ2[โˆ’โ„Ž,0]. To do this, we will compare this sequence of functions with the cosine Fourier sequence of functions.
Consider ๐œ‘(๐‘›)โˆถ[โˆ’โ„Ž,0]โ†’โ„œ where ๐œ‘(๐‘›)(๐œƒ)=๐‘›๎“๐‘–=0๐‘Ž๐‘–๎‚€cos2๐œ‹๐‘–๐œƒโ„Ž๎‚.(5.25) We would like ๐œ‘(๐‘›)(๐œƒ)=๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๐‘“๐‘–(๐œƒ)(5.26) for some set {๐‘ฅ๐‘–}๐‘›+1๐‘–=0 of real numbers. By the Fourier expansion, ๐œ‘(๐‘›)(๐œƒ)=๐‘Ž0+๐‘Ž1๎‚€cos2๐œ‹๐œƒโ„Ž๎‚+โ‹ฏ+๐‘Ž๐‘›๎‚€cos2๐œ‹๐‘›๐œƒโ„Ž๎‚.(5.27) We want {๐‘ฅ๐‘–}๐‘›+1๐‘–=0 where ๐œ‘(๐‘›)(๐œƒ)=๐‘ฅ0+๐‘ฅ1๎€ท๐›ผ1,1+๐›ผ1,2๎€ธ๐‘ž(๐œƒ)+๐‘ฅ2๎‚€๐›ผ2,1๐‘ž(๐œƒ)+๐›ผ2,2๎‚€cos2๐œ‹๐œƒโ„Ž๎‚๎‚+๐‘ฅ3๎‚€๐›ผ3,1๎‚€cos2๐œ‹๐œƒโ„Ž๎‚+๐›ผ3,2๎‚€cos4๐œ‹๐œƒโ„Ž๎‚๎‚+โ‹ฏ+๐‘ฅ๐‘›๎‚ต๐›ผ๐‘›,1๎‚ตcos2๐œ‹(๐‘›โˆ’2)๐œƒโ„Ž๎‚ถ+๐›ผ๐‘›,2๎‚ตcos2๐œ‹(๐‘›โˆ’1)๐œƒโ„Ž๎‚ถ๎‚ถ+๐‘ฅ๐‘›+1๎‚ต๐›ผ๐‘›+1,1๎‚ตcos2๐œ‹(๐‘›โˆ’1)๐œƒโ„Ž๎‚ถ+๐›ผ๐‘›+1,2๎‚€cos2๐œ‹๐‘›๐œƒโ„Ž๎‚๎‚ถ=๎€ท๐‘ฅ0+๐‘ฅ1๐›ผ1,1๎€ธ๎€ท๐‘ฅ+๐‘ž(๐œƒ)1๐›ผ1,2+๐‘ฅ2๐›ผ2,1๎€ธ๎‚€+cos2๐œ‹๐œƒโ„Ž๎‚๎€ท๐‘ฅ2๐›ผ2,2+๐‘ฅ3๐›ผ3,1๎€ธ๎‚€+โ‹ฏ+cos2๐œ‹๐‘–๐œƒโ„Ž๎‚๎€ท๐‘ฅ๐‘–+1๐›ผ๐‘–+1,2+๐‘ฅ๐‘–+2๐›ผ๐‘–+2,1๎€ธ๎‚ต+โ‹ฏ+cos2๐œ‹(๐‘›โˆ’1)๐œƒโ„Ž๎‚ถ๎€ท๐‘ฅ๐‘›๐›ผ๐‘›,2+๐‘ฅ๐‘›+1๐›ผ๐‘›+1,1๎€ธ๎‚€+cos2๐œ‹๐‘›๐œƒโ„Ž๎‚๎€ท๐‘ฅ๐‘›+1๐›ผ๐‘›+1,2๎€ธ.(5.28) Equating the last coefficients gives ๐‘ฅ๐‘›+1=๐‘Ž๐‘›๐›ผ๐‘›+1,2,๐‘ฅ๐‘›=๐‘Ž๐‘›โˆ’1โˆ’๐‘ฅ๐‘›+1๐›ผ๐‘›+1,1๐›ผ๐‘›,2.(5.29) Continuing, ๐‘ฅ๐‘–=๐‘Ž๐‘–โˆ’1โˆ’๐‘ฅ๐‘–+1๐›ผ๐‘–+1,1๐›ผ๐‘–,2,(5.30) and finally ๐‘ฅ1๐‘ฅ=โˆ’2๐›ผ2,1๐›ผ1,2,๐‘ฅ0=๐‘Ž0โˆ’๐‘ฅ1๐›ผ1,1.(5.31) Hence, with the above choice of {๐‘ฅ๐‘–}๐‘›+1๐‘–=0, ๐‘›๎“๐‘–=0๐‘Ž๐‘–๎‚€cos2๐œ‹๐‘–๐œƒโ„Ž๎‚=๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๐‘“๐‘–(๐œƒ),(5.32) and so โ€–โ€–โ€–โ€–๐œ‘โˆ’๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๐‘“๐‘–โ€–โ€–โ€–โ€–2=โ€–โ€–โ€–โ€–๐œ‘โˆ’๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๐‘“๐‘–+๐‘›๎“๐‘–=0๐‘Ž๐‘–๎‚€cos2๐œ‹๐‘–โ‹…โ„Ž๎‚โˆ’๐‘›๎“๐‘–=0๐‘Ž๐‘–๎‚€cos2๐œ‹๐‘–โ‹…โ„Ž๎‚โ€–โ€–โ€–โ€–2=โ€–โ€–โ€–โ€–๐œ‘โˆ’๐‘›๎“๐‘–=0๐‘Ž๐‘–๎‚€cos2๐œ‹๐‘–โ‹…โ„Ž๎‚+๐‘›๎“๐‘–=0๐‘Ž๐‘–๎‚€cos2๐œ‹๐‘–โ‹…โ„Ž๎‚โˆ’๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๐‘“๐‘–โ€–โ€–โ€–โ€–2โ‰คโ€–โ€–โ€–โ€–๐œ‘โˆ’๐‘›๎“๐‘–=0๐‘Ž๐‘–๎‚€cos2๐œ‹๐‘–โ‹…โ„Ž๎‚โ€–โ€–โ€–โ€–2+โ€–โ€–โ€–โ€–๐‘›๎“๐‘–=0๐‘Ž๐‘–๎‚€cos2๐œ‹๐‘–โ‹…โ„Ž๎‚โˆ’๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๐‘“๐‘–โ€–โ€–โ€–โ€–2=โ€–โ€–โ€–โ€–๐œ‘โˆ’๐‘›๎“๐‘–=0๐‘Ž๐‘–๎‚€cos2๐œ‹๐‘–โ‹…โ„Ž๎‚โ€–โ€–โ€–โ€–2โŸถ0(5.33) as ๐‘›โ†’โˆž.

To find an approximate solution to the generalized Black-Scholes equation we start by letting ๐‘‹๐œ‹โˆ—๎‚(๐‘ก)=๐„[๐‘“(๐‘†๐‘‡)โˆฃ๐‘†๐‘ก=๐œ‘] (from [6]) and approximating ๐œ‘ by ๐œ‘(๐‘›)=๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๐‘“๐‘–.(5.34) We define the space ๐‚๐‘› as the set of all continuous functions that can be represented by this summation for some {๐‘ฅ๐‘–}โˆž๐‘–=0. Note that ๐‚๐‘›โŠ‚๐ฟ2๐‘›[โˆ’โ„Ž,0]. Also define ๐‘’๐‘›โˆถโ„œ๐‘›+2โ†’โ„œ by ๐‘’๐‘›๎€ท๎€ธ=โƒ—๐‘ฅ๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๐‘“๐‘–,(5.35) so that ฮจ(๐‘ก,๐œ‘(๐‘›))=ฮจ(๐‘ก,๐‘’๐‘›(โƒ—๐‘ฅ)). Define ฮจ๐‘›โˆถ[0,๐‘‡]ร—โ„œ๐‘›+2โ†’โ„œ by ฮจ๐‘›(๐‘ก,โƒ—๐‘ฅ)=ฮจ(๐‘ก,๐œ‘(๐‘›)) provided that the โƒ—๐‘ฅ is formed by the coefficients of ๐œ‘(๐‘›) in the spanning set {๐‘“๐‘–}โˆž๐‘–=0. In general, โƒ—๐‘ฅ(๐‘ก) is formed by the coefficients of ๐œ‘๐‘ก(๐‘›) in the spanning set {๐‘“๐‘–}โˆž๐‘–=0. Also, define ๐‘ฃ๐‘›โˆถ[โˆ’โ„Ž,0]โ†’โ„œ by ๐‘ฃ๐‘›โŽงโŽชโŽจโŽชโŽฉ๎‚ƒ1(๐œƒ)=0,for๐œƒโˆˆโˆ’โ„Ž,โˆ’๐‘›๎‚,๎‚ƒโˆ’1๐‘›๐œƒ+1,for๐œƒ=๐‘›๎‚„.,0(5.36) Last, let ๐‘”๐‘›โˆถ[0,๐‘‡]ร—๐‚๐‘›ร—โ„œ๐‘›+1ร—๐ถ1,2([0,๐‘‡]ร—๐‚)โ†’โ„œ be defined by ๐‘”๐‘›๎€ท๐‘ก,๐œ‘(๐‘›)๎€ธ๎ƒฉ,โƒ—๐‘ฅ,ฮจ=๐‘Ÿ๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๎ƒช๎ƒฌ๐ทฮจ(๐‘ก,๐œ‘(๐‘›))๎€ท๐Ÿ{0}๎€ธโˆ’๐‘›+1๎“๐‘–=0๐‘˜๐‘–๐œ•๐œ•๐‘ฅ๐‘–ฮจ๐‘›๎€ท๎€ธ๎ƒญ+๐›ฟ๐‘ก,โƒ—๐‘ฅ22๎ƒฉ๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๎ƒช2๎ƒฌ๐ท2ฮจ(๐‘ก,๐œ‘(๐‘›))๎€ท๐Ÿ{0},๐Ÿ{0}๎€ธโˆ’๐‘›+1๎“๐‘–,๐‘—=0๐‘˜๐‘–๐‘˜๐‘—๐œ•2๐œ•๐‘ฅ๐‘–๐œ•๐‘ฅ๐‘—ฮจ๐‘›๎€ท๎€ธ๎ƒญ,๐‘ก,โƒ—๐‘ฅ(5.37) where the ๐‘˜๐‘– are the coefficients of ๐‘ฃ๐‘› using the spanning set {๐‘“๐‘–}โˆž๐‘–=0. Finally, define the operator (โ‹…)๐‘›โˆถ๐‚โ†’๐‚๐‘› by (๐œ‘)๐‘›=๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๐‘“๐‘–,(5.38) where the right-hand side is the first ๐‘›+2 terms of the {๐‘“๐‘–}-expansion of ๐œ‘.

We are now ready for a theorem which enables us to approximate the solution of the infinite-dimensional Black-Scholes equation by solving a first-order real-valued partial differential equation and an equation similar to the generalized Black-Scholes equation but without the ๐†(ฮจ)(๐‘ก,๎‚๐œ‘๐‘ก) term. The lack of this term allows approximate solutions to be found using traditional techniques.

Theorem 5.2. Let ๐‘†0=๐œ“โˆˆ๐‚+ and ๐‘กโˆˆ[0,๐‘‡]. Let ๐‘“ be a ๐ถ2(๐‚) function satisfying the conditions of Theorem 4.1 and let ฮ›=๐‘“(๐‘†๐‘‡). Then ๎€ท๐‘Ÿฮจ๐‘ก,๐œ‘(๐‘›)๎€ธ=๐œ•ฮจ๎€ท๐œ•๐‘ก๐‘ก,๐œ‘(๐‘›)๎€ธ๎€ท๎€ท+๐†(ฮจ)๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ+๎€ท๐ทฮจ๐‘ก,๐œ‘(๐‘›)๎€ธ๎€ท๐‘Ÿ๐œ‘(๐‘›)(0)๐Ÿ{0}๎€ธ+12๐ท2ฮจ๎€ท๐‘ก,๐œ‘(๐‘›)๐‘๎€ท๐œ‘๎€ธ๎€ท(๐‘›)๎€ธ๐Ÿ{0}๎€ท๐œ‘,๐‘(๐‘›)๎€ธ๐Ÿ{0}๎€ธ๎€ท,โˆ€๐‘ก,๐œ‘(๐‘›)๎€ธโˆˆ[0,๐‘‡)ร—๐‚๐‘›,(5.39) where ฮจ๎€ท๐‘‡,๐œ‘(๐‘›)๎€ธ๎€ท๐œ‘=๐‘“(๐‘›)๎€ธโˆ€๐œ‘(๐‘›)โˆˆ๐‚๐‘›(5.40) has a solution of the form ๐‘‰๎€ท๎€ท๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๐น๎€ท๐‘ก,๐œ‘(๐‘›)๎€ธ.(5.41) Here, ๐‘‰(๐‘ก,(๎‚๐œ‘๐‘ก)๐‘›)=๐‘ค๐‘›(๐‘ก,0) is a solution to ๐น๎€ท๐‘ก,๐œ‘(๐‘›)๎€ธ๐œ•๐‘ค๐‘›๎€ท๎€ท๐œ•๐‘ก(๐‘ก,๐‘ข)+๐น๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๐œ•๐‘ค๐‘›๎€ท๎€ท๐œ•๐‘ข(๐‘ก,๐‘ข)(5.42)+๐†(๐น)๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๐‘ค๐‘›(๐‘ก,๐‘ข)=0(5.43) for ๐‘กโˆˆ[0,๐‘‡] and ๐‘ขโˆˆ[0,๐œ–) for some ๐œ–>0 and ๐‘ค๐‘›(๐‘‡,0)=1, and ๐นโˆถโ„œ+ร—๐‚๐‘›โ†’โ„œ is a solution of ๎€ท๐‘Ÿ๐น๐‘ก,๐œ‘(๐‘›)๎€ธ=๐œ•๐น๎€ท๐œ•๐‘ก๐‘ก,๐œ‘(๐‘›)๎€ธ+๎€ท๐ท๐น๐‘ก,๐œ‘(๐‘›)๎€ธ๎€ท๐‘Ÿ๐œ‘(๐‘›)(0)๐Ÿ{0}๎€ธ+12๐ท2๐น๎€ท๐‘ก,๐œ‘(๐‘›)๐‘๎€ท๐œ‘๎€ธ๎€ท(๐‘›)๎€ธ๐Ÿ{0}๎€ท๐œ‘,๐‘(๐‘›)๎€ธ๐Ÿ{0}๎€ธโˆ€๎€ท๐‘ก,๐œ‘(๐‘›)๎€ธโˆˆ[0,๐‘‡)ร—๐‚๐‘›,(5.44) where ๐น๎€ท๐‘‡,๐œ‘(๐‘›)๎€ธ๎€ท๐œ‘=๐‘“(๐‘›)๎€ธโˆ€๐œ‘(๐‘›)โˆˆ๐‚๐‘›,(5.45) and ๐‘“ is a uniformly bounded ๐ถ2(๐‚) function satisfying the conditions of Theorem 4.1.

Proof. We assume a solution of the form ฮจ(๐‘ก,๐œ‘(๐‘›))=๐‘‰(๐‘ก,(๎‚๐œ‘๐‘ก)๐‘›)๐น(๐‘ก,๐œ‘(๐‘›)), then๎€ท๎€ท๐‘Ÿ๐‘‰๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๐น๎€ท๐‘ก,๐œ‘(๐‘›)๎€ธ=๐œ•๎€ท๐‘‰๎€ท๎€ท๐œ•๐‘ก๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๐น๎€ท๐‘ก,๐œ‘(๐‘›)๎€ท๎€ท๎€ธ๎€ธ+๐†(๐‘‰๐น)๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ+๎€ท๐ท(๐‘‰๐น)๐‘ก,๐œ‘(๐‘›)๎€ธ๎€ท๐‘Ÿ๐œ‘(๐‘›)(0)๐Ÿ{0}๎€ธ+12๐ท2๎€ท(๐‘‰๐น)๐‘ก,๐œ‘(๐‘›)๐‘๎€ท๐œ‘๎€ธ๎€ท(๐‘›)๎€ธ๐Ÿ{0}๎€ท๐œ‘,๐‘(๐‘›)๎€ธ๐Ÿ{0}๎€ธ๎€ท=๐น๐‘ก,๐œ‘(๐‘›)๎€ธ๐œ•๐‘‰๎€ท๎€ท๐œ•๐‘ก๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๎€ท๎€ท+๐‘‰๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๐œ•๐น๎€ท๐œ•๐‘ก๐‘ก,๐œ‘(๐‘›)๎€ธ๎€ท+๐น๐‘ก,๐œ‘(๐‘›)๎€ธ๎€ท๎€ท๐†(๐‘‰)๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๎€ท๎€ท+๐‘‰๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๎€ท๐†(๐น)๐‘ก,๐œ‘(๐‘›)๎€ธ๎€ท๎€ท+๐‘‰๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธร—๎‚†๎€ท๐ท(๐น)๐‘ก,๐œ‘(๐‘›)๎€ธ๎€ท๐‘Ÿ๐œ‘(๐‘›)(0)๐Ÿ{0}๎€ธ+12๐ท2๎€ท(๐น)๐‘ก,๐œ‘(๐‘›)๐‘๎€ท๐œ‘๎€ธ๎€ท(๐‘›)๎€ธ๐Ÿ{0}๎€ท๐œ‘,๐‘(๐‘›)๎€ธ๐Ÿ{0}๎€ธ๎‚‡๎€ท,โˆ€๐‘ก,๐œ‘(๐‘›)๎€ธโˆˆ[0,๐‘‡)ร—๐‚๐‘›.(5.46) If ๐น(๐‘ก,๐œ‘(๐‘›)) is the solution to (5.44), then ๐น๎€ท๐‘ก,๐œ‘(๐‘›)๎€ธ๐œ•๐‘‰๎€ท๎€ท๐œ•๐‘ก๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๎€ท๎€ท+๐น๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๎€ท๎€ท๐†(๐‘‰)๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๎€ท๎€ท+๐†(๐น)๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๐‘‰๎€ท๎€ท๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ=0.(5.47) Define ๐‘‡๐‘ขโˆถ๐‚โ†’๐‚ by ๐‘‡๐‘ข(๐œ‘)=๐œ‘๐‘ข, that is, ๐‘‡๐‘ข is a shift operator. Now let ๐‘‰(๐‘ก,(๎‚๐œ‘๐‘ก+๐‘ข)๐‘›)=๐‘‰(๐‘ก,(๐‘‡๐‘ข(๎‚๐œ‘๐‘ก))๐‘›)=๐‘ค๐‘›(๐‘ก,๐‘ข) for a fixed ๐œ‘โˆˆ๐‚. Then ๎€ท๎€ท๐†(๐‘‰)๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ=๐œ•๐‘ค๐‘›๎€ท๐œ•๐‘ข๐‘ก,0+๎€ธ,(5.48) where the superscript + denotes a right-hand derivative with respect to ๐‘ข. Then ๐น๎€ท๐‘ก,๐œ‘(๐‘›)๎€ธ๐œ•๐‘ค๐‘›๎€ท๎€ท๐œ•๐‘ก(๐‘ก,0)+๐น๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๐œ•๐‘ค๐‘›๎€ท๐œ•๐‘ข๐‘ก,0+๎€ธ๎€ท๎€ท+๐†(๐น)๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๐‘ค๐‘›(๐‘ก,0)=0.(5.49)
A slightly more restrictive, but more familiar form is ๐น๎€ท๐‘ก,๐œ‘(๐‘›)๎€ธ๐œ•๐‘ค๐‘›๎€ท๎€ท๐œ•๐‘ก(๐‘ก,๐‘ข)+๐น๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๐œ•๐‘ค๐‘›๎€ท๎€ท๐œ•๐‘ข(๐‘ก,๐‘ข)+๐†(๐น)๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘›๎€ธ๐‘ค๐‘›(๐‘ก,๐‘ข)=0,(5.50) where ๐‘กโˆˆ[0,โ„Ž] and ๐‘ขโˆˆ[0,๐œ–) for some ๐œ–>0. There is the additional requirement that ๐‘ค๐‘›(๐‘‡,0)=1 so that (5.44) holds.

Remark 5.3. It can be easily shown that ๐‘†๐‘ก is ๐›ผ-Hรถlder continuous a.s. for 0<๐›ผ<1/2 provided that ๐‘†0 is ๐›ผ-Hรถlder continuous for the same ๐›ผ. Therefore, ||๐น๐‘›๎€ท๎€ธ||๐‘ก,โƒ—๐‘ฅโˆ’๐น(๐‘ก,๐œ‘)โŸถ0(5.51) for each ๐‘ก as ๐‘›โ†’โˆž where ๐น(๐‘ก,๐œ‘) is a solution to (5.44) and ๐น๐‘›(๐‘ก,โƒ—๐‘ฅ)=๐น(๐‘ก,๐œ‘(๐‘›)) is an approximate solution, since ๐น is ๐ถ2(๐‚) in its second variable and ๐น๐‘›๎€ท๎€ธ๎€ท๐‘ก,โƒ—๐‘ฅ=๐น๐‘ก,๐‘’๐‘›๎€ท๎€ทโƒ—๐‘ฅ๎€ธ๎€ธ=๐น๐‘ก,๐œ‘(๐‘›)๎€ธ.(5.52)

The proof of the following corollary is identical to that of Theorem 5.2, with the use of Remark 5.3 to obtain ฮจ(๐‘ก,๐œ‘).

Corollary 5.4. If ๐‘†0 is Hรถlder continuous, then ๐œ•๐‘Ÿฮจ(๐‘ก,๐œ‘)=๎€ท๐œ•๐‘กฮจ(๐‘ก,๐œ‘)+๐†(ฮจ)๐‘ก,๎‚๐œ‘๐‘ก๎€ธ+๎€ท๐ทฮจ(๐‘ก,๐œ‘)๐‘Ÿ๐œ‘(0)๐Ÿ{0}๎€ธ+12๐ท2๎€ทฮจ(๐‘ก,๐œ‘)๐‘(๐œ‘)๐Ÿ{0},๐‘(๐œ‘)๐Ÿ{0}๎€ธ[,โˆ€(๐‘ก,๐œ‘)โˆˆ0,๐‘‡)ร—๐‚+,(5.53) where ฮจ(๐‘‡,๐œ‘)=๐‘“(๐œ‘)โˆ€๐œ‘โˆˆ๐‚+(5.54) has a solution of the form ๐‘‰(๐‘ก,๎‚๐œ‘๐‘ก)๐น(๐‘ก,๐œ‘). Here, ๐‘‰(๐‘ก,๎‚๐œ‘๐‘ก)=๐‘ค(๐‘ก,0) is a solution to ๐น(๐‘ก,๐œ‘)๐œ•๐‘ค๎€ท๐œ•๐‘ก(๐‘ก,๐‘ข)+๐น๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐œ•๐‘ค๎€ท๐œ•๐‘ข(๐‘ก,๐‘ข)+๐†(๐น)๐‘ก,๎‚๐œ‘๐‘ก๎€ธ๐‘ค(๐‘ก,๐‘ข)=0(5.55) for ๐‘กโˆˆ[0,๐‘‡] and ๐‘ขโˆˆ[0,๐œ–) for some ๐œ–>0. ๐น(๐‘ก,๐œ‘) is the solution to (5.44) where one lets ๐‘›โ†’โˆž. In addition, ๐‘ค(๐‘‡,0)=1.

Now we must solve (5.44), which is done in the following theorem. With this solution, the first-order partial differential equation can be solved by traditional means.

Theorem 5.5. Let ๎€ท๐‘Ÿฮจ๐‘ก,๐œ‘(๐‘›)๎€ธ=๐œ•ฮจ๎€ท๐œ•๐‘ก๐‘ก,๐œ‘(๐‘›)๎€ธ+๎€ท๐ทฮจ๐‘ก,๐œ‘(๐‘›)๎€ธ๎€ท๐‘Ÿ๐œ‘(๐‘›)(0)๐Ÿ{0}๎€ธ+12๐ท2ฮจ๎€ท๐‘ก,๐œ‘(๐‘›)๐‘๎€ท๐œ‘๎€ธ๎€ท(๐‘›)๎€ธ๐Ÿ{0}๎€ท๐œ‘,๐‘(๐‘›)๎€ธ๐Ÿ{0}๎€ธโˆ€๎€ท๐‘ก,๐œ‘(๐‘›)๎€ธโˆˆ[0,๐‘‡)ร—๐‚๐‘›,(5.56) where ฮจ๎€ท๐‘‡,๐œ‘(๐‘›)๎€ธ๎€ท๐œ‘=๐‘“(๐‘›)๎€ธโˆ€๐œ‘(๐‘›)โˆˆ๐‚๐‘›,(5.57) and ๐‘“ is a uniformly bounded ๐ถ2(๐‚) function satisfying the conditions of Theorem 4.1. Let ๐‘“๐‘›โˆถโ„œ๐‘›+2โ†’โ„œ be defined by ๐‘“๐‘›=๐‘“โˆ˜๐‘’๐‘›, then ฮจ๐‘›๎€ท๎€ธ=๐‘’๐‘ก,โƒ—๐‘ฅโˆ’๐‘Ÿ(๐‘‡โˆ’๐‘ก)โˆš๎€œ2๐œ‹โˆžโˆ’โˆž๐‘“๐‘›๎‚ต๐›ฟexp๎‚ธ๎‚ต๐‘Ÿ๐ตโˆ’22๐ต2๎‚ถโˆš(๐‘‡โˆ’๐‘ก)+๐›ฟ๐ต๐‘ฆ๎‚น๎‚ถ๐‘’๐‘‡โˆ’๐‘กโƒ—๐‘ฅโˆ’๐‘ฆ2/2+๎€œ๐‘‘๐‘ฆ๐‘‡๐‘ก๐‘”๐‘›๎‚€๐‘ ,๐œ‘๐‘ (๐‘›)๎‚๐‘’,โƒ—๐‘ฅ(๐‘ ),ฮจโˆ’๐‘Ÿ(๐‘ โˆ’๐‘ก)๐‘‘๐‘ .(5.58) Here, โŽกโŽขโŽขโŽขโŽฃ๐‘˜๐ต=0๐‘˜0โ‹ฏ๐‘˜0๐‘˜1๐‘˜1โ‹ฏ๐‘˜1๐‘˜โ‹ฎโ‹ฎโ‹ฎโ‹ฎ๐‘›+1๐‘˜๐‘›+1โ‹ฏ๐‘˜๐‘›+1โŽคโŽฅโŽฅโŽฅโŽฆ,๐‘ฃ๐‘›=๐‘›+1๎“๐‘–=0๐‘˜๐‘–๐‘“๐‘–(5.59) from (5.36).

Proof. Since ฮจ๐‘›โˆถ[0,๐‘‡]ร—โ„œ๐‘›+2โ†’โ„œ, the definition of the Frรฉchet derivatives and the properties of the set {๐‘“๐‘–}โˆž๐‘–=0 give ๐‘Ÿฮจ๐‘›๎€ท๎€ธ=๐œ•๐‘ก,โƒ—๐‘ฅฮจ๐œ•๐‘ก๐‘›๎€ท๎€ธ๎ƒฉ๐‘ก,โƒ—๐‘ฅ+๐‘Ÿ๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๎ƒช๐‘›+1๎“๐‘–=0๐‘˜๐‘–๐œ•๐œ•๐‘ฅ๐‘–ฮจ๐‘›๎€ท๎€ธ+๐›ฟ๐‘ก,โƒ—๐‘ฅ22๎ƒฉ๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๎ƒช2๐‘›+1๎“๐‘–,๐‘—=0๐‘˜๐‘–๐‘˜๐‘—๐œ•2๐œ•๐‘ฅ๐‘–๐œ•๐‘ฅ๐‘—ฮจ๐‘›๎€ท๎€ธ๐‘ก,โƒ—๐‘ฅ+๐‘”๐‘›๎‚€๐‘ก,๐œ‘๐‘ (๐‘›)๎‚๎€ท๎€ธโˆˆ[,โƒ—๐‘ฅ,ฮจ,โˆ€๐‘ก,โƒ—๐‘ฅ0,๐‘‡)ร—โ„œ๐‘›+2,ฮจ๎€ท๎€ธ๐‘‡,โƒ—๐‘ฅ(๐‘‡)=๐‘“๐‘›๎€ท๎€ธ.โƒ—๐‘ฅ(๐‘‡)(5.60) The โƒ—๐‘ฅ(๐‘‡) consists of the first ๐‘›+2 coefficients of ๐‘†๐‘‡ in the set of functions {๐‘“๐‘–}. By the Feynman-Kac theorem (see [15, Theoremโ€‰โ€‰5.7.6]), ฮจ๐‘›๎€ท๎€ธ๐‘ก,โƒ—๐‘ฅ=๐‘’โˆ’๐‘Ÿ(๐‘‡โˆ’๐‘ก)๎‚๐„๎€บ๐‘“๎€ท๎€ธ๎€ป+๎€œโƒ—๐‘ฅ(๐‘‡)โˆฃโƒ—๐‘ฅ(๐‘ก)=โƒ—๐‘ฅ๐‘‡๐‘ก๐‘”๐‘›๎‚€๐‘ ,๐œ‘๐‘ (๐‘›)๎‚๐‘’,โƒ—๐‘ฅ(๐‘ ),ฮจโˆ’๐‘Ÿ(๐‘ โˆ’๐‘ก)๐‘‘๐‘ ,(5.61) where โƒ—๐‘ฅ(๐‘ก) is the solution to ๐‘‘๐‘ฅ๐‘–(๐‘ก)=๐‘๐‘–๎€ท๎€ธ๐‘ก,โƒ—๐‘ฅ(๐‘ก)๐‘‘๐‘ก+๐‘›+1๎“๐‘—=0๐œŽ๐‘–๐‘—๎€ท๎€ธ๐‘‘๎‚‹๐‘Š๐‘ก,โƒ—๐‘ฅ(๐‘ก)(๐‘—)(๐‘ก)(5.62) for ๐‘–=0,1,โ€ฆ,๐‘›+1. Noting that โƒ—๐‘ฅ(๐‘ก)=โƒ—๐‘ฅ, ๐‘๐‘–๎€ท๎€ธ๐‘ก,โƒ—๐‘ฅ=๐‘Ÿ๐‘˜๐‘–๎ƒฉ๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๎ƒช,๐›ฟ2๐‘˜๐‘–๐‘˜๐‘—๎ƒฉ๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๎ƒช2=๐‘›+1๎“๐‘˜=0๐œŽ๐‘–๐‘˜๎€ท๎€ธ๐œŽ๐‘ก,โƒ—๐‘ฅ๐‘—๐‘˜๎€ท๎€ธ๐‘ก,โƒ—๐‘ฅ(5.63) with 0โ‰ค๐‘–,๐‘—โ‰ค๐‘›+1. Hence, โƒ—๐‘๎€ท๎€ธโŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ๐‘˜๐‘ก,โƒ—๐‘ฅ=๐‘Ÿ0๎ƒฉ๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๎ƒช๐‘˜1๎ƒฉ๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๎ƒชโ‹ฎ๐‘˜๐‘›+1๎ƒฉ๐‘›+1๎“๐‘–=0๐‘ฅ๐‘–๎ƒชโŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ,(5.64) and so โƒ—๐‘๎€ท