International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 782572 |

Mou-Hsiung Chang, Roger K. Youree, "Spectral Approximation of Infinite-Dimensional Black-Scholes Equations with Memory", International Journal of Stochastic Analysis, vol. 2009, Article ID 782572, 37 pages, 2009.

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

Academic Editor: Kambiz Farahmand
Received02 Sep 2009
Accepted02 Dec 2009
Published14 Jan 2010


This paper considers the pricing of a European option using a (𝐡,𝑆)-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 𝐢[βˆ’β„Ž,0] for the Black-Scholes equation. It is also shown that the 𝑛th 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., [1–5]), the equations that describe the bank account and the price of the stock are typically written, respectively, as𝑑𝐡(𝑑)=π‘Ÿπ΅(𝑑)𝑑𝑑,𝐡(0)=π‘₯,𝑑𝑆(𝑑)=𝛼𝑆(𝑑)𝑑𝑑+πœŽπ‘†(𝑑)π‘‘π‘Š(𝑑),𝑆(0)=𝑦,(1.1) where π‘Š={π‘Š(𝑑),𝑑β‰₯0} is a one-dimensional standard Brownian motion defined on a complete filtered probability space (Ξ©,𝐹,𝐏;{𝐹(𝑑),𝑑β‰₯0}) and π‘Ÿ, 𝛼, and 𝜎 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 (𝐡,𝑆)-market, where 𝐡 stands for the bank account and 𝑆 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 (𝐡,𝑆)-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 (𝐡,𝑆)-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., [7–14]).

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 (𝐡,𝑆)-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 [12–14] for a financial market that is more restricted than those of [10, 11].

This paper considers the pricing of a European option using a (𝐡,𝑆)-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 𝐢[βˆ’β„Ž,0] for the Black-Scholes equation. It is also shown that the 𝑛th 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 FrΓ©chet derivative and extended FrΓ©chet 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<β„Ž<∞ be a fixed constant. This constant will be the length of the time window in which the hereditary information is contained. If π‘Ž,π‘βˆˆβ„œ with π‘Ž<𝑏, denote the space of continuous functions πœ™βˆΆ[π‘Ž,𝑏]β†’β„œ by 𝐢[π‘Ž,𝑏]. Define 𝐢+[][][]π‘Ž,𝑏={πœ™βˆˆπΆπ‘Ž,π‘βˆ£πœ™(πœƒ)β‰₯0βˆ€πœƒβˆˆπ‘Ž,𝑏}.(2.1) Note that 𝐢[π‘Ž,𝑏] is a real separable Banach space equipped with the uniform topology defined by the sup-norm β€–πœ™β€–=supπ‘‘βˆˆ[π‘Ž,𝑏]|πœ™(𝑑)| and 𝐢+[π‘Ž,𝑏] is a closed subset of 𝐢[π‘Ž,𝑏]. Throughout the end of this paper, we let 𝐂=𝐢[βˆ’β„Ž,0] and 𝐂+[]}={πœ™βˆˆπ‚βˆ£πœ™(πœƒ)β‰₯0βˆ€πœƒβˆˆβˆ’β„Ž,0(2.2) for simplicity. If πœ“βˆˆπΆ[βˆ’β„Ž,∞) and π‘‘βˆˆ[0,∞), let πœ“π‘‘βˆˆπ‚ be defined by πœ“π‘‘(πœƒ)=πœ“(𝑑+πœƒ), πœƒβˆˆ[βˆ’β„Ž,0].

Let Ξ©=𝐢[βˆ’β„Ž,∞), the space of continuous functions πœ”βˆΆ[βˆ’β„Ž,∞)β†’β„œ, and let 𝐹=𝐡(𝐢[βˆ’β„Ž,∞)), the Borel 𝜎-algebra of subsets of 𝐢[βˆ’β„Ž,∞) under the topology defined by the metric π‘‘βˆΆΞ©Γ—Ξ©β†’β„œ, whereπ‘‘ξ€·πœ”,πœ”ξ…žξ€Έ=βˆžξ“π‘›=112𝑛supβˆ’β„Žβ‰€π‘‘β‰€π‘›||πœ”(𝑑)βˆ’πœ”ξ…ž||(𝑑)1+supβˆ’β„Žβ‰€π‘‘β‰€π‘›||πœ”(𝑑)βˆ’πœ”ξ…ž||.(𝑑)(2.3)

Let 𝐏 be the Wiener measure defined on (Ξ©,𝐹) with []𝐏{πœ”βˆˆΞ©βˆ£πœ”(πœƒ)=0βˆ€πœƒβˆˆβˆ’β„Ž,0}=1.(2.4) Note that the probability space (Ξ©,𝐹,𝐏) is the canonical Wiener space under which the coordinate maps π‘Š={π‘Š(𝑑),𝑑β‰₯0}, π‘Š(𝑑)∢𝐢[βˆ’β„Ž,∞)β†’β„œ, defined by π‘Š(𝑑)(πœ”)=πœ”(𝑑) for all 𝑑β‰₯βˆ’β„Ž and πœ”βˆˆΞ© is a standard Brownian motion and 𝐏{π‘Š0=0}=1. Let the filtration πΉπ‘Š={𝐹(𝑑),𝑑β‰₯βˆ’β„Ž} be the 𝐏-augmentation of the natural filtration of the Brownian motion π‘Š, defined by 𝐹(𝑑)={βˆ…,Ξ©} for all π‘‘βˆˆ[βˆ’β„Ž,0] and 𝐹(𝑑)=𝜎(π‘Š(𝑠),0≀𝑠≀𝑑),𝑑β‰₯0.(2.5)

Equivalently, 𝐹(𝑑) is the smallest sub-𝜎-algebra of subsets of Ξ© with respect to which the mappings π‘Š(𝑠)βˆΆΞ©β†’β„œ are measurable for all 0≀𝑠≀𝑑. It is clear that the filtration πΉπ‘Š defined above is right continuous in the sense of [15].

Consider the 𝐂-valued process {π‘Šπ‘‘,𝑑β‰₯0}, where π‘Š0=0 and π‘Šπ‘‘(πœƒ)=π‘Š(𝑑+πœƒ), πœƒβˆˆ[βˆ’β„Ž,0] for all 𝑑β‰₯0. That is, for each 𝑑β‰₯0, π‘Šπ‘‘(πœ”)=πœ”π‘‘ and π‘Š0=0. In [10], it is shown that 𝐹0=𝐹(𝑑) for π‘‘βˆˆ[βˆ’β„Ž,0] and 𝐹(𝑑)=𝐹𝑑 for 𝑑β‰₯0, where πΉπ‘‘ξ€·π‘Š=πœŽπ‘ ξ€Έ,0≀𝑠≀𝑑,𝑑β‰₯0.(2.6)

The new model for the (𝐡,𝑆)-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 𝐡 and that of the stock account 𝑆 depend not only on the current unit price but also on their historical prices. Specifically, we assume that 𝐡 and 𝑆 evolve according to the following two linear functional differential equations: 𝐡𝑑𝐡(𝑑)=𝐿𝑑𝑆𝑑𝑑,𝑑β‰₯0,(2.7)𝑑𝑆(𝑑)=𝑀𝑑𝑆𝑑𝑑+π‘π‘‘ξ€Έπ‘‘π‘Š(𝑑),𝑑β‰₯0,(2.8) with initial price functions 𝐡0=πœ™ and 𝑆0=πœ“, where πœ™ and πœ“ are given functions in 𝐂+. In the model, 𝐿, 𝑀, and 𝑁 are bounded linear functionals on the real Banach space 𝐂. The bounded linear functionals 𝐿,𝑀,π‘βˆΆπ‚β†’β„œ can be represented as (see [6]) ξ€œπΏ(πœ™)=0βˆ’β„Žξ€œπœ™(πœƒ)π‘‘πœ‚(πœƒ),𝑀(πœ™)=0βˆ’β„Žπ‘ξ€œπœ™(πœƒ)π‘‘πœ‰(πœƒ),(2.9)(πœ™)=0βˆ’β„Žπœ™(πœƒ)π‘‘πœ(πœƒ),πœ™βˆˆπ‚,(2.10) where the above integrals are to be interpreted as Lebesgue-Stieltjes integrals and πœ‚, πœ‰, and 𝜁 are functions that are assumed to satisfy the following conditions.

Assumption 2.1. The functions πœ‚,πœ‰βˆΆ[βˆ’β„Ž,0]β†’β„œ, are nondecreasing functions on [βˆ’β„Ž,0] such that πœ‚(0)βˆ’πœ‚(βˆ’β„Ž)>0 and πœ‰(0)βˆ’πœ‰(βˆ’β„Ž)>0, and the function 𝜁∢[βˆ’β„Ž,0]β†’β„œ is a function of bounded variation on [βˆ’β„Ž,0] such that ∫0βˆ’β„Žπœ™(πœƒ)π‘‘πœ(πœƒ)β‰₯𝜎>0 for every πœ™βˆˆπ‚+.

We will, throughout the end, extend the domain of the above three functions to 𝑅 by defining πœ‚(πœƒ)=πœ‚(βˆ’β„Ž) for πœƒβ‰€βˆ’β„Ž and πœ‚(πœƒ)=πœ‚(0) for πœƒβ‰₯0, 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 𝐡(𝐿;πœ™)={𝐡(𝑑),βˆ’β„Žβ‰€π‘‘<∞} of (2.7) with the initial function πœ™βˆˆπ‚+ takes the following form: 𝐡(𝑑)=πœ™(0)π‘’π‘Ÿπ‘‘,𝑑β‰₯0,(2.11) and 𝐡0=πœ™βˆˆπ‚+. Then the constant π‘Ÿ satisfies the following equation: ξ€œπ‘Ÿ=0βˆ’β„Žπ‘’π‘Ÿπœƒπ‘‘πœ‚(πœƒ).(2.12) The existence and uniqueness of a positive number π‘Ÿ that satisfies the above equation is shown in [6].

Throughout the end, we will fix the initial unit price functions πœ™, and πœ“βˆˆπ‚+, and the functional π‘βˆΆπ‚β†’β„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 π‘€βˆΆπ‚β†’β„œ and πΏβˆΆπ‚β†’β„œ in (2.8), we write the stock price process 𝑆(𝑀,𝑁;πœ“) as 𝑆(𝑀)={𝑆(𝑑),𝑑β‰₯βˆ’β„Ž} for simplicity. And, when the functional πΎβˆΆπ‚β†’β„œ, 𝐾(πœ™π‘‘)=π‘Ÿπœ™(𝑑) is used in place of π‘€βˆΆπ‚β†’β„œ in (2.8), its solution process will be written as 𝑆(𝐾)={𝑆(𝑑),𝑑β‰₯βˆ’β„Ž}.

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

A trading strategy in the (𝐡,𝑆)-market is a progressively measurable vector process πœ‹={(πœ‹1(𝑑),πœ‹2(𝑑)),0≀𝑑<∞} defined on (Ξ©,𝐹,𝐏;πΉπ‘Š) such that for each π‘Ž>0, ξ€œπ‘Ž0π„ξ€Ίπœ‹2𝑖(𝑑)𝑑𝑑<∞,𝑖=1,2,(2.13) where πœ‹1(𝑑) and πœ‹2(𝑑) represent, respectively, the number of units of the bank account and the number of shares of the stock owned by the writer at time 𝑑β‰₯0, and 𝐄 is the expectation with respect to 𝐏.

The writer's total asset is described by the wealth process π‘‹πœ‹(𝑀)={π‘‹πœ‹(𝑑),0≀𝑑<∞} defined by π‘‹πœ‹(𝑑)=πœ‹1(𝑑)𝐡(𝑑)+πœ‹2(𝑑)𝑆(𝑑),0≀𝑑<∞,(2.14) where again 𝐡(𝐿;πœ™) and 𝑆(𝑀,𝑁;π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 (πœ‹1(𝑑),πœ‹2(𝑑))βˆˆβ„œ2.

We will make the following basic assumption throughout this paper.

Assumption 2.2 (self-financing condition). In the (𝐡,𝑆)-market, it is assumed that all trading strategies πœ‹ satisfy the following self-financing condition: π‘‹πœ‹(𝑑)=π‘‹πœ‹(ξ€œ0)+𝑑0πœ‹1(ξ€œπ‘ )𝑑𝐡(𝑠)+𝑑0πœ‹2(𝑠)𝑑𝑆(𝑠),0≀𝑑<∞,a.s.(2.15) or equivalently, π‘‘π‘‹πœ‹(𝑑)=πœ‹1(𝑑)𝑑𝐡(𝑑)+πœ‹2(𝑑)𝑑𝑆(𝑑),0≀𝑑<∞.(2.16)

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

For the unit price of the bank account 𝐡(𝐿;πœ™)={𝐡(𝑑),𝑑β‰₯0} and the stock 𝑆(𝑀,𝑁;πœ“)={𝑆(𝑑),𝑑β‰₯0} described in (2.7) and (2.8), define ξ‚‹ξ€œπ‘Š(𝑑)=π‘Š(𝑑)+𝑑0𝛾𝐡𝑠,𝑆𝑠𝑑𝑠,𝑑β‰₯0,(2.17) where π›ΎβˆΆπ‚+×𝐂+β†’β„œ is defined by 𝛾(πœ™,πœ“)=πœ™(0)𝑀(πœ“)βˆ’πœ“(0)𝐿(πœ™).πœ™(0)𝑁(πœ“)(2.18)

Define the process 𝑍(𝐿,𝑀,𝑁;πœ™,πœ“)={𝑍(𝑑),𝑑β‰₯0} by ξ‚»ξ€œπ‘(𝑑)=exp𝑑0𝛾𝐡𝑠,𝑆𝑠1π‘‘π‘Š(𝑠)βˆ’2ξ€œπ‘‘0||𝛾𝐡𝑠,𝑆𝑠||2𝑑𝑠,𝑑β‰₯0.(2.19) The following results are proven in [6].

Lemma 2.3. The process 𝑍(𝐿,𝑀,𝑁;πœ™,πœ“)={𝑍(𝑑),𝑑β‰₯0} defined by (2.19) is a martingale defined on (Ξ©,𝐹,𝐏;πΉπ‘Š).

Lemma 2.4. There exists a unique probability measure 𝐏 defined on the canonical measurable space (Ξ©,𝐹) such that ξ‚ξ€ΊπŸπ(𝐴)=𝐄𝐴𝑍(𝑇)βˆ€π΄βˆˆπΉπ‘‡,0<𝑇<∞,(2.20) where 𝟏𝐴 is the indicator function of π΄βˆˆπΉπ‘‡.

Lemma 2.5. The process ξ‚‹π‘Š defined by (2.17) is a standard Brownian motion defined on the filtered probability space (Ξ©,𝐹,𝐏;πΉπ‘Š).

From the above, it has been shown (see [6, equation (14)]) that 𝑆𝑑𝑆(𝑑)=π‘Ÿπ‘†(𝑑)𝑑𝑑+π‘π‘‘ξ€Έπ‘‘ξ‚‹π‘Š(𝑑),(2.21) with 𝑆0=πœ“βˆˆπ‚+. It is also clear that the probabilistic behavior of 𝑆(𝑀) under the probability measure 𝐏 is the same as that of 𝑆(𝐾) under the probability measure 𝐏; that is, they have the same distribution.

Define the process π‘Œπœ‹(𝐿,𝑀,𝑁;πœ™,πœ“)={π‘Œπœ‹(𝑑),𝑑β‰₯0}, called the discounted wealth process, by π‘Œπœ‹π‘‹(𝑑)=πœ‹(𝑑)𝐡(𝑑),𝑑β‰₯0.(2.22) We say that a trading strategy πœ‹ from SF(𝐿,𝑀,𝑁;πœ™,πœ“) belongs to a subclass SFπœβŠ‚SF if 𝐏 a.s. π‘Œπœ‹ξ‚π„ξ€Ί(𝑑)β‰₯βˆ’πœβˆ£πΉπ‘‘ξ€»,𝑑β‰₯0,(2.23) where 𝐄 is the expectation with respect to 𝐏, 𝜍 is a nonnegative 𝐹-measurable random variable such that 𝐄[𝜍]<∞. We say that πœ‹ belongs to SF+βŠ‚SF if 𝜍β‰₯0.

In [6, 10], it is shown that π‘Œπœ‹ is a local martingale; for πœ‹βˆˆSF𝜍, π‘Œπœ‹ is a supermartingale, and is a nonnegative supermartingale if πœ‹βˆˆSF+.

Throughout, we assume the reward function Ξ› is an 𝐹𝑇-measurable nonnegative random variable satisfying the following condition: 𝐄Λ1+πœ–ξ€»<∞,(2.24) for some πœ–>0. Here, 𝑇>0 is the expiration time. (Note that the above condition on Ξ› implies that 𝐄[Ξ›]<∞.)

Let Ξ› be a nonnegative 𝐹𝑇-measurable random variable satisfying (2.24). A trading strategy πœ‹βˆˆSF is a (𝑀;Ξ›,π‘₯)-hedge of European type if π‘‹πœ‹(0)=πœ‹1(0)πœ™(0)+πœ‹2(0)πœ“(0)=π‘₯(2.25) and 𝐏 a.s. π‘‹πœ‹(𝑇)β‰₯Ξ›.(2.26) We say that a (𝑀;Ξ›,π‘₯)-hedge trading strategy πœ‹βˆ—βˆˆSF(𝑀) is minimal if π‘‹πœ‹(𝑇)β‰₯π‘‹πœ‹βˆ—(𝑇)(2.27) for any (𝑀;Ξ›,π‘₯)-hedge strategy πœ‹βˆˆSF(𝑀).

Let Ξ (𝑀;Ξ›,π‘₯) be the set of (𝑀;Ξ›,π‘₯)-hedge strategies from SF+(𝑀). Define 𝐢(𝑀;Ξ›)=inf{π‘₯β‰₯0∢Π(𝑀;Ξ›,π‘₯)β‰ βˆ…}.(2.28) The value 𝐢(𝑀;Ξ›) defined above is called the rational price of the contingent claim of European type. If the infimum in (2.28) is achieved, then 𝐢(𝑀;Ξ›) is the minimal possible initial capital for which there exists a trading strategy πœ‹βˆˆSF+(𝑀) possessing the property that 𝐏 a.s. π‘‹πœ‹(𝑇)β‰₯Ξ›.

Let π‘Œ(𝑀)={π‘Œ(𝑑),0≀𝑑≀𝑇} be defined by ξ‚π„ξ‚ΈΞ›π‘Œ(𝑑)=βˆ£ξ‚πΉπ΅(𝑇)𝑑,0≀𝑑≀𝑇,(2.29) where ξ‚πΉπ‘‘ξ‚‹π‘Š=𝜎(𝑠,0≀𝑠≀𝑑). In [10], it is shown that the process π‘Œ(𝑀) is a martingale defined on ξ‚ξ‚‹π‘Š)(Ξ©,𝐹,𝐏;𝐹 and can be represented by ξ€œπ‘Œ(𝑑)=π‘Œ(0)+𝑑0𝛽(𝑠)π‘‘π‘Š(𝑠),(2.30) where 𝛽={𝛽(𝑑),0≀𝑑≀𝑇} that is πΉξ‚‹π‘Š-adapted and βˆ«π‘‡0𝛽2(𝑑)𝑑𝑑<∞ (𝐏 a.s.).

The following lemma and theorem provide the main results of [6, 10]. Let πœ‹βˆ—={(πœ‹βˆ—1(𝑑),πœ‹βˆ—2(𝑑)),0≀𝑑≀𝑇} be a trading strategy, where πœ‹βˆ—2(𝑑)=𝛽(𝑑)𝐡(𝑑)𝑁𝑆𝑑,πœ‹βˆ—1𝑆(𝑑)=π‘Œ(𝑑)βˆ’(𝑑)πœ‹π΅(𝑑)βˆ—2[].(𝑑),π‘‘βˆˆ0,𝑇(2.31)

Lemma 2.6. πœ‹βˆ—βˆˆSF(𝑀) and for each π‘‘βˆˆ[0,𝑇], π‘Œ(𝑑)=π‘Œπœ‹βˆ—(𝑑) for each π‘‘βˆˆ[0,𝑇] where again π‘Œπœ‹βˆ— is the process defined in (2.22) with the minimal strategy πœ‹βˆ— defined in (2.31).

Theorem 2.7. Let Ξ› be an 𝐹𝑇-measurable random variable defined on the filtered probability space (Ξ©,𝐹,𝐏;πΉπ‘Š) that satisfies (2.24). Then the rational price 𝐢(𝑀;Ξ›) defined in (2.28) is given by 𝐄𝑒𝐢(𝑀;Ξ›)=βˆ’π‘Ÿπ‘‡Ξ›ξ€»,(2.32) where π‘Ÿ is the positive constant that satisfies (2.12). Furthermore, there exists a minimal hedge πœ‹βˆ—={(πœ‹βˆ—1(𝑑),πœ‹βˆ—2(𝑑)),0≀𝑑≀𝑇}, where πœ‹βˆ—2(𝑑)=𝛽(𝑑)𝐡(𝑑)𝑁𝑆𝑑,πœ‹βˆ—1(𝑑)=π‘Œπœ‹βˆ—(𝑑)βˆ’πœ‹βˆ—2𝑆(𝑑)(𝑑),𝐡(𝑑)(2.33) and the process 𝛽={𝛽(𝑑),0≀𝑑≀𝑇} is given by (2.30).
If in addition, the reward Ξ› is intrinsic, that is, Ξ›=Ξ“(𝑆(𝑀)) for some measurable function Ξ“βˆΆπ‚+β†’β„œ, then the rational price 𝐢(𝑀;Ξ›) does not depend on the mean growth rate 𝑀 of the stock and 𝐄𝑒𝐢(Ξ›)=βˆ’π‘Ÿπ‘‡Ξ›ξ€».(2.34)

3. FrΓ©chet and Extended FrΓ©chet 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 𝐹𝑇-measurable reward function Ξ› has the explicit expression Ξ›=𝑓(𝑆𝑇), where again 𝑆𝑇(πœƒ)=𝑆(𝑇+πœƒ), πœƒβˆˆ[βˆ’β„Ž,0] and 𝑆(𝐾)={𝑆(𝑑),𝑑β‰₯0} is the unit price of the stock described by the following equation: 𝑆𝑑𝑆(𝑑)=π‘Ÿπ‘†(𝑑)𝑑𝑑+π‘π‘‘ξ€Έπ‘‘ξ‚‹π‘Š(𝑑),𝑑β‰₯0,(3.1) where 𝑆0=πœ“βˆˆπ‚+. Throughout this section, we assume that 𝑆(𝑑), and therefore 𝑁(𝑆𝑑), are uniformly bounded almost surely. This assumption is realistic for the price of a stock during time interval [0,𝑇] in a financial system with finite total wealth.

The remaining sections make extensive use of FrΓ©chet derivatives. Let π‚βˆ— be the space of bounded linear functionals Ξ¦βˆΆπ‚β†’β„œ. π‚βˆ— is a real separable Banach space under the supremum operator norm β€–Ξ¦β€–=supπœ™β‰ 0||||Ξ¦(πœ™).β€–πœ™β€–(3.2) For Ψ∢[0,𝑇]Γ—π‚β†’β„œ, we denote the FrΓ©chet derivative of Ξ¨ at πœ™βˆˆπ‚ by 𝐷Ψ(𝑑,πœ™). The second FrΓ©chet derivative at πœ™ is denoted as 𝐷2Ξ¨(𝑑,πœ™).

Let Ξ“ be the vector space of all simple functions of the form π‘£πŸ{0}, where π‘£βˆˆπ‘… and 𝟏{0}∢[βˆ’β„Ž,0]β†’β„œ is defined by 𝟏{0}ξƒ―[(πœƒ)=0,forπœƒβˆˆβˆ’β„Ž,0),1,forπœƒ=0.(3.3) Form the direct sum π‚βŠ•Ξ“ and equip it with the complete norm β€–β€–πœ™+π‘£πŸ{0}β€–β€–=supπœƒβˆˆ[βˆ’β„Ž,0]||||πœ™(πœƒ)+|𝑣|,πœ™βˆˆπ‚,π‘£βˆˆβ„œ.(3.4) Then 𝐷Ψ(𝑑,πœ™) has a unique continuous linear extension from π‚βŠ•Ξ“ to β„œ which we will denote by 𝐷Ψ(𝑑,πœ™), and similarly for 𝐷2Ξ¨(𝑑,πœ™); see [16] or [17] for more details.

Finally, we define 𝐆(Ξ¨)𝑑,ξ‚πœ“π‘‘ξ€Έ=lim𝑒→0+1𝑒Ψ𝑑,ξ‚πœ“π‘‘+π‘’ξ€Έξ€·βˆ’Ξ¨π‘‘,ξ‚πœ“π‘‘ξ€Έξ€»(3.5) for all π‘‘βˆˆ[0,∞) and πœ“βˆˆπ‚+, where ξ‚πœ“βˆΆ[βˆ’β„Ž,∞)β†’β„œ is defined by ξƒ―[ξ‚πœ“(𝑑)=πœ“(𝑑)ifπ‘‘βˆˆβˆ’β„Ž,0)πœ“(0)if𝑑β‰₯0.(3.6) Let π‘“βˆΆπ‚β†’β„œ. We say that π‘“βˆˆπΆ1(𝐂) if 𝑓 has a continuous FrΓ©chet derivative. Similarly, π‘“βˆˆπΆπ‘›(𝐂) if 𝑓 has a continuous 𝑛th FrΓ©chet derivative. For π‘“βˆΆπ‘…+Γ—π‚β†’β„œ, we say that π‘“βˆˆπΆβˆž,𝑛([0,∞)×𝐂) if 𝑓 is infinitely differentiable in its first variable and has a continuous 𝑛th partial derivative in its second variable.

Proposition 3.1. Let πœ‘βˆˆπ‚ and π‘“βˆΆπ‚β†’β„œ with π‘“βˆˆπΆ2(𝐂). Define Ψ∢[0,𝑇]Γ—π‚β†’β„œ by Ξ¨(𝑑,πœ‘)=π‘’βˆ’π‘Ÿ(π‘‡βˆ’π‘‘)ξ‚π„ξ€Ίπ‘“ξ€·π‘†π‘‡ξ€Έβˆ£π‘†π‘‘ξ€».=πœ‘(3.7) Then Ψ∈𝐢∞,2([0,∞)×𝐂).

Proof. That π‘’βˆ’π‘Ÿ(π‘‡βˆ’π‘‘) is 𝐢∞[0,∞) is clear, so we have only to show that Ξ₯∈𝐢2(𝐂), where Ξ₯(πœ‘)=𝐄[𝑓(𝑆𝑇)βˆ£π‘†π‘‘=πœ‘] given that π‘“βˆˆπΆ2(𝐂).
We have that 𝑆𝑑𝑆(𝑑)=π‘Ÿπ‘†(𝑑)𝑑𝑑+π‘π‘‘ξ€Έπ‘‘ξ‚‹π‘Š(𝑑),𝑑β‰₯0,(3.8) with 𝑆0=πœ“βˆˆπ‚+. Under Assumption 2.1 on π‘βˆΆπ‚β†’β„œ and the properties of Ξ₯, it can be shown that there exists π»βˆΆβ„œΓ—β„œΓ—π‚β†’π‚ such that 𝑆𝑑=𝐻(𝑑,π‘Š(𝑑),πœ“). Therefore, 𝐄𝑓𝑆Ξ₯(πœ‘)=π‘‡ξ€Έβˆ£π‘†π‘‘ξ€»=1=πœ‘βˆšξ€œ2πœ‹βˆžβˆ’βˆžπ‘“(𝐻(π‘‡βˆ’π‘‘,𝑦,πœ‘))π‘’βˆ’π‘¦2/2𝑑𝑦.(3.9) By Theorem 3.2, Chapter 2 of [16], 𝐻(𝑑,𝑦,β‹…)∈𝐢1(𝐂) as a function of πœ“. By a second application of the same theorem (since π‘“βˆˆπΆ2(𝐂)), we have that 𝐻(𝑑,𝑦,β‹…)∈𝐢2(𝐂) as a function of πœ“. Define π‘”βˆΆβ„œΓ—β„œΓ—π‚β†’β„œ by 𝑔=π‘“βˆ˜π». Since π‘“βˆˆπΆ2(𝐂) and 𝐻(𝑑,𝑦,β‹…)∈𝐢2(𝐂) in its third variable, 𝑔(𝑑,𝑦,β‹…)∈𝐢2(𝐂). Hence, for πœ‘,πœ™βˆˆπ‚, ξ‚π„ξ€Ίπ‘“ξ€·π‘†π‘‡ξ€Έβˆ£π‘†π‘‘ξ€»βˆ’ξ‚π„ξ€Ίπ‘“ξ€·π‘†=πœ‘+πœ™π‘‡ξ€Έβˆ£π‘†π‘‘ξ€»=1=πœ‘βˆšξ€œ2πœ‹βˆžβˆ’βˆž[]𝑒𝑓(𝐻(π‘‡βˆ’π‘‘,𝑦,πœ‘+πœ™))βˆ’π‘“(𝐻(π‘‡βˆ’π‘‘,𝑦,πœ‘))βˆ’π‘¦2/2=1π‘‘π‘¦βˆšξ€œ2πœ‹βˆžβˆ’βˆžπ·π‘”(π‘‡βˆ’π‘‘,𝑦,πœ‘)(πœ™)π‘’βˆ’π‘¦2/21𝑑𝑦+βˆšξ€œ2πœ‹βˆžβˆ’βˆžπ‘œ(πœ™)π‘’βˆ’π‘¦2/2𝑑𝑦,(3.10) where π‘œ(πœ™) is a function mapping continuous functions into the reals such that π‘œ(πœ™)β€–πœ™β€–βŸΆ0asβ€–πœ™β€–βŸΆ0.(3.11) The last integral is clearly π‘œ(πœ™) and 1βˆšξ€œ2πœ‹βˆžβˆ’βˆžπ·π‘”(π‘‡βˆ’π‘‘,𝑦,πœ‘)(πœ™)π‘’βˆ’π‘¦2/2𝑑𝑦(3.12) is bounded and linear in πœ™, so this integral is the first FrΓ©chet derivative with respect to πœ‘. Since 𝑔(𝑑,𝑦,β‹…)∈𝐢2(𝐂), the process can be repeated, giving a second FrΓ©chet derivative with respect to πœ‘ and so Ξ₯∈𝐢2(𝐂).

Proposition 3.2. Let πœ‘βˆˆπ‚ and π‘“βˆΆπ‚β†’β„œ. Further assume π‘“βˆˆπΆ2(𝐂) and let Ψ∢[0,𝑇]Γ—π‚β†’β„œ be defined by Ξ¨(𝑑,πœ‘)=π‘’βˆ’π‘Ÿ(π‘‡βˆ’π‘‘)ξ‚πΈξ€Ίπ‘“ξ€·π‘†π‘‡ξ€Έβˆ£π‘†π‘‘ξ€».=πœ‘(3.13) Then if 𝐷𝑓 and 𝐷2𝑓 are globally Lipschitz, then so is 𝐷2Ξ¨.

Recall from Proposition 3.1 that π‘”βˆΆπ‘…Γ—π‘…Γ—π‚β†’π‘… is π‘“βˆ˜π» where 𝑆𝑑=𝐻(𝑑,π‘Š(𝑑),πœ“) with 𝑆0=πœ“βˆˆπ‚+.

Proposition 3.3. Let πœ‘βˆˆπ‚ and π‘“βˆΆπ‚β†’β„œ. Further assume π‘“βˆˆπΆ2(𝐂) and let Ψ∢[0,𝑇]Γ—π‚β†’β„œ be defined by Ξ¨(𝑑,πœ‘)=π‘’βˆ’π‘Ÿ(π‘‡βˆ’π‘‘)ξ‚πΈξ€Ίπ‘“ξ€·π‘†π‘‡ξ€Έβˆ£π‘†π‘‘ξ€».=πœ‘(3.14) Then if 𝑓 and 𝐆(𝑔)(π‘‡βˆ’π‘‘,𝑦,ξ‚πœ“π‘‘) are globally bounded, then so is 𝐆(Ξ¨)(𝑑,ξ‚πœ“π‘‘).

Proof. We have that 𝐆(Ξ¨)𝑑,ξ‚πœ“π‘‘ξ€Έ=lim𝑒→0+1𝑒Ψ𝑑,ξ‚πœ“π‘‘+π‘’ξ€Έξ€·βˆ’Ξ¨π‘‘,ξ‚πœ“π‘‘ξ€Έξ€»=lim𝑒→0+1𝑒1βˆšξ€œ2πœ‹βˆžβˆ’βˆžξ€Ίπ‘”ξ€·π‘‡βˆ’π‘‘,𝑦,ξ‚πœ“π‘‘+π‘’ξ€Έξ€·βˆ’π‘”π‘‡βˆ’π‘‘,𝑦,ξ‚πœ“π‘‘π‘’ξ€Έξ€»βˆ’π‘¦2/2=1π‘‘π‘¦βˆšξ€œ2πœ‹βˆžβˆ’βˆžlim𝑒→0+1π‘’ξ€Ίπ‘”ξ€·π‘‡βˆ’π‘‘,𝑦,ξ‚πœ“π‘‘+π‘’ξ€Έξ€·βˆ’π‘”π‘‡βˆ’π‘‘,𝑦,ξ‚πœ“π‘‘π‘’ξ€Έξ€»βˆ’π‘¦2/2≀1π‘‘π‘¦βˆšξ€œ2πœ‹βˆžβˆ’βˆžπ‘€π‘’βˆ’π‘¦2/2𝑑𝑦=𝑀<∞,(3.15) where we used the assumption that 𝑓 and hence 𝑔 are globally bounded to move the limit inside the integral and 𝐆(𝑔)(π‘‡βˆ’π‘‘,𝑦,ξ‚πœ“π‘‘)≀𝑀<∞.

Remark 3.4. Note that since 𝐆(Ξ¨)(ξ‚πœ“π‘ ) is bounded for all π‘ βˆˆ[0,𝑇], βˆ«π‘‘0𝐷Ψ(𝑠,ξ‚πœ“π‘ )(π‘‘ξ‚πœ“π‘ ) exits. Also, if 𝐷2𝑓 is bounded, βˆ«π‘‘0𝐷2Ξ¨(𝑠,ξ‚πœ“π‘ )(π‘‘ξ‚πœ“π‘ ,π‘‘ξ‚πœ“π‘ ) 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 (𝐡,𝑆)-market model is (2.7) and (2.8). The following theorem is a restatement of Theorem 3.1 in [10].

Theorem 4.1. Let Ξ¨(𝑑,πœ‘)=π‘’βˆ’π‘Ÿ(π‘‡βˆ’π‘‘)𝐄[𝑓(𝑆𝑇)βˆ£π‘†π‘‘=πœ‘], where 𝑆0=πœ“βˆˆπ‚+ and π‘‘βˆˆ[0,𝑇]. Let 𝑓 be a 𝐢2(𝐂) function with 𝐷𝑓 and 𝐷2𝑓 globally Lipschitz and let Ξ›=𝑓(𝑆𝑇) and π‘₯=π‘‹πœ‹βˆ—(0). Finally, let 𝑓 and 𝐆(𝑔)(π‘‡βˆ’π‘‘,𝑦,ξ‚πœ“π‘‘) be globally bounded. Then if π‘‹πœ‹βˆ—(𝑑)=Ξ¨(𝑑,𝑆𝑑) is the wealth process for the minimal (Ξ›,π‘₯)-hedge, one has πœ•π‘ŸΞ¨(𝑑,πœ‘)=ξ€·πœ•π‘‘Ξ¨(𝑑,πœ‘)+𝐆(Ξ¨)𝑑,ξ‚πœ‘π‘‘ξ€Έ+𝐷Ψ(𝑑,πœ‘)π‘Ÿπœ‘(0)𝟏{0}ξ€Έ+12𝐷2ξ€·Ξ¨(𝑑,πœ‘)𝑁(πœ‘)𝟏{0},𝑁(πœ‘)𝟏{0}ξ€Έ[,a.s.βˆ€(𝑑,πœ‘)∈0,𝑇)×𝐂+,(4.1) where Ξ¨(𝑇,πœ‘)=𝑓(πœ‘)βˆ€πœ‘βˆˆπ‚+,(4.2) and the trading strategy (πœ‹βˆ—1(𝑑),πœ‹βˆ—2(𝑑)) is defined by πœ‹βˆ—2(𝑑)=ξ€·πŸπ·Ξ¨(𝑑,πœ‘){0}ξ€Έπœ‹a.s.,βˆ—11(𝑑)=𝑋𝐡(𝑑)πœ‹βˆ—(𝑑)βˆ’πœ‘(0)πœ‹βˆ—2ξ€».(𝑑)(4.3) Furthermore, if (4.1) and (4.2) hold, then Ξ¨(𝑑,𝑆𝑑) is the wealth process for the (Ξ›,π‘₯)-hedge with πœ‹βˆ—2(𝑑)=𝐷Ψ(𝑑,𝑆𝑑)(𝟏{0}) and πœ‹βˆ—1(𝑑)=(1/𝐡(𝑑))[π‘‹πœ‹βˆ—(𝑑)βˆ’π‘†(𝑑)πœ‹βˆ—2(𝑑)].

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

Equations (4.1) and (4.2) are the generalized Black-Scholes equation for the (𝐡,𝑆)-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 Ψ∢[0,𝑇]Γ—π‚β†’β„œ satisfying the following conditions:

(i)Ψ∈𝐢1,2([0,𝑇]×𝐂),(ii)Ξ¨(𝑇,πœ‘)=𝑓(πœ‘) for all πœ‘βˆˆπ‚,(iii)Ξ¨ satisfies (4.1).

The sequence of approximate solutions is constructed by looking at finite-dimensional subspaces of 𝐂, 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 π‚βŠ‚πΏ2[βˆ’β„Ž,0] where 𝐿2[βˆ’β„Ž,0] is the space of all square-integrable functions on the interval [βˆ’β„Ž,0]. Furthermore, 𝐂 is dense in 𝐿2[βˆ’β„Ž,0]. It is well known (e.g., [19]) that even extensions of a function πœ‘ in 𝐿2[βˆ’β„Ž,0] may be represented by a cosine Fourier series where β€–β€–β€–β€–πœ‘βˆ’π‘ξ“π‘–=0π‘Žπ‘–ξ‚€cos2πœ‹π‘–β‹…β„Žξ‚β€–β€–β€–β€–2⟢0(5.1) as π‘β†’βˆž where π‘Ž0=1β„Žξ€œ0βˆ’β„Žπ‘Žπœ‘(πœƒ)π‘‘πœƒ,𝑖=2β„Žξ€œ0βˆ’β„Žξ‚€πœ‘(πœƒ)cos2πœ‹π‘–πœƒβ„Žξ‚π‘‘πœƒ,𝑖=1,2,3,….(5.2) Here, ‖𝑓‖22=ξ€œ0βˆ’β„Žπ‘“2(πœƒ)π‘‘πœƒ(5.3) for π‘“βˆˆπΏ2[βˆ’β„Ž,0]. If πœ‘ is HΓΆlder-continuous, then the convergence is also point wise (see, e.g., [20]).

Throughout this section, we let 𝐿2𝑛[βˆ’β„Ž,0] be the subspace of 𝐿2[βˆ’β„Ž,0] consisting of functions that can be represented as a finite Fourier series, that is, πœ‘(𝑛)∈𝐿2𝑛[βˆ’β„Ž,0] if πœ‘(𝑛)(πœƒ)=𝑛𝑖=0π‘Žπ‘–ξ‚€cos2πœ‹π‘–πœƒβ„Žξ‚(5.4) for all πœƒβˆˆ[βˆ’β„Ž,0].

We will see that it is convenient having a spanning set {𝑓𝑖}βˆžπ‘–=0 for 𝐿2𝑛[βˆ’β„Ž,0] where π‘“π‘–βˆΆ[βˆ’β„Ž,0]β†’β„œ for 𝑖=0,1,… such that 𝑓𝑖(0)=1 for all 𝑖 and 𝑁(π‘“π‘–βˆ«)=0βˆ’β„Žπ‘“π‘–(πœƒ)π‘‘πœ(πœƒ)=𝛿 for all 𝑖. Here, βˆ«π›Ώ=0βˆ’β„Žπ‘‘πœ(πœƒ)βˆˆβ„œ. Let π‘žβˆΆ[βˆ’β„Ž,0]β†’β„œ be any function such that 𝑁(π‘ž)=π‘ž(0)β‰ 0. For example, let π‘ž(πœƒ)=1+1βˆ’π›Ώβˆ’π‘‘2𝑑1πœƒ+πœƒ2,(5.5) where 𝑑1=∫0βˆ’β„Žπœƒπ‘‘πœ(πœƒ) and 𝑑2=∫0βˆ’β„Žπœƒ2π‘‘πœ(πœƒ). To this end, we define the following functions. Let 𝑓0[],𝑓(πœƒ)=1βˆ€πœƒβˆˆβˆ’β„Ž,01(πœƒ)=𝛼1,1+𝛼1,2π‘ž[],𝑓(πœƒ)βˆ€πœƒβˆˆβˆ’β„Ž,02(πœƒ)=𝛼2,1π‘ž(πœƒ)+𝛼2,2ξ‚€cos2πœ‹πœƒβ„Žξ‚[],βˆ€πœƒβˆˆβˆ’β„Ž,0(5.6) and for 𝑖=3,4,…, 𝑓𝑖(πœƒ)=𝛼𝑖,1ξ‚΅cos2πœ‹(π‘–βˆ’2)πœƒβ„Žξ‚Ά+𝛼𝑖,2ξ‚΅cos2πœ‹(π‘–βˆ’1)πœƒβ„Žξ‚Ά[]βˆ€πœƒβˆˆβˆ’β„Ž,0.(5.7) Recall that π‘βˆΆπΏ2[βˆ’β„Ž,0]β†’β„œ is defined by ξ€œπ‘(πœ‘)=0βˆ’β„Žπœ‘(πœƒ)π‘‘πœ(πœƒ),(5.8) and let 𝑐𝑖=𝑁cos2πœ‹π‘–β‹…β„Ž=ξ€œξ‚ξ‚0βˆ’β„Žξ‚€cos2πœ‹π‘–πœƒβ„Žξ‚π‘‘πœ(πœƒ).(5.9) Here again π‘žβˆΆ[βˆ’β„Ž,0]β†’β„œ is any function such that 𝑁(π‘ž)=π‘ž(0)β‰ 0. For example, π‘ž can be chosen as in (5.5). In this case, the constant 𝛼1,2 is nonzero but otherwise arbitrary, 𝛼1,1=1βˆ’π›Ό1,2π›Όπ‘ž(0),2,1=π›Ώβˆ’π‘1ξ€·π‘ž(0)1βˆ’π‘1ξ€Έ,𝛼2,2=1βˆ’π›Ό2,1π‘ž(0),(5.10) and so on with 𝛼𝑖,2=π›Ώβˆ’π‘π‘–βˆ’2π‘π‘–βˆ’1βˆ’π‘π‘–βˆ’2,𝛼𝑖,1=1βˆ’π›Όπ‘–,2(5.11) for 𝑖β‰₯3.

Lemma 5.1. The set {𝑓𝑖}βˆžπ‘–=0 defined in (5.6) and (5.7) forms a spanning set for 𝐿2[βˆ’β„Ž,0] in the sense that β€–β€–β€–β€–πœ‘βˆ’π‘›+1𝑖=0π‘₯𝑖𝑓𝑖‖‖‖‖2⟢0(5.12) as π‘›β†’βˆž, where the π‘₯𝑖 are defined by π‘₯𝑛+1=π‘Žπ‘›π›Όπ‘›+1,2,π‘₯𝑛=π‘Žπ‘›βˆ’1βˆ’π‘₯𝑛+1𝛼𝑛+1,1𝛼𝑛,2,(5.13) and continuing using π‘₯𝑖=π‘Žπ‘–βˆ’1βˆ’π‘₯𝑖+1𝛼𝑖+1,1𝛼𝑖,2(5.14) until π‘₯1π‘₯=βˆ’2𝛼2,1𝛼1,2,π‘₯0=π‘Ž0βˆ’π‘₯1𝛼1,1.(5.15) This set of functions has the properties that 𝑓𝑖(0)=1 and 𝑁(𝑓𝑖)=𝛿 for all 𝑖=0,1,….

Proof. For any πœ‘βˆˆπΏ2[βˆ’β„Ž,0], we can construct an even extension πœ™βˆˆπΏ2[βˆ’β„Ž,β„Ž] where πœ™(πœƒ)=πœ‘(πœƒ) for all πœƒβˆˆ[βˆ’β„Ž,0] and πœ™(πœƒ)=πœ‘(βˆ’πœƒ) for all πœƒβˆˆ[0,β„Ž]. The function πœ™ may be represented by a Fourier series of cosine functions πœ™(πœƒ)βˆΌπ‘ξ“π‘–=0π‘Žπ‘–ξ‚€cos2πœ‹π‘–πœƒβ„Žξ‚,(5.16) where the β€œβˆΌβ€™β€™ is used to indicate that β€–β€–β€–β€–πœ™βˆ’π‘ξ“π‘–=0π‘Žπ‘–ξ‚€cos2πœ‹π‘–β‹…β„Žξ‚β€–β€–β€–β€–2⟢0(5.17) as π‘β†’βˆž. In what mentioned before, π‘Ž0=1β„Žξ€œ0βˆ’β„Žπ‘Žπœ™(πœƒ)π‘‘πœƒ,𝑖=2β„Žξ€œ0βˆ’β„Žξ‚€πœ™(πœƒ)cos2πœ‹π‘–πœƒβ„Žξ‚π‘‘πœƒ(5.18) for all 𝑖=1,2,…. For simplicity, we will replace the β€œβˆΌβ€™β€™ with an equality sign knowing that mean-square convergence is implied.
For the Fourier series, the basis is cos2πœ‹π‘–πœƒβ„Žξ‚ξ‚‡βˆžπ‘–=0,(5.19) so the first term of this basis and {𝑓𝑖}βˆžπ‘–=0 are the same, namely, the constant β€œ1.’’ Clearly 𝑓0(0)=1 and 𝑁(𝑓0)=𝛿. The first part of this proof is to show that for all 𝑖=0,1,…, 𝑓𝑖(0)=1 and 𝑁(𝑓𝑖)=𝛿.
For 𝑓1, we have that 𝑓1(0)=𝛼1,1+𝛼1,2π‘ž(0)=1 which implies that 𝛼1,1=1βˆ’π›Ό1,2π‘ž(0).(5.20) Also, 𝑁(𝑓1)=𝛼1,1𝛿+𝛼1,2𝑁(π‘ž)=𝛿. Since we do not want 𝛼1,2=0, we require that 𝑁(π‘ž)=π‘ž(0).(5.21) There are no restrictions on 𝛼1,2 other than 𝛼1,2β‰ 0.
For 𝑓2, 𝛼2,1π‘ž(0)+𝛼2,2=1 requires that 𝛼2,2=1βˆ’π›Ό2,1π‘ž(0).(5.22) Since we want 𝛼2,1𝑁(π‘ž)+𝛼2,2𝑐1=𝛿, then 𝛼2,1=π›Ώβˆ’π‘1𝑁(π‘ž)βˆ’π‘ž(0)𝑐1=π›Ώβˆ’π‘1ξ€·π‘ž(0)1βˆ’π‘1ξ€Έ.(5.23)
The rest of the 𝑓𝑖, that is, where 𝑖β‰₯3, are handled alike. In order that 𝑓𝑖(0)=1, we require that 𝛼𝑖,1=1βˆ’π›Όπ‘–,2. To ensure that 𝑁(𝑓𝑖)=𝛿, 𝛼𝑖,2=π›Ώβˆ’π‘π‘–βˆ’2π‘π‘–βˆ’1βˆ’π‘π‘–βˆ’2.(5.24) We have now shown that the sequence of functions {𝑓𝑖}βˆžπ‘–=0 is such that 𝑓𝑖(0)=1 and 𝑁(𝑓𝑖)=𝛿 for all 𝑖=0,1,…. Now it must be shown that this sequence is a spanning set for 𝐿2[βˆ’β„Ž,0]. To do this, we will compare this sequence of functions with the cosine Fourier sequence of functions.
Consider πœ‘(𝑛)∢[βˆ’β„Ž,0]β†’β„œ where πœ‘(𝑛)(πœƒ)=𝑛𝑖=0π‘Žπ‘–ξ‚€cos2πœ‹π‘–πœƒβ„Žξ‚.(5.25) We would like πœ‘(𝑛)(πœƒ)=𝑛+1𝑖=0π‘₯𝑖𝑓𝑖(πœƒ)(5.26) for some set {π‘₯𝑖}𝑛+1𝑖=0 of real numbers. By the Fourier expansion, πœ‘(𝑛)(πœƒ)=π‘Ž0+π‘Ž1ξ‚€cos2πœ‹πœƒβ„Žξ‚+β‹―+π‘Žπ‘›ξ‚€cos2πœ‹π‘›πœƒβ„Žξ‚.(5.27) We want {π‘₯𝑖}𝑛+1𝑖=0 where πœ‘(𝑛)(πœƒ)=π‘₯0+π‘₯1𝛼1,1+𝛼1,2ξ€Έπ‘ž(πœƒ)+π‘₯2𝛼2,1π‘ž(πœƒ)+𝛼2,2ξ‚€cos2πœ‹πœƒβ„Žξ‚ξ‚+π‘₯3𝛼3,1ξ‚€cos2πœ‹πœƒβ„Žξ‚+𝛼3,2ξ‚€cos4πœ‹πœƒβ„Žξ‚ξ‚+β‹―+π‘₯𝑛𝛼𝑛,1ξ‚΅cos2πœ‹(π‘›βˆ’2)πœƒβ„Žξ‚Ά+𝛼𝑛,2ξ‚΅cos2πœ‹(π‘›βˆ’1)πœƒβ„Žξ‚Άξ‚Ά+π‘₯𝑛+1𝛼𝑛+1,1ξ‚΅cos2πœ‹(π‘›βˆ’1)πœƒβ„Žξ‚Ά+𝛼𝑛+1,2ξ‚€cos2πœ‹π‘›πœƒβ„Žξ‚ξ‚Ά=ξ€·π‘₯0+π‘₯1𝛼1,1ξ€Έξ€·π‘₯+π‘ž(πœƒ)1𝛼1,2+π‘₯2𝛼2,1ξ€Έξ‚€+cos2πœ‹πœƒβ„Žξ‚ξ€·π‘₯2𝛼2,2+π‘₯3𝛼3,1ξ€Έξ‚€+β‹―+cos2πœ‹π‘–πœƒβ„Žξ‚ξ€·π‘₯𝑖+1𝛼𝑖+1,2+π‘₯𝑖+2𝛼𝑖+2,1ξ€Έξ‚΅+β‹―+cos2πœ‹(π‘›βˆ’1)πœƒβ„Žξ‚Άξ€·π‘₯𝑛𝛼𝑛,2+π‘₯𝑛+1𝛼𝑛+1,1ξ€Έξ‚€+cos2πœ‹π‘›πœƒβ„Žξ‚ξ€·π‘₯𝑛+1𝛼𝑛+1,2ξ€Έ.(5.28) Equating the last coefficients gives π‘₯𝑛+1=π‘Žπ‘›π›Όπ‘›+1,2,π‘₯𝑛=π‘Žπ‘›βˆ’1βˆ’π‘₯𝑛+1𝛼𝑛+1,1𝛼𝑛,2.(5.29) Continuing, π‘₯𝑖=π‘Žπ‘–βˆ’1βˆ’π‘₯𝑖+1𝛼𝑖+1,1𝛼𝑖,2,(5.30) and finally π‘₯1π‘₯=βˆ’2𝛼2,1𝛼1,2,π‘₯0=π‘Ž0βˆ’π‘₯1𝛼1,1.(5.31) Hence, with the above choice of {π‘₯𝑖}𝑛+1𝑖=0, 𝑛𝑖=0π‘Žπ‘–ξ‚€cos2πœ‹π‘–πœƒβ„Žξ‚=𝑛+1𝑖=0π‘₯𝑖𝑓𝑖(πœƒ),(5.32) and so β€–β€–β€–β€–πœ‘βˆ’π‘›+1𝑖=0π‘₯𝑖𝑓𝑖‖‖‖‖2=β€–β€–β€–β€–πœ‘βˆ’π‘›+1𝑖=0π‘₯𝑖𝑓𝑖+𝑛𝑖=0π‘Žπ‘–ξ‚€cos2πœ‹π‘–β‹…β„Žξ‚βˆ’π‘›ξ“π‘–=0π‘Žπ‘–ξ‚€cos2πœ‹π‘–β‹…β„Žξ‚β€–β€–β€–β€–2=β€–β€–β€–β€–πœ‘βˆ’π‘›ξ“π‘–=0π‘Žπ‘–ξ‚€cos2πœ‹π‘–β‹…β„Žξ‚+𝑛𝑖=0π‘Žπ‘–ξ‚€cos2πœ‹π‘–β‹…β„Žξ‚βˆ’π‘›+1𝑖=0π‘₯𝑖𝑓𝑖‖‖‖‖2β‰€β€–β€–β€–β€–πœ‘βˆ’π‘›ξ“π‘–=0π‘Žπ‘–ξ‚€cos2πœ‹π‘–β‹…β„Žξ‚β€–β€–β€–β€–2+‖‖‖‖𝑛𝑖=0π‘Žπ‘–ξ‚€cos2πœ‹π‘–β‹…β„Žξ‚βˆ’π‘›+1𝑖=0π‘₯𝑖𝑓𝑖‖‖‖‖2=β€–β€–β€–β€–πœ‘βˆ’π‘›ξ“π‘–=0π‘Žπ‘–ξ‚€cos2πœ‹π‘–β‹…β„Žξ‚β€–β€–β€–β€–2⟢0(5.33) as π‘›β†’βˆž.

To find an approximate solution to the generalized Black-Scholes equation we start by letting π‘‹πœ‹βˆ—ξ‚(𝑑)=𝐄[𝑓(𝑆𝑇)βˆ£π‘†π‘‘=πœ‘] (from [6]) and approximating πœ‘ by πœ‘(𝑛)=𝑛+1𝑖=0π‘₯𝑖𝑓𝑖.(5.34) We define the space 𝐂𝑛 as the set of all continuous functions that can be represented by this summation for some {π‘₯𝑖}βˆžπ‘–=0. Note that π‚π‘›βŠ‚πΏ2𝑛[βˆ’β„Ž,0]. Also define π‘’π‘›βˆΆβ„œπ‘›+2β†’β„œ by 𝑒𝑛=βƒ—π‘₯𝑛+1𝑖=0π‘₯𝑖𝑓𝑖,(5.35) so that Ξ¨(𝑑,πœ‘(𝑛))=Ξ¨(𝑑,𝑒𝑛(βƒ—π‘₯)). Define Ξ¨π‘›βˆΆ[0,𝑇]Γ—β„œπ‘›+2β†’β„œ by Ψ𝑛(𝑑,βƒ—π‘₯)=Ξ¨(𝑑,πœ‘(𝑛)) provided that the βƒ—π‘₯ is formed by the coefficients of πœ‘(𝑛) in the spanning set {𝑓𝑖}βˆžπ‘–=0. In general, βƒ—π‘₯(𝑑) is formed by the coefficients of πœ‘π‘‘(𝑛) in the spanning set {𝑓𝑖}βˆžπ‘–=0. Also, define π‘£π‘›βˆΆ[βˆ’β„Ž,0]β†’β„œ by π‘£π‘›βŽ§βŽͺ⎨βŽͺβŽ©ξ‚ƒ1(πœƒ)=0,forπœƒβˆˆβˆ’β„Ž,βˆ’π‘›ξ‚,ξ‚ƒβˆ’1π‘›πœƒ+1,forπœƒ=𝑛.,0(5.36) Last, let π‘”π‘›βˆΆ[0,𝑇]Γ—π‚π‘›Γ—β„œπ‘›+1×𝐢1,2([0,𝑇]×𝐂)β†’β„œ be defined by 𝑔𝑛𝑑,πœ‘(𝑛),βƒ—π‘₯,Ξ¨=π‘Ÿπ‘›+1𝑖=0π‘₯𝑖ξƒͺ𝐷Ψ(𝑑,πœ‘(𝑛))ξ€·πŸ{0}ξ€Έβˆ’π‘›+1𝑖=0π‘˜π‘–πœ•πœ•π‘₯𝑖Ψ𝑛+𝛿𝑑,βƒ—π‘₯22𝑛+1𝑖=0π‘₯𝑖ξƒͺ2𝐷2Ξ¨(𝑑,πœ‘(𝑛))ξ€·πŸ{0},𝟏{0}ξ€Έβˆ’π‘›+1𝑖,𝑗=0π‘˜π‘–π‘˜π‘—πœ•2πœ•π‘₯π‘–πœ•π‘₯𝑗Ψ𝑛,𝑑,βƒ—π‘₯(5.37) where the π‘˜π‘– are the coefficients of 𝑣𝑛 using the spanning set {𝑓𝑖}βˆžπ‘–=0. Finally, define the operator (β‹…)π‘›βˆΆπ‚β†’π‚π‘› by (πœ‘)𝑛=𝑛+1𝑖=0π‘₯𝑖𝑓𝑖,(5.38) where the right-hand side is the first 𝑛+2 terms of the {𝑓𝑖}-expansion of π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 𝐆(Ξ¨)(𝑑,ξ‚πœ‘π‘‘) term. The lack of this term allows approximate solutions to be found using traditional techniques.

Theorem 5.2. Let 𝑆0=πœ“βˆˆπ‚+ and π‘‘βˆˆ[0,𝑇]. Let 𝑓 be a 𝐢2(𝐂) function satisfying the conditions of Theorem 4.1 and let Ξ›=𝑓(𝑆𝑇). Then ξ€·π‘ŸΞ¨π‘‘,πœ‘(𝑛)ξ€Έ=πœ•Ξ¨ξ€·πœ•π‘‘π‘‘,πœ‘(𝑛)ξ€Έξ€·ξ€·+𝐆(Ξ¨)𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έ+𝐷Ψ𝑑,πœ‘(𝑛)ξ€Έξ€·π‘Ÿπœ‘(𝑛)(0)𝟏{0}ξ€Έ+12𝐷2Ψ𝑑,πœ‘(𝑛)π‘ξ€·πœ‘ξ€Έξ€·(𝑛)ξ€ΈπŸ{0}ξ€·πœ‘,𝑁(𝑛)ξ€ΈπŸ{0}ξ€Έξ€·,βˆ€π‘‘,πœ‘(𝑛)ξ€Έβˆˆ[0,𝑇)×𝐂𝑛,(5.39) where Ψ𝑇,πœ‘(𝑛)ξ€Έξ€·πœ‘=𝑓(𝑛)ξ€Έβˆ€πœ‘(𝑛)βˆˆπ‚π‘›(5.40) has a solution of the form 𝑉𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€ΈπΉξ€·π‘‘,πœ‘(𝑛)ξ€Έ.(5.41) Here, 𝑉(𝑑,(ξ‚πœ‘π‘‘)𝑛)=𝑀𝑛(𝑑,0) is a solution to 𝐹𝑑,πœ‘(𝑛)ξ€Έπœ•π‘€π‘›ξ€·ξ€·πœ•π‘‘(𝑑,𝑒)+𝐹𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έπœ•π‘€π‘›ξ€·ξ€·πœ•π‘’(𝑑,𝑒)(5.42)+𝐆(𝐹)𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έπ‘€π‘›(𝑑,𝑒)=0(5.43) for π‘‘βˆˆ[0,𝑇] and π‘’βˆˆ[0,πœ–) for some πœ–>0 and 𝑀𝑛(𝑇,0)=1, and πΉβˆΆβ„œ+Γ—π‚π‘›β†’β„œ is a solution of ξ€·π‘ŸπΉπ‘‘,πœ‘(𝑛)ξ€Έ=πœ•πΉξ€·πœ•π‘‘π‘‘,πœ‘(𝑛)ξ€Έ+𝐷𝐹𝑑,πœ‘(𝑛)ξ€Έξ€·π‘Ÿπœ‘(𝑛)(0)𝟏{0}ξ€Έ+12𝐷2𝐹𝑑,πœ‘(𝑛)π‘ξ€·πœ‘ξ€Έξ€·(𝑛)ξ€ΈπŸ{0}ξ€·πœ‘,𝑁(𝑛)ξ€ΈπŸ{0}ξ€Έβˆ€ξ€·π‘‘,πœ‘(𝑛)ξ€Έβˆˆ[0,𝑇)×𝐂𝑛,(5.44) where 𝐹𝑇,πœ‘(𝑛)ξ€Έξ€·πœ‘=𝑓(𝑛)ξ€Έβˆ€πœ‘(𝑛)βˆˆπ‚π‘›,(5.45) and 𝑓 is a uniformly bounded 𝐢2(𝐂) function satisfying the conditions of Theorem 4.1.

Proof. We assume a solution of the form Ξ¨(𝑑,πœ‘(𝑛))=𝑉(𝑑,(ξ‚πœ‘π‘‘)𝑛)𝐹(𝑑,πœ‘(𝑛)), thenξ€·ξ€·π‘Ÿπ‘‰π‘‘,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€ΈπΉξ€·π‘‘,πœ‘(𝑛)ξ€Έ=πœ•ξ€·π‘‰ξ€·ξ€·πœ•π‘‘π‘‘,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€ΈπΉξ€·π‘‘,πœ‘(𝑛)ξ€·ξ€·ξ€Έξ€Έ+𝐆(𝑉𝐹)𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έ+𝐷(𝑉𝐹)𝑑,πœ‘(𝑛)ξ€Έξ€·π‘Ÿπœ‘(𝑛)(0)𝟏{0}ξ€Έ+12𝐷2ξ€·(𝑉𝐹)𝑑,πœ‘(𝑛)π‘ξ€·πœ‘ξ€Έξ€·(𝑛)ξ€ΈπŸ{0}ξ€·πœ‘,𝑁(𝑛)ξ€ΈπŸ{0}ξ€Έξ€·=𝐹𝑑,πœ‘(𝑛)ξ€Έπœ•π‘‰ξ€·ξ€·πœ•π‘‘π‘‘,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έξ€·ξ€·+𝑉𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έπœ•πΉξ€·πœ•π‘‘π‘‘,πœ‘(𝑛)ξ€Έξ€·+𝐹𝑑,πœ‘(𝑛)𝐆(𝑉)𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έξ€·ξ€·+𝑉𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έξ€·π†(𝐹)𝑑,πœ‘(𝑛)ξ€Έξ€·ξ€·+𝑉𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€ΈΓ—ξ‚†ξ€·π·(𝐹)𝑑,πœ‘(𝑛)ξ€Έξ€·π‘Ÿπœ‘(𝑛)(0)𝟏{0}ξ€Έ+12𝐷2ξ€·(𝐹)𝑑,πœ‘(𝑛)π‘ξ€·πœ‘ξ€Έξ€·(𝑛)ξ€ΈπŸ{0}ξ€·πœ‘,𝑁(𝑛)ξ€ΈπŸ{0},βˆ€π‘‘,πœ‘(𝑛)ξ€Έβˆˆ[0,𝑇)×𝐂𝑛.(5.46) If 𝐹(𝑑,πœ‘(𝑛)) is the solution to (5.44), then 𝐹𝑑,πœ‘(𝑛)ξ€Έπœ•π‘‰ξ€·ξ€·πœ•π‘‘π‘‘,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έξ€·ξ€·+𝐹𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έξ€·ξ€·π†(𝑉)𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έξ€·ξ€·+𝐆(𝐹)𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έπ‘‰ξ€·ξ€·π‘‘,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έ=0.(5.47) Define π‘‡π‘’βˆΆπ‚β†’π‚ by 𝑇𝑒(πœ‘)=πœ‘π‘’, that is, 𝑇𝑒 is a shift operator. Now let 𝑉(𝑑,(ξ‚πœ‘π‘‘+𝑒)𝑛)=𝑉(𝑑,(𝑇𝑒(ξ‚πœ‘π‘‘))𝑛)=𝑀𝑛(𝑑,𝑒) for a fixed πœ‘βˆˆπ‚. Then 𝐆(𝑉)𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έ=πœ•π‘€π‘›ξ€·πœ•π‘’π‘‘,0+ξ€Έ,(5.48) where the superscript + denotes a right-hand derivative with respect to 𝑒. Then 𝐹𝑑,πœ‘(𝑛)ξ€Έπœ•π‘€π‘›ξ€·ξ€·πœ•π‘‘(𝑑,0)+𝐹𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έπœ•π‘€π‘›ξ€·πœ•π‘’π‘‘,0+ξ€Έξ€·ξ€·+𝐆(𝐹)𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έπ‘€π‘›(𝑑,0)=0.(5.49)
A slightly more restrictive, but more familiar form is 𝐹𝑑,πœ‘(𝑛)ξ€Έπœ•π‘€π‘›ξ€·ξ€·πœ•π‘‘(𝑑,𝑒)+𝐹𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έπœ•π‘€π‘›ξ€·ξ€·πœ•π‘’(𝑑,𝑒)+𝐆(𝐹)𝑑,ξ‚πœ‘π‘‘ξ€Έπ‘›ξ€Έπ‘€π‘›(𝑑,𝑒)=0,(5.50) where π‘‘βˆˆ[0,β„Ž] and π‘’βˆˆ[0,πœ–) for some πœ–>0. There is the additional requirement that 𝑀𝑛(𝑇,0)=1 so that (5.44) holds.

Remark 5.3. It can be easily shown that 𝑆𝑑 is 𝛼-HΓΆlder continuous a.s. for 0<𝛼<1/2 provided that 𝑆0 is 𝛼-HΓΆlder continuous for the same 𝛼. Therefore, ||𝐹𝑛||𝑑,βƒ—π‘₯βˆ’πΉ(𝑑,πœ‘)⟢0(5.51) for each 𝑑 as π‘›β†’βˆž where 𝐹(𝑑,πœ‘) is a solution to (5.44) and 𝐹𝑛(𝑑,βƒ—π‘₯)=𝐹(𝑑,πœ‘(𝑛)) is an approximate solution, since 𝐹 is 𝐢2(𝐂) in its second variable and 𝐹𝑛𝑑,βƒ—π‘₯=𝐹𝑑,𝑒𝑛⃗π‘₯ξ€Έξ€Έ=𝐹𝑑,πœ‘(𝑛)ξ€Έ.(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 Ξ¨(𝑑,πœ‘).

Corollary 5.4. If 𝑆0 is HΓΆlder continuous, then πœ•π‘ŸΞ¨(𝑑,πœ‘)=ξ€·πœ•π‘‘Ξ¨(𝑑,πœ‘)+𝐆(Ξ¨)𝑑,ξ‚πœ‘π‘‘ξ€Έ+𝐷Ψ(𝑑,πœ‘)π‘Ÿπœ‘(0)𝟏{0}ξ€Έ+12𝐷2ξ€·Ξ¨(𝑑,πœ‘)𝑁(πœ‘)𝟏{0