Viscosity Solutions and American Option Pricing in a Stochastic Volatility Model of the Ornstein-Uhlenbeck Type

Abstract

We study the valuation of American-type derivatives in the stochastic volatility model of Barndorff-Nielsen and Shephard (2001). We characterize the value of such derivatives as the unique viscosity solution of an integral-partial differential equation when the payoff function satisfies a Lipschitz condition.

1. Introduction

In their seminal paper, Barndorff-Nielsen and Shephard [1] introduced a model that has been shown to describe particularly well financial assets for which log-returns have heavy tail distributions and display long-range dependence. In this model, the volatility of the asset is described by an Ornstein-Uhlenbeck-type process with a pure jump Lévy process acting as the background driving process. An empirical study was made in [1] and showed from exchange rate data that suitable distributions for the Lévy process are the so-called generalized inverse gaussian distributions from which well-understood examples are the normal inverse gaussian (studied in [2]) and the gamma distribution.

The BNS model has been studied from different points of view. Benth et al. [3] considered the problem of optimal portfolio selection. Nicolato and Vernados [4] have studied European option pricing and described the set of equivalent martingale measures under this model. To evaluate these types of options, the authors propose the transform-based method and a simple Monte Carlo method.

In this paper, we consider the pricing of American options with the use of integral-partial differential equations (IPDEs). Although our technique can be simplified and used for European options and certain path-dependent options such as barrier options (see [5] for a definition and examples), we will mainly concentrate on American type derivatives which have not been studied for this model. The main difficulty in this case is the lack of Lipschitz continuity of some of the coefficients of the IPDE.

The question of whether observed option prices can be calibrated and shown to reproduce stylized features such as smiles using this model is of great practical and theoretical relevance. However the majority of exchange-traded options are of the American type and there do not exist any established numerical methods to compute option prices for this model. One could use Monte-Carlo methods; however it is generally more efficient to characterize the option value function as a solution of a variational inequality and to discretize this inequality in order to get an approximation of the value function. Whereas the existence of a solution to the IPDE suggests the use of finite difference schemes, the uniqueness of the solution is a particularly crucial property which insures the convergence of any such numerical scheme to the correct value function. The characterization of the value function as the unique solution of an IPDE is thus the first step to achieve this goal. The design and implementation of a numerical scheme are beyond the scope of this paper; however we refer the reader to the paper of Levendorskiĭ et al. [6] and references therein for some ideas on how this problem could be approached.

The connection between viscosity solutions of IPDEs and Lévy processes has been studied in the literature by various authors. Pham [7] considered a general stopping time problem of a controlled jump diffusion processes. However, his results do not apply here because the Lipschitz condition on the coefficients is not satisfied in our current setting. Cont and Voltchkova [5] studied barrier options and Barles et al. [8] established the connection between viscosity solutions and backward stochastic differential equations. In these papers, the stock price considered is modeled by a stochastic differential equation with jumps driven by a Lévy process. The main difference between the BNS model and these models is the presence of stochastic volatility. However, we will see that the lack of smoothness of the solution to our IPDE will also lead us to consider the notion of viscosity solutions as presented in [9].

The rest of the paper is organized as follows. In Section 2, we present the model and recall the results of Nicolato and Vernados [4] regarding the set of equivalent martingale measures. Section 3 is devoted to the continuity of the value function. In Section 4, we prove that the value function is the viscosity solution of the associated IPDE, and the uniqueness of the solution is presented in Section 5.

2. Lévy Processes and the BNS Model

Let 𝑇>0. We consider the stochastic volatility model of Barndorff-Nielsen and Shephard [1] for the price process of an asset, denoted by 𝑆={𝑆_𝑡}_{0≤𝑡≤𝑇} and defined on a filtered probability space (Ω,ℱ,{ℱ_𝑡}_{0≤𝑡≤𝑇},ℙ). We thus assume that the log-return 𝑋_𝑡=log(𝑆_𝑡) of the asset satisfies the following stochastic differential equation:𝑑𝑋_𝑡=(𝜇+𝛽𝑉_𝑡)𝑑𝑡+𝑉_𝑡𝑑𝐵_𝑡+𝜌𝑑𝑍_𝜆𝑡 (2.1) with𝑑𝑉_𝑡=−𝜆𝑉_𝑡𝑑𝑡+𝑑𝑍_𝜆𝑡 (2.2) in which 𝜇,𝛽∈ℝ, 𝜆>0 and 𝜌≤0.𝐵={𝐵_𝑡}_{0≤𝑡≤𝑇} is a Brownian motion, and 𝑍={𝑍_𝑡}_{0≤𝑡≤𝑇} is the background driving Lévy process (BDLP) under the physical measure ℙ. In this model, 𝑍 has no gaussian component and the increments are positive. 𝑍 and 𝐵 are assumed to be independent, and 𝔽={ℱ_𝑡}_{0≤𝑡≤𝑇} is the usual filtration generated by the pair (𝐵,𝑍). The positivity of the jumps of 𝑍 insure that the process 𝑉 is always positive. We denote by 𝑊 the Lévy measure of 𝑍.

Suppose that ℚ is a probability measure equivalent to ℙ under which 𝑆 is a martingale. We are interested in American-type derivatives of the form 𝑈_𝑡=esssup_{𝜏∈𝒯𝑇,𝜏≥𝑡}𝐄_ℚ[𝑒^{−𝑟(𝜏−𝑡)}ℎ(𝑋_𝜏)∣ℱ_𝑡] (2.3) in which ℎ is the payoff function, and 𝒯_𝑇 is the set of all stopping times with values less or equal to 𝑇. Since {𝑋_𝑡}_{0≤𝑡≤𝑇} and {𝑉_𝑡}_{0≤𝑡≤𝑇} are Markov processes, 𝑈_𝑡 can be written as a function of (𝑥,𝑣,𝑡), say, 𝑈_𝑡=𝑢(𝑥,𝑣,𝑡)=sup_{𝜏∈𝒯𝑇−𝑡}𝐄_ℚ(𝑒^−𝑟𝜏ℎ(𝑋_𝜏^𝑥,𝑣)) (2.4) in which (𝑋_𝑡^𝑥,𝑣)_𝑡≥0 is the process 𝑋 for which 𝑋₀=𝑥 and 𝑉₀=𝑣. We also denote by (𝑉^𝑣_𝑡)_𝑡≥0 the process 𝑉 starting at 𝑉₀=𝑣 and at 𝑡=0.

2.1. Equivalent Martingale Measures

We start by summarizing the results of Nicolato and Vernados [4] concerning the set of equivalent martingale measures. In order to do so, we define the set 𝒴^′={𝑦∶[0,∞)→[0,∞);^∞₀(𝑦(𝑥)−1)²𝑤(𝑥)𝑑𝑥<∞} (2.5) and ℳ^′ as the set of all equivalent martingale measures ℚ such that 𝑍 is still a Lévy process under ℚ independent of 𝐵, possibly with a different marginal distribution.

As in [4], we impose the following conditions on the process 𝑍: (C1)the process 𝑍 is given by the characteristic triplet (0,0,𝑊) so that the cumulant transform is given by 𝜅(𝜃)=log{𝐄[exp(𝜃𝑍₁)]}=^∞₀(𝑒^𝜃𝑧−1)𝑊(𝑑𝑧), (2.6) for values of 𝜃, for which this expression is defined; (C2)𝜃=sup{𝜃∈ℝ∣𝜅(𝜃)<∞}>0; (C3)lim_𝜃→𝜃𝜅(𝜃)=∞.

Remark 2.1. Assumption (C2) implies that there exists 𝜃₀>0 such that (𝑒𝜃^∞_00𝑧−1)𝑊(𝑑𝑧)<∞. (2.7) For 𝑧>0 and 𝑛≥1, we have 0<𝑧^𝑛≤(𝑛!/𝜃^𝑛₀)(𝑒𝜃_0𝑧−1), so that 𝜇_𝑛∶=∫^∞₀𝑧^𝑛𝑊(𝑑𝑧)<∞. Furthermore, Assumption (C2) is a sufficient condition for the process 𝑍 to have finite moments of all orders.

The following theorem was proved in [4].

Theorem 2.2. For all ℚ∈ℳ^′, there exists 𝑦∈𝒴^′ such that 𝑑𝑋_𝑡=(𝑟−𝜆𝜅^𝑦1(𝜌)−2𝑉_𝑡)𝑑𝑡+𝑉_t𝑑𝐵^ℚ_𝑡+𝜌𝑑𝑍_𝜆𝑡, (2.8) in which 𝜅^𝑦(𝜃)=^∞₀(𝑒^𝜃𝑥−1)𝑦(𝑥)𝑤(𝑥)𝑑𝑥, (2.9) and 𝐵^ℚ_𝑡=𝐵_𝑡−∫^𝑡₀(𝑉_𝑠)⁻¹(𝑟−𝜇−(𝛽+1/2)𝑉_𝑠−𝜆𝜅^𝑦(𝜌))𝑑𝑠 and 𝑍_𝜆𝑡 are, respectively, a Brownian motion and a Lévy process under ℚ. 𝑤^𝑦(𝑥)=𝑦(𝑥)𝑤(𝑥) is the Lévy density of 𝑍₁ under ℚ and 𝜅^𝑦(𝜃) is the cumulant function.

In the remaining part of this paper, all expectations will be with respect to a chosen EMM ℚ, unless specified otherwise, and 𝑊 and 𝐵 will denote the associated Lévy measure and the Brownian motion associated to ℚ.

Let 𝒪=ℝ×ℝ₊×[0,𝑇) and assume for a moment that 𝑢 is Lipschitz in (𝑥,𝑣) and𝑢∈𝐶^2,1,1(𝒪), (2.10) that is 𝑢 is differentiable with respect to 𝑣 and 𝑡, and twice differentiable with respect to 𝑥. We can then apply Itô's formula to 𝑈 to find𝑑𝑈_𝑡=(𝜕𝑢𝜕𝑡+ℒ[𝑢])𝑑𝑡+𝜕𝑢𝜕𝑥𝑉_𝑡𝑑𝐵_𝑡+𝑑𝒱_𝑡, (2.11) in which 1ℒ[𝑢]=(𝑟−2𝑣−𝜆𝜅^𝑦(𝜌)+𝜆𝜌𝜇₁)𝜕𝑢𝜕𝑥−𝜆(𝑣−𝜇₁)𝜕𝑢+1𝜕𝑣2𝑣𝜕²𝑢𝜕𝑥²+𝜆^∞₀(𝑢(𝑥+𝜌𝑧,𝑣+𝑧,𝑡)−𝑢(𝑥_,𝑣,𝑡)−(𝜌𝑧𝜕𝑢𝜕𝑥+𝑧𝜕𝑢𝜕𝑣))𝑊(𝑑𝑧), (2.12) and 𝒱_𝑡 is the ℚ-martingale given by 𝑑𝒱_𝑡=^∞₀(𝑢(𝑋_𝑡−+𝜌𝑧,𝑉_𝑡−+𝑧,𝑡)−𝑢(𝑋_𝑡−,𝑉_𝑡−,𝑡)−(𝜌𝑧𝜕𝑢𝜕𝑥(𝑋_𝑡−,𝑉_𝑡−,𝑡)+𝑧𝜕𝑢𝜕𝑣(𝑋_𝑡−,𝑉_𝑡−+,𝑡)))𝑁(𝑑𝑧,𝜆𝑑𝑡)_∞0(𝜌𝑧𝜕𝑢𝜕𝑥(𝑋_𝑡−,𝑉_𝑡−,𝑡)+𝑧𝜕𝑢𝜕𝑣(𝑋_𝑡−,𝑉_𝑡−,𝑡))𝑁(𝑑𝑧,𝜆𝑑𝑡) (2.13) in which 𝑁(𝑑𝑧,𝑑𝑡)=𝑁(𝑑𝑧,𝑑𝑡)−𝑊(𝑑𝑧)𝑑𝑡, and 𝑁(𝑑𝑧,𝑑𝑡) is the random measure of the process 𝑍. Since ∫^𝑡₀(𝜕𝑢/𝜕𝑥)𝑉_𝑡𝑑𝐵_𝑡 is a ℚ-martingale, if it can be shown that 𝑒^−𝑟𝑡𝑈_𝑡 is also a martingale, we can then expect 𝑢 to satisfy the following integral-partial differential equation (IPDE): 𝜕𝑢𝜕𝑡(𝑥,𝑣,𝑡)+ℒ[𝑢](𝑥,𝑣,𝑡)−𝑟𝑢(𝑥,𝑣,𝑡)=0 (2.14) if 𝑢(𝑥,𝑣,𝑡)>ℎ(𝑥). Otherwise 𝑢(𝑥,𝑣,𝑡)=ℎ(𝑥) and this IPDE can be written asmax(𝜕𝑢𝜕𝑡(𝑥,𝑣,𝑡)+ℒ[𝑢](𝑥,𝑣,𝑡)−𝑟𝑢(𝑥,𝑣,𝑡),ℎ(𝑥)−𝑢(𝑥,𝑣,𝑡))=0. (2.15) It is clear also that the function satisfies𝑢(𝑥,𝑣,𝑡)=ℎ(𝑥)for𝑣=0or𝑡=𝑇. (2.16)

Condition (2.10) is in fact very restrictive and most of the time not satisfied. Despite this problem, we will see that 𝑢 can still be regarded as a solution of this equation in a weaker sense.

3. Continuity of the Value Function

Recall the definition of the value of an American option with payoff ℎ:𝑢(𝑥,𝑣,𝑡)=sup_{𝜏∈𝒯𝑇−𝑡}𝐄(𝑒^−𝑟𝜏ℎ(𝑋_𝜏^𝑥,𝑣)). (3.1) In the rest of this paper, we will assume that ℎ is positive and satisfies the Lipschitz condition, in other words ∃𝐾>0 such that forall(𝑥₁,𝑥₂)∈ℝ²|ℎ(𝑥₁)−ℎ(𝑥₂)|≤𝐾|𝑥₁−𝑥₂|. (3.2) For instance, the payoff function for an American put with strike 𝑋>0 is ℎ(𝑥)=max(𝑋−exp(𝑥),0) and satisfies this condition.

Our goal is to show that the function 𝑢 satisfies the IPDE (2.15) in some weak sense. In order to give meaning to this IPDE for a function 𝑢 that does not satisfy basic differentiability conditions, we introduce the idea of viscosity solutions following Crandall and Lions [10]. Let 𝒲 be the set of functions 𝑓∶𝒪→ℝ that satisfy sup_{(𝑥,𝑣),(𝑥′,𝑣′)∈ℝ×ℝ+}|𝑓(𝑥,𝑣,𝑡)−𝑓(𝑥^′,𝑣^′,𝑡)|1+|𝑥−𝑥^′|+|𝑣−𝑣^′|<∞∀𝑡∈[0,𝑇]. (3.3)

Definition 3.1. The function 𝑢∈𝐶⁰(𝒪)∩𝒲 is a viscosity subsolution (supersolution) of (2.15)-(2.16) if forall(𝑥,𝑣,𝑡)∈𝒪 and forall𝜓∈𝒲∩𝐶^2,1,1(𝒪) such that (i)𝜓(𝑥,𝑣,𝑡)=𝑢(𝑥,𝑣,𝑡) and (ii)forall(𝑥^′,𝑣^′,𝑡^′)∈𝒪𝜓(𝑥^′,𝑣^′,𝑡^′)≥𝑢(𝑥^′,𝑣^′,𝑡^′)(≤),max(𝜕𝜓(𝑥,𝑣,𝑡)𝜕𝑡+ℒ[𝜓](𝑥,𝑣,𝑡)−𝑟𝜓(𝑥,𝑣,𝑡);ℎ(𝑥)−𝑢(𝑥,𝑣,𝑡))≥0(≤),𝑢(𝑥,𝑣,𝑡)=ℎ(𝑥)for𝑣=0or𝑡=𝑇. (3.4)(3.5) The function 𝑢 is a viscosity solution if it is both subsolution and supersolution.

Remark 3.2. As noted in [5, page 317] the condition 𝜓∈𝒲 is sufficient to have a well-defined integral term in ℒ[𝜓]. In fact if 𝜓∈𝒲∩𝐶^2,1,1, then ^∞₀(𝜓(𝑥+𝜌𝑧,𝑣+𝑧,𝑡)−𝜓(𝑥,𝑣,𝑡)−(𝜌𝑧𝜕𝜓𝜕𝑥(𝑥,𝑣,𝑡)+𝑧𝜕𝜓≤𝜕𝑣(𝑥,𝑣,𝑡)))𝑊(𝑑𝑧)_𝑧<𝜂𝐶𝑧²𝑊(𝑑𝑧)+^∞_𝜂𝐶(1+|𝑧|)𝑊(𝑑𝑧)<∞ (3.6) for any 𝜂>0.

An important property of viscosity solutions is the continuity of the function. It is the content of the following proposition.

Proposition 3.3. When ℎ satisfies the Lipschitz condition (3.2), the function 𝑢 is continuous and in 𝒲.

Proof. In this proof, we will assume for simplicity that 𝑟=0. The generalization to 𝑟>0 is straightforward. Throughout, 𝐶 is a positive constant that can change from line to line.
We start by showing the continuity of 𝑢 with respect to (𝑥,𝑣), uniformly in 𝑡. We have the following representation of the volatility process:𝑉^𝑣_𝑡=𝑣𝑒^−𝜆𝑡+^𝑡₀𝑒^−𝜆𝑠𝑑𝑍_𝜆𝑠. (3.7) We define the integrated variance process started with 𝑉₀=𝑣 by 𝑉_𝑡^𝑣,∗=^𝑡₀𝑉^𝑣_𝑠𝑑𝑠. (3.8) By (2.2), we find that 𝑉^𝑣_𝑡1𝑑𝑡=𝜆(−𝑑𝑉^𝑣_𝑡+𝑑𝑍_𝜆𝑡), (3.9) so that we have the following representation of the integrated variance process 𝑉_𝑡^𝑣,∗=1𝜆(𝑣−𝑉^𝑣_𝑡1)+𝜆^𝑡₀𝑑𝑍_𝜆𝑠=𝑣𝜀(𝑡)+^𝑡₀𝜀(𝑠)𝑑𝑍_𝜆𝑠, (3.10)(3.11) in which 𝜀(𝑡)=(1−𝑒^−𝜆𝑡)/𝜆.
We also have the following identities:𝑋_𝑡^𝑥,𝑣1=𝑥−𝜆𝜅(𝜌)𝑡−2𝑉_𝑡^𝑣,∗+^𝑡₀𝑉^𝑣_𝑠𝑑𝐵_𝑠+𝜌𝑍_𝜆𝑡,Δ𝑉_𝑡∶=𝑉^𝑣′_𝑡−𝑉^𝑣_𝑡=Δ𝑣𝑒^−𝜆𝑡,Δ𝑉^∗_𝑡∶=𝑉^𝑣′_𝑡^,∗−𝑉_𝑡^𝑣,∗=Δ𝑣𝜀(𝑡),Δ𝑋_𝑡∶=𝑋^{𝑥′,𝑣′}_𝑡−𝑋_𝑡^𝑥,𝑣=𝑥^′1−𝑥−2Δ𝑣𝜀(𝑡)+^𝑡₀(𝑉^𝑣′_𝑠−𝑉^𝑣_𝑠)𝑑𝐵_𝑠1∶=Δ𝑥−2Δ𝑣𝜀(𝑡)+𝑀^𝑣,𝑣′_𝑡, (3.12) with Δ𝑥=𝑥^′−𝑥 and Δ𝑣=𝑣^′−𝑣.
Using the Lipschitz condition on ℎ, we obtain|𝑢(𝑥^′,𝑣^′,𝑡)−𝑢(𝑥,𝑣,𝑡)|=|sup_{𝜏∈𝒯𝑇−𝑡}𝐄ℎ(𝑋^{𝑥′,𝑣′}_𝜏)−sup_{𝜏∈𝒯𝑇−𝑡}𝐄ℎ(𝑋_𝜏^𝑥,𝑣)|≤sup_{𝜏∈𝒯𝑇−𝑡}𝐄|ℎ(𝑋^{𝑥′,𝑣′}_𝜏)−ℎ(𝑋_𝜏^𝑥,𝑣)|≤𝐶sup_{𝜏∈𝒯𝑇−𝑡}𝐄|𝑋^{𝑥′,𝑣′}_𝜏−𝑋_𝜏^𝑥,𝑣|. (3.13)
Then,|𝑢(𝑥^′,𝑣^′,𝑡)−𝑢(𝑥,𝑣,𝑡)|≤𝐶(|Δ𝑥|+|Δ𝑣|+sup_{𝜏∈𝒯𝑇−𝑡}𝐄|𝑀^𝑣,𝑣′_𝜏|). (3.14)
Letting 𝒢=𝜎({𝑍_𝑠}_{0≤𝑠≤𝑇}), the 𝜎-field generated by the BDLP 𝑍 up to the maturity 𝑇, we find that {𝑀^𝑣,𝑣′_𝑡}_𝑡≥0 is a 𝒢∨ℱ_𝑡-martingale. Thus, {|𝑀^𝑣,𝑣′_𝑡|}_𝑡≥0 is a 𝒢∨ℱ_𝑡-submartingale and Doob's theorem applies. In other words,sup_{𝜏∈𝒯𝑇−𝑡}𝐄|𝑀^𝑣,𝑣′_𝜏|≤𝐄(sup_{𝜏∈𝒯𝑇−𝑡}𝐄(|𝑀^𝑣,𝑣′_𝜏|∣𝒢))≤𝐄(𝐄(|𝑀^𝑣,𝑣′_𝑇−𝑡|∣𝒢))≤𝐄(𝐄((𝑀^𝑣,𝑣′_𝑇−𝑡)²∣𝒢)). (3.15) Also, 𝐄((𝑀^𝑣,𝑣′_𝑇−𝑡)²∣𝒢)=₀^𝑇−𝑡(𝑉^𝑣′_𝑠−2𝑉^𝑣′_𝑠𝑉^𝑣_𝑠+𝑉^𝑣_𝑠=)𝑑𝑠_0𝑇−𝑡Δ𝑣𝑒^−𝜆𝑠𝑑𝑠+2₀^𝑇−𝑡𝑉^𝑣_𝑠−(𝑉^𝑣_𝑠)²+𝑉^𝑣_𝑠Δ𝑣𝑒^−𝜆𝑠=𝑑𝑠_0𝑇−𝑡Δ𝑣𝑒^−𝜆𝑠𝑑𝑠+2₀^𝑇−𝑡−𝑉^𝑣_𝑠Δ𝑣𝑒^−𝜆𝑠𝑉^𝑣_𝑠+(𝑉^𝑣_𝑠)²+𝑉^𝑣_𝑠Δ𝑣𝑒^−𝜆𝑠≤𝑑𝑠_0𝑇−𝑡3|Δ𝑣|𝑒^−𝜆𝑠𝑑𝑠=3|Δ𝑣|𝜀(𝑇−𝑡)≤3|Δ𝑣|𝑇. (3.16)
And thus we proved the continuity of 𝑢 in (𝑥,𝑣) uniformly in 𝑡 since|𝑢(𝑥^′,𝑣^′,𝑡)−𝑢(𝑥,𝑣,𝑡)|≤𝐶(|Δ𝑥|+|Δ𝑣|+|Δ𝑣|). (3.17) In particular 𝑢∈𝒲 because of the following inequality: |𝑢(𝑥^′,𝑣^′,𝑡)−𝑢(𝑥,𝑣,𝑡)|≤𝐶(|Δ𝑥|+|Δ𝑣|+|Δ𝑣|)≤2𝐶(1+|Δ𝑥|+|Δ𝑣|). (3.18)
The next step of the proof is to show that𝐄sup_{𝑡≤𝑠≤𝑡′}|𝑋_𝑠^𝑥,𝑣−𝑋_𝑡^𝑥,𝑣|→0,𝐄sup_{𝑡≤𝑠≤𝑡′}|𝑉^𝑣_𝑠−𝑉^𝑣_𝑡|→0 (3.19) as |𝑡−𝑡^′|→0. This is easily obtained by first observing that 𝐄sup_{𝑡≤𝑠≤𝑡′}|𝑋_𝑠^𝑥,𝑣−𝑋_𝑡^𝑥,𝑣1|≤2𝐄sup_{𝑡≤𝑠≤𝑡′}|𝑉_𝑠^𝑣,∗−𝑉_𝑡^𝑣,∗|+𝐄sup_{𝑡≤𝑠≤𝑡′}|^𝑠_𝑡𝑉^𝑣_𝑦𝑑𝐵_𝑦|+𝜌𝐄sup_{𝑡≤𝑠≤𝑡′}|𝑍_𝜆𝑠−𝑍_𝜆𝑡|≤12𝑣|𝜀(𝑡^′)−𝜀(𝑡)|+𝐄|^𝑡′_𝑡𝑒^{−𝜆(𝑡′−𝑠)}𝑑𝑍_𝜆𝑠|+𝐶𝐄|𝑉_𝑡^𝑣,∗_′−𝑉_𝑡^𝑣,∗|+𝜌𝐄|𝑍_𝜆𝑡′−𝑍_𝜆𝑡|≤12𝑣|𝜀(𝑡^′)−𝜀(𝑡)|+(1+𝜌)𝐄|𝑍_𝜆𝑡′−𝑍_𝜆𝑡|+12𝑣|𝜀(𝑡^′)−𝜀(𝑡)|+𝐄|𝑍_𝜆𝑡′−𝑍_𝜆𝑡|. (3.20) As for the process 𝑉, 𝐄sup_{𝑡≤𝑠≤𝑡′}|𝑉^𝑣_𝑠−𝑉^𝑣_𝑡|≤|1−𝑒^{−𝜆(𝑡′−𝑡)}|𝐄|𝑉^𝑣_𝑡|+𝐄|^𝑡′_𝑡𝑒^{−𝜆(𝑡′−𝑠)}𝑑𝑍_𝜆𝑠|≤|1−𝑒^{−𝜆(𝑡′−𝑡)}|𝐄|𝑉^𝑣_𝑡|+𝐄|𝑍_𝜆𝑡′−𝑍_𝜆𝑡|. (3.21) Since 𝑉^𝑣_𝑡≤𝑣+𝑍_𝜆𝑇 for all 𝑡≤𝑇, 𝐄sup_{𝑡≤𝑠≤𝑡′}|𝑉^𝑣_𝑠−𝑉^𝑣_𝑡|≤𝐶(𝑣+𝐄𝑍_𝜆𝑇)|𝑡^′−𝑡|+𝐄|𝑍_𝜆𝑡′−𝑍_𝜆𝑡|, (3.22) and we need to show that 𝐄|𝑍_𝜆𝑡′−𝑍_𝜆𝑡|→0 when |𝑡^′−𝑡|→0.
We mentioned earlier that condition (C2) implies that the moments of 𝑍_𝑡 are finite for all orders. Thus 𝑍 is uniformly integrable. Since 𝑍 is also continuous in probability, it is continuous in ℒ₁, and the conclusion follows.
Let us now show continuity with respect to time. Let 0≤𝑡≤𝑡^′≤𝑇. Take 𝜏∈𝒯_𝑇−𝑡 and define 𝜏^′=𝜏∧(𝑇−𝑡^′). Then,𝐄(𝑒^−𝑟𝜏ℎ(𝑋_𝜏^𝑥,𝑣))=𝐄(𝑒^{−𝑟𝜏′}ℎ(𝑋_𝜏^𝑥,𝑣_′))+𝐄(𝑒^−𝑟𝜏ℎ(𝑋_𝜏^𝑥,𝑣)−𝑒^{−𝑟𝜏′}ℎ(𝑋_𝜏^𝑥,𝑣_′))≤𝑢(𝑥,𝑣,𝑡^′)+𝐄(𝑒^−𝑟𝜏ℎ(𝑋_𝜏^𝑥,𝑣)−𝑒^{−𝑟𝜏′}ℎ(𝑋_𝜏^𝑥,𝑣_′)). (3.23) From this inequality, we readily find that |𝑢(𝑥,𝑣,𝑡^′)−𝑢(𝑥,𝑣,𝑡)|≤𝐄sup_{𝑇−𝑡′≤𝑠≤𝑇−𝑡}|𝑋_𝑠^𝑥,𝑣−𝑋^𝑥,𝑣_{𝑇−𝑡′}| (3.24) which converges to zero as |𝑡−𝑡^′|→0.
Global continuity follows from the following inequality:|𝑢(𝑥^′,𝑣^′,𝑡^′)−𝑢(𝑥,𝑣,𝑡)|≤|𝑢(𝑥^′,𝑣^′,𝑡^′)−𝑢(𝑥,𝑣,𝑡^′)|+|𝑢(𝑥,𝑣,𝑡^′)−𝑢(𝑥,𝑣,𝑡)| (3.25) and the fact that the first bound is independent of 𝑡^′.

4. Viscosity Solutions

This section is devoted to the viscosity solution property of the value function 𝑢. In order to prove that 𝑢 is a viscosity solution of (2.15), we need the following dynamic programming principle. It is a consequence of the martingale property of the Snell envelope stopped before its optimal stopping time and it is the key property needed in the proof of the subsolution property.

Lemma 4.1. Let 𝜖>0, (𝑥,𝑣,𝑡)∈𝒪, and define the stopping time 𝜏^𝜖=inf{0≤𝑠≤𝑇−𝑡∣𝑒^−𝑟𝑠𝑢(𝑋_𝑠^𝑥,𝑣,𝑉^𝑣_𝑠,𝑡+𝑠)−𝜖≤𝑒^−𝑟𝑠ℎ(𝑋_𝑠^𝑥,𝑣)}. (4.1) Then, 𝑢(𝑥,𝑣,𝑡)=𝐄[𝑒^{−𝑟𝜏𝜖}𝑢(𝑋_𝜏^𝑥,𝑣_𝜖,𝑉^𝑣_𝜏𝜖,𝑡+𝜏^𝜖)]. (4.2)

Proof. For some constant 𝐶, we have that 𝑛𝐄(|𝟏_{{ℎ(𝑋𝜏)≥𝑛}}ℎ(𝑋_𝜏)|)≤𝐄(ℎ(𝑋_𝜏)²)≤𝐶+𝐶𝐄(𝑋²_𝜏) (4.3) for all 𝜏∈𝒯_𝑇. We know that 𝑋_𝜏=𝑋₀+𝑟𝜏+𝑉^∗_𝜏+∫^𝜏₀𝑉_𝑠𝑑𝐵_𝑠+𝜌𝑍_𝜆𝜏 and that 0≤𝑉^∗_𝜏≤𝑉^∗_𝑇≤(1/𝜆)(𝑉₀+𝑍_𝜆𝑇) from (3.11). As a result, 𝑋²_𝜏≤4(𝑋₀+𝑟𝑇)² +4(1/𝜆²)(𝑉₀+𝑍_𝜆𝑇)² +4(∫^𝜏₀𝑉_𝑠𝑑𝐵_𝑠)² +4𝜌²𝑍²_𝜆𝑇≤𝐶+𝐶𝑍²_𝜆𝑇 +𝐶(∫^𝜏₀𝑉_𝑠𝑑𝐵_𝑠)² for some constant 𝐶 large enough. Hence 𝐄𝑋²_𝜏≤𝐶+𝐶𝐄𝑍²_𝜆𝑇+𝐶𝐄𝑉^∗_𝑇<∞ for all 𝜏∈𝒯_𝑇. As a consequence, sup_{𝜏∈𝒯𝑇}𝐄(|𝟏_{{ℎ(𝑋𝜏)≥𝑛}}ℎ(𝑋_𝜏)|) converges to 0 as 𝑛 grows to infinity, that is, the collection {𝑒^−𝑟𝜏ℎ(𝑋_𝜏)∶𝜏∈𝒯_𝑇} is uniformly integrable. Hence we find that the process (𝑒^−𝑟𝑠ℎ(𝑋_𝑠))_{0≤𝑠≤𝑇} is of Class D, and we can apply the results of [11] to get the result.

The proof of the solution property of 𝑢 makes use of the following lemma.

Lemma 4.2. Let 𝑡≤𝑇 and 𝜖>0. Suppose that 𝑢(𝑥,𝑣,𝑡)−ℎ(𝑥)>𝜖. Then ℚ(𝜏^𝜖<𝑠)→0 when 𝑠→0.

Proof. Let 𝜂>𝜖 such that 𝜂<𝑢(𝑥,𝑣,𝑡)−ℎ(𝑥).
First we show that 𝑒^{−𝑟𝜏𝜖}𝑢(𝑋_𝜏^𝑥,𝑣_𝜖,𝑉^𝑣_𝜏𝜖,𝜏^𝜖)−𝑒^{−𝑟𝜏𝜖}ℎ(𝑋_𝜏^𝑥,𝑣_𝜖)≤𝜖 almost surely. For some sequence 𝑠_𝑛↓𝜏^𝜖, 𝑒^{−𝑟𝑠𝑛}𝑢(𝑋_𝑠^𝑥,𝑣_𝑛,𝑉^𝑣_𝑠𝑛,𝑠_𝑛)≤𝑒^{−𝑟𝑠𝑛}ℎ(𝑋_𝑠^𝑥,𝑣_𝑛)+𝜖 for 𝑛 large enough. In this case, since (𝑋_𝑠^𝑥,𝑣_𝑛,𝑉^𝑣_𝑠𝑛) converges to (𝑋_𝜏^𝑥,𝑣_𝜖,𝑉^𝑣_𝜏𝜖) in ℒ¹, we can take a subsequence if necessary and find that |𝑢(𝑋_𝑠^𝑥,𝑣_𝑛,𝑉^𝑣_𝑠𝑛,𝑠_𝑛)−𝑢(𝑋_𝜏^𝑥,𝑣_𝜖,𝑉^𝑣_𝜏𝜖,𝜏_𝜖)|→0 and ℎ(𝑋_𝑠^𝑥,𝑣_𝑛)→ℎ(𝑋_𝜏^𝑥,𝑣_𝜖) a.s. with 𝑛→∞. Taking the limit, we find that𝑒^{−𝑟𝜏𝜖}𝑢(𝑋_𝜏^𝑥,𝑣_𝜖,𝑉^𝑣_𝜏𝜖,𝜏^𝜖)=lim_𝑛→∞𝑒^{−𝑟𝑠𝑛}𝑢(𝑋_𝑠^𝑥,𝑣_𝑛,𝑉^𝑣_𝑠𝑛,𝑠_𝑛)≤lim_𝑛→∞𝑒^{−𝑟𝑠𝑛}ℎ(𝑋_𝑠^𝑥,𝑣_𝑛)+𝜖=𝑒^{−𝑟𝜏𝜖}ℎ(𝑋_𝜏^𝑥,𝑣_𝜖)+𝜖a.s. (4.4)
Since 𝑢 is continuous with respect to 𝑡, we find that 𝜂<𝑒^−𝑟𝑠𝑢(𝑥,𝑣,𝑡+𝑠)−𝑒^−𝑟𝑠ℎ(𝑥) for 𝑠 small enough. Then, for 𝑠 small enough,ℚ(𝜏^𝜖<𝑠)≤ℚ(𝑒^{−𝑟𝜏𝜖}(𝑢(𝑥,𝑣,𝜏^𝜖)−ℎ(𝑥))+𝑒^{−𝑟𝜏𝜖}(ℎ(𝑋_𝜏^𝑥,𝑣_𝜖)−𝑢(𝑋_𝜏^𝑥,𝑣_𝜖,𝑉^𝑣_𝜏𝜖,𝜏^𝜖))>𝜂−𝜖)≤ℚ(𝑒^{−𝑟𝜏𝜖}|𝑢(𝑥,𝑣,𝜏^𝜖)−𝑢(𝑋_𝜏^𝑥,𝑣_𝜖,𝑉^𝑣_𝜏𝜖,𝜏^𝜖)|+𝑒^{−𝑟𝜏𝜖}|ℎ(𝑋_𝜏^𝑥,𝑣_𝜖)−ℎ(𝑥)|>𝜂−𝜖)≤ℚ(|𝑉^𝑣_𝜏𝜖−𝑣|>𝛿₂)+ℚ(|𝑋_𝜏^𝑥,𝑣_𝜖−𝑥|>𝛿₃) (4.5) for some constants 𝛿₂>0 and 𝛿₃>0. By the continuity in probability of the processes 𝑋 and 𝑉, we know that this expression goes to zero when 𝑠→0.

We can now show that 𝑢 is a viscosity solution.

Theorem 4.3. When ℎ satisfies the Lipschitz condition (3.2), 𝑢 is a viscosity solution of IPDE (2.15).

Proof. We already know that 𝑢 is continuous and in 𝒲.
Let us start by showing that 𝑢 is a supersolution of (2.15). Let (𝑥,𝑣,𝑡)∈𝒪 and 𝜓 satisfy the conditions given in the above definition of supersolutions. By definition, for any Δ𝑡>0,0≥𝑒^{−𝑟Δ𝑡}𝐄(𝑢(𝑋^𝑥,𝑣_Δ𝑡,𝑉^𝑣_Δ𝑡,𝑡+Δ𝑡))−𝑢(𝑥,𝑣,𝑡)≥𝐄(𝑒^{−𝑟Δ𝑡}𝜓(𝑋^𝑥,𝑣_Δ𝑡,𝑉^𝑣_Δ𝑡,𝑡+Δ𝑡)−𝜓(𝑋₀^𝑥,𝑣,𝑉^𝑣₀,𝑡))=𝐄(₀^Δ𝑡𝑒^−𝑟𝑠(−𝑟𝜓+𝜕𝜓𝜕𝑡+ℒ[𝜓])(𝑋_𝑠^𝑥,𝑣,𝑉^𝑣_𝑠+,𝑡+𝑠)𝑑𝑠_0Δ𝑡𝜕𝜓𝜕𝑥(𝑋_𝑠^𝑥,𝑣,𝑉^𝑣_𝑠,𝑡+𝑠)𝑒^−𝑟𝑠𝑉^𝑣_𝑠𝑑𝐵_𝑠+Ψ^𝑥,𝑣_Δ𝑡−Ψ₀^𝑥,𝑣), (4.6) in which Ψ^𝑥,𝑣 is the martingale defined by 𝑑Ψ_𝑠^𝑥,𝑣=𝑒^{−𝑟𝑠∞}₀(𝜓(𝑋^𝑥,𝑣_𝑠−+𝜌𝑧,𝑉^𝑣_𝑠−+𝑧,𝑡+𝑠)−𝜓(𝑋^𝑥,𝑣_𝑠−,𝑉^𝑣_𝑠−,𝑡+𝑠)−𝑧(𝜌𝜕𝜓+𝜕𝑥𝜕𝜓𝜕𝑣)(𝑋^𝑥,𝑣_𝑠−,𝑉^𝑣_𝑠−,𝑡+𝑠))𝑁(𝑑𝑧,𝜆𝑑𝑠)+𝑒^{−𝑟𝑠∞}₀𝑧(𝜌𝜕𝜓+𝜕𝑥𝜕𝜓𝜕𝑣)(𝑋^𝑥,𝑣_𝑠−,𝑉^𝑣_𝑠−,𝑡+𝑠)𝑁(𝑑𝑧,𝜆𝑑𝑠). (4.7) Since ₀^Δ𝑡𝜕𝜓𝜕𝑥(𝑋_𝑠^𝑥,𝑣,𝑉^𝑣_𝑠,𝑡+𝑠)𝑒^−𝑟𝑠𝑉^𝑣_𝑠𝑑𝐵_𝑠 (4.8) is also a martingale, we have the following inequality: 0≥₀^Δ𝑡𝐄(𝑒^−𝑟𝑠(−𝑟𝜓+𝜕𝜓𝜕𝑡+ℒ[𝜓])(𝑋_𝑠^𝑥,𝑣,𝑉^𝑣_𝑠,𝑡+𝑠))𝑑𝑠, (4.9) in other words, dividing by Δ𝑡 and taking the limit as Δ𝑡→0, 0≥−𝑟𝜓(𝑥,𝑣,𝑡)+𝜕𝜓𝜕𝑡(𝑥,𝑣,𝑡)+ℒ[𝜓](𝑥,𝑣,𝑡). (4.10) Since, by definition, 𝑢(𝑥,𝑣,𝑡)≥ℎ(𝑥), 𝑢 satisfies (2.15). To prove that 𝑢 is a viscosity subsolution of (2.15), let (𝑥,𝑣,𝑡)∈𝒪 and 𝜓 satisfy the conditions of the above definition for subsolutions. If 𝑢(𝑥,𝑣,𝑡)=ℎ(𝑥), inequality (3.4) is satisfied. Otherwise, let 0<𝜖<𝑢(𝑥,𝑣,𝑡)−ℎ(𝑥). (4.11) We know from Lemma 4.1 that 0=𝐄(𝑒^{−𝑟(Δ𝑡∧𝜏𝜖)}𝑢(𝑋^𝑥,𝑣_{Δ𝑡∧𝜏𝜖},𝑉^𝑣_{Δ𝑡∧𝜏𝜖},𝑡+(Δ𝑡∧𝜏^𝜖)))−𝑢(𝑥,𝑣,𝑡)≤𝐄(𝑒^{−𝑟(Δ𝑡∧𝜏𝜖)}𝜓(𝑋^𝑥,𝑣_{Δ𝑡∧𝜏𝜖},𝑉^𝑣_{Δ𝑡∧𝜏𝜖},𝑡+(Δ𝑡∧𝜏^𝜖)))−𝜓(𝑥,𝑣,𝑡)=𝐄(^{Δ𝑡∧𝜏𝜖}₀𝑒^−𝑟𝑠(−𝑟𝜓+𝜕𝜓𝜕𝑡+ℒ[𝜓])(𝑋_𝑠^𝑥,𝑣,𝑉^𝑣_𝑠,𝑡+𝑠)𝑑𝑠) (4.12) for any Δ𝑡>0. Knowing that ℚ(𝜏^𝜖<Δ𝑡)→0 when Δ𝑡→0 by Lemma 4.2, dividing the preceding inequality by Δ𝑡 and taking the limit to 0, we get the desired result by Lebesgue's dominated convergence theorem.

5. Comparison Principles and Uniqueness of the Solution

In this section, we prove a comparison result from which we obtain the uniqueness of the solution of the IPDE. In proving comparison results for viscosity solutions, the notion of parabolic superjet and subjet as defined by Crandall et al. [10] is particularly useful. Setting 𝑦=(𝑥,𝑣), we define the parabolic superjet and its closure by𝒥^2,+𝑢(𝑦,𝑡)={(𝑝,𝑞,𝐴)∈ℝ×ℝ²×𝒮₂suchthat𝑢(𝑦^′,𝑡^′)−𝑢(𝑦,𝑡)≤𝑝(𝑡^′−𝑡)+𝑞⋅(𝑦^′+1−𝑦)2(𝑦^′−𝑦)^𝑇⋅𝐴⋅(𝑦^′−𝑦)+𝑜(|𝑡^′−𝑡|+|𝑦^′−𝑦|²)as(𝑡^′,𝑦^′)→(𝑡,𝑦)},𝒥^2,+𝑢(𝑦,𝑡)={(𝑝,𝑞,𝐴)=lim_𝑛→∞(𝑝_𝑛,𝑞_𝑛,𝐴_𝑛)suchthat(𝑝_𝑛,𝑞_𝑛,𝐴_𝑛)∈𝒥^2,+𝑢(𝑦_𝑛,𝑡_𝑛)and(𝑦_𝑛,𝑡_𝑛)→(𝑦,𝑡)0𝑥021𝑓3(lim_𝑛→∞)}, (5.1) The subjet and its closure are then defined similarly by 𝒥^2,−𝑢(𝑦,𝑡)=−𝒥^2,+(−𝑢)(𝑦,𝑡),𝒥^2,−𝑢(𝑦,𝑡)=−𝒥^2,+(−𝑢)(𝑦,𝑡). (5.2)

We then have the following lemma which is essentially proved in [8] (Lemma 3.3).

Lemma 5.1. If the function 𝑢∈𝐶⁰(ℝ×ℝ₊×[0,𝑇]) is a viscosity subsolution (resp. supersolution) of (2.15), then for all(𝑥,𝑣,𝑡)∈ℝ×ℝ₊×[0,𝑇) and for all(𝑝,𝑞,𝐴)∈𝒥^2,+𝑢(𝑥,𝑣,𝑡) (resp., 𝒥^2,−𝑢(𝑥,𝑣,𝑡)) max(𝑝+ℒ_𝜉^𝑞,𝐴[𝑢,𝜓](𝑥,𝑣,𝑡)−𝑟𝑢(𝑥,𝑣,𝑡);ℎ(𝑥)−𝑢(𝑥,𝑣,𝑡))≥0(≤), (5.3) in which ℒ_𝜉^𝑞,𝐴1[𝑢,𝜓](𝑥,𝑣,𝑡)=(𝑟−2𝑣−𝜆𝜅^𝑦(𝜌)+𝜆𝜌𝜇^𝜂)𝑞⁽¹⁾−𝜆(𝑣−𝜇^𝜂)𝑞⁽²⁾+12𝑣𝐴₁₁+𝜆^𝜉₀(𝜓(𝑥+𝜌𝑧,𝑣+𝑧,𝑡)−𝜓(𝑥,𝑣,𝑡)−(𝜌𝑧𝜕𝜓𝜕𝑥+𝑧𝜕𝜓𝜕𝑣))𝑊(𝑑𝑧)+𝜆^∞_𝜉(𝑢(𝑥+𝜌𝑧,𝑣+𝑧,𝑡)−𝑢(𝑥,𝑣,𝑡)−(𝜌𝑧𝜕𝜓𝜕𝑥+𝑧𝜕𝜓𝜕𝑣))𝑊(𝑑𝑧) (5.4) for some 𝜓∈𝐶^2,1,1 and 0<𝜉<1.

Pham [7] obtains the uniqueness of the solution when the coefficients of ℒ satisfy Lipschitz conditions on ℝ²×[0,𝑇]. For 𝛿>0, define 𝒪^𝛿=ℝ×(𝛿,∞)×[0,𝑇). Then, the coefficients of ℒ satisfy the Lipschitz conditions on 𝒪^𝛿, and using the ideas of uniqueness proofs in the literature, we can show a comparison principle on 𝒪^𝛿. This result will then be used to show the uniqueness of the solution on 𝒪.

Theorem 5.2. Let 𝜖≥0, and let 𝑢₁ be a subsolution and 𝑢₂ a supersolution of (2.15) on 𝒪^𝛿 such that 𝑢₁(𝑥,𝑣,𝑡)≤𝑢₂(𝑥,𝑣,𝑡)+𝜖 (5.5) for 𝑡=𝑇 or 𝑣=𝛿. Then 𝑢₁(𝑥,𝑣,𝑡)≤𝑢₂(𝑥,𝑣,𝑡)+𝜖𝑒^{𝑟(𝑇−𝑡)} for all (𝑥,𝑣,𝑡)∈𝒪^𝛿.

Proof. An IPDE of the form (𝜕𝜓(𝑥,𝜗,𝑡)/𝜕𝑡)+ℒ[𝜓](𝑥,𝜗,𝑡)−𝑟𝜓(𝑥,𝜗,𝑡)=0 for (𝑥,𝜗,𝑡)∈𝒪 and 𝜓(𝑥,𝜗,𝑇)=ℎ(𝑥) was shown to have a unique solution in [8] when the coefficients of ℒ satisfy some given Lipschitz conditions. In fact when (𝑥,𝜗,𝑡) and (𝑥^′,𝜗^′,𝑡^′)∈𝒪^𝛿, we have |𝜗^′−√𝜗|≤(1/2𝛿)|𝜗^′−𝜗|, and so the operator ℒ satisfies the assumptions made in [8]. The extension of the uniqueness result to our current setting is straightforward, and we only give the main details.
We first show that 𝑢₁−𝑢₂ is a subsolution of a related IPDE. Suppose that 𝜓∈𝒲∩𝐶² and 𝑢₁−𝑢₂−𝜓 attains a maximum at (𝑦₀,𝑡₀)∈𝒪^𝛿. Set𝑤(𝑦₁,𝑦₂,𝑡,𝑠)=𝑢₁(𝑦₁,𝑡)−𝑢₂(𝑦₂,𝑠),𝜙(𝑦₁,𝑦₂1,𝑡,𝑠)=2𝜖|𝑦₁−𝑦₂|²+12𝛼|𝑡−𝑠|²+𝜓(𝑦₁,𝑡). (5.6) Since 𝑢₁ and 𝑢₂ are in 𝒲, the function 𝑤−𝜙 attains its maximum (𝑦^∗₁,𝑦^∗₂,𝑡^∗,𝑠^∗) (which depends on 𝜖,𝛼) in 𝒪^𝛿×𝒪^𝛿. By a classical argument in the theory of viscosity solutions, we can show that (1/𝜖)|𝑦^∗₁−𝑦^∗₂|²,(1/𝛼)|𝑡^∗−𝑠^∗|²→0 when 𝜖,𝛼→0 and (𝑦^∗₁,𝑦^∗₂,𝑡^∗,𝑠^∗)→(𝑦₀,𝑦₀,𝑡₀,𝑡₀) (5.7) when 𝜖,𝛼→0.
Applying Theorem 8.3 of Crandall et al. [9] to the functions 𝑤 and 𝜙, we find matrices 𝑌₁,𝑌₂ such that(𝑎+𝜕𝜓𝜕𝑡(𝑦^∗₁,𝑡^∗),𝑏+𝐷𝜓(𝑦^∗₁,𝑡^∗),𝑌₁)∈𝒥^2,+𝑢₁(𝑦^∗₁,𝑡^∗)(−𝑎,−𝑏,−𝑌₂)∈𝒥^2,+(−𝑢₁)(𝑦^∗₂,𝑠^∗), (5.8) with a=(1/𝛼)(𝑡^∗−𝑠^∗) and 𝑏=(1/𝜖)(𝑦^∗₁−𝑦^∗₂) and for 0<𝜉<1 the inequalities max(𝑎+ℒ^𝑏,𝑌1_𝜉[𝑢₁,𝜓](𝑦^∗₁,𝑡^∗)+𝜕𝜓𝜕𝑡(𝑦^∗₁,𝑡^∗)+𝑣^∗₁𝜕²𝜓𝜕𝑥²(𝑦^∗₁,𝑡^∗)−𝑟𝑢₁(𝑦^∗₁,𝑡^∗);ℎ(𝑥^∗₁)−𝑢₁(𝑦^∗₁,𝑡^∗))≥0,max(𝑎+ℒ^𝑏,𝑌2_𝜉[𝑢₂,𝜓](𝑦^∗₂,𝑠^∗)−𝑟𝑢₂(𝑦^∗₂,𝑠^∗);ℎ(𝑥^∗₂)−𝑢₂(𝑦^∗₂,𝑠^∗))≤0 (5.9) are satisfied. Write these two expressions as max(𝐴,𝐵)≥0 and max(𝐶,𝐷)≤0. Then max(𝐴−𝐶,𝐵−𝐷)≥0. Now, 𝐵−𝐷=ℎ(𝑥^∗₁)−𝑢₁(𝑦^∗₁,𝑡^∗)−ℎ(𝑥^∗₂)+𝑢₂(𝑦^∗₂,𝑠^∗), and because ℎ is Lipschitz |ℎ(𝑥^∗₁)−ℎ(𝑥^∗₂)|→0 when 𝜖,𝛼→0. Thus 𝐵−𝐷→𝑢₂(𝑦₀,𝑡₀)−𝑢₁(𝑦₀,𝑡₀). On the other hand it was shown in [8] that 𝐴−𝐶≤𝑟(𝑢₂(𝑦^∗₂,𝑠^∗)−𝑢₁(𝑦^∗₁,𝑡^∗1))+𝜖(121−𝜆+4𝛿)|𝑦^∗₁−𝑦^∗₂|²+𝜕𝜓𝜕𝑡+(𝑟^′−12𝑣^∗₁,𝑣^∗₁)𝐷𝜓(𝑦^∗₁,𝑡^∗)+𝑣^∗₁𝜕²𝜓𝜕𝑥²(𝑦^∗₁,𝑡^∗)+𝜆^∞₀(𝜓(𝑥^∗₁+𝜌𝑧,𝑣^∗₁+𝑧,𝑡^∗)−𝜓(𝑦^∗₁,𝑡^∗)−𝑧(𝜌,1)⋅𝐷𝜓)𝑊(𝑑𝑧)+𝜆^𝜉₀(𝜙(𝑥^∗₁+𝜌𝑧,𝑣^∗₁+𝑧,𝑥^∗₂,𝑣^∗₂,𝑡^∗,𝑠^∗)−𝜙(𝑥^∗₁,𝑣^∗₁,𝑥^∗₂,𝑣^∗₂,𝑡^∗,𝑠^∗)−𝑧(𝜌,1)⋅(𝑏+𝐷𝜓(𝑦^∗₁,𝑡^∗)))𝑊(𝑑𝑧)−𝜆^𝜉₀(𝜙(𝑥^∗₁,𝑣^∗₁,𝑥^∗₂+𝜌𝑧,𝑣^∗₂+𝑧,𝑡^∗,𝑠^∗)−𝜙(𝑥^∗₁,𝑣^∗₁,𝑥^∗₂,𝑣^∗₂,𝑡^∗,𝑠^∗)−𝑧(𝜌,1)⋅𝑏₂)𝑊(𝑑𝑧), (5.10) in which 𝑟^′=(𝑟−𝜆𝜅^𝑦(𝜌)+𝜆𝜇₂). Using the fact that 𝜙∈𝒲∩𝒞² we find by letting 𝜉→0 and then 𝜖,𝛼→0 that 𝐴−𝐶≤−𝑟(𝑢₁(𝑦₀,𝑡₀)−𝑢₂(𝑦₀,𝑡₀))+𝜕𝜓𝜕𝑡+ℒ𝜓. (5.11) Consequently, max(−𝑟(𝑢₁−𝑢₂)(𝑦₀,𝑡₀)+𝜕𝜓𝜕𝑡(𝑦₀,𝑡₀)+ℒ𝜓(𝑦₀,𝑡₀),−(𝑢₁−𝑢₂)(𝑦₀,𝑡₀))≥0. (5.12)
As shown in [8] (see Lemma 3.8), there exists a function 𝜒≥1 such that𝜕𝜒𝜕𝑡+ℒ𝜒−𝑟𝜒<0 (5.13) for which the maximum 𝑀=sup_{ℝ×ℝ+×[𝑡1,𝑇]}((𝑢₁−𝑢₂)(𝑦,𝑡)−𝛽𝜒(𝑦,𝑡))𝑒^{𝑟(𝑡−𝑇)} (5.14) is attained at some point (𝑦₀,𝑡₀). Then (𝑢₁−𝑢₂−𝛽𝜒)(𝑦,𝑡)≤(𝑢₁−𝑢₂−𝛽𝜒)(𝑦₀,𝑡₀)𝑒^{𝑟(𝑡0−𝑡)}. (5.15) Let 𝜓(𝑦,𝑡)=𝛽𝜒(𝑦,𝑡)−(𝑢₁−𝑢₂−𝛽𝜒)(𝑦₀,𝑡₀)𝑒^{𝑟(𝑡0−𝑡)}. Then 𝜓 satisfies the properties in the subsolution definition, hence it satisfies (5.12). But (𝜕𝜓𝜕𝑡+ℒ𝜓)(𝑦₀,𝑡₀)=(𝛽𝜕𝜒𝜕𝑡+𝑟(𝑢₁−𝑢₂−𝛽𝜒)+𝛽ℒ𝜒)(𝑦₀,𝑡₀)<𝑟(𝑢₁−𝑢₂)(𝑦₀,𝑡₀). (5.16) Hence, either (𝑢₁−𝑢₂)(𝑦₀,𝑡₀)≤0, or 𝑣₀=𝛿 or 𝑡₀=𝑇 and, in this case, (𝑢₁−𝑢₂)(𝑦₀,𝑡₀)≤𝜖 by assumption. Hence, we conclude that (𝑢₁−𝑢₂)(𝑦,𝑡)≤𝛽𝜒(𝑦,𝑡)−𝛽𝜒(𝑦₀,𝑡₀)𝑒^{𝑟(𝑡0−𝑡)}+(𝑢₁−𝑢₂)(𝑦₀,𝑡₀)𝑒^{𝑟(𝑡0−𝑡)}≤𝛽𝜒(𝑦,𝑡)+𝜖𝑒^{𝑟(𝑡0−𝑡)}. (5.17) Sending 𝛽 to zero we get 𝑢₁≤𝑢₂+𝜖𝑒^{𝑟(𝑇−𝑡)} on ℝ×(𝛿,∞)×[𝑡₁,𝑇]. As done in [8], we can repeat this argument as many times as needed to get 𝑢₁≤𝑢₂+𝜖𝑒^{𝑟(𝑇−𝑡)} on 𝒪^𝛿.

A solution of (2.15)-(2.16) is said to be minimal if it is less or equal to any other solution of (2.15)-(2.16). To prove uniqueness, we first show that the solution 𝑢 is minimal.

Theorem 5.3. 𝑢 is the minimal viscosity solution of (2.15)-(2.16).

Proof. Let 𝛿>0 and define 𝑢^𝛿(𝑥,𝑣,𝑡)=sup_{𝜏∈𝒯𝑇−𝑡,𝜏≤𝜏𝛿}𝐄(𝑒^−𝑟𝜏ℎ(𝑋_𝜏^𝑥,𝑣)) (5.18) in which 𝜏_𝛿=inf{𝑠≥0∶𝑉^𝑣_𝑠≤𝛿}. (5.19) Then 𝑢^𝛿 is a viscosity solution of (2.15) on 𝒪^𝛿 with boundary conditions 𝑢^𝛿(𝑥,𝑣,𝑡)=ℎ(𝑥)for𝑡=𝑇or𝑣=𝛿. (5.20) The proof of this statement is essentially the same as the proof for the viscosity solution property of 𝑢. The main difference is that the maturity 𝑇 is replaced by 𝜏_𝛿. Note that 𝑉^𝛿′_𝑠>𝛿 for 𝛿^′>𝛿𝑒^𝜆𝑇, hence 𝑢^𝛿(𝑥,𝑣,𝑡)=𝑢(𝑥,𝑣,𝑡) for all 𝑥∈ℝ,𝑡<𝑇 and 𝑣>𝛿𝑒^𝜆𝑇.
Let 𝑢 be another viscosity solution of (2.15)-(2.16). Then 𝑢 is a viscosity solution of (2.15) on 𝒪^𝛿 with boundary values 𝑢(𝑥,𝑣,𝑡) for 𝑡=𝑇 or 𝑣=𝛿. Also, 𝑢(𝑥,𝑣,𝑡)≥ℎ(𝑥)=𝑢^𝛿(𝑥,𝑣,𝑡) for 𝑡=𝑇 or 𝑣=𝛿. By Theorem 5.2, we find that 𝑢≥𝑢^𝛿 on 𝒪^𝛿. In particular, 𝑢(𝑥,𝑣,𝑡)≥𝑢(𝑥,𝑣,𝑡) for 𝑥∈ℝ,𝑡<𝑇 and 𝑣>𝛿𝑒^𝜆𝑇. Since 𝛿 is arbitrary, 𝑢≥𝑢 on 𝒪.

Following Pham [7], we denote by 𝑈𝐶_𝑥,𝑣(𝒪) the set of functions defined on 𝒪 uniformly continuous in (𝑥,𝑣), uniformly in 𝑡. We have already shown that the function 𝑢 satisfies |𝑢(𝑥^′,𝑣^′,𝑡)−𝑢(𝑥,𝑣,𝑡)|≤𝐶(|𝑥^′−𝑥|+|𝑣^′−𝑣|+|𝑣^′−𝑣|). (5.21) Hence, 𝑢∈𝑈𝐶_𝑥,𝑣(𝒪). Using the two previous theorems, we can show the uniqueness in 𝑈𝐶_𝑥,𝑣(𝒪).

Theorem 5.4. 𝑢 is the unique viscosity solution of (2.15)-(2.16) in 𝑈𝐶_𝑥,𝑣(𝒪).

Proof. Let 𝑢∈𝑈𝐶_𝑥,𝑣(𝒪) be another viscosity solution of (2.15)-(2.16). Let 𝜖>0. Then there exists 𝛿>0 such that 0≤𝑢(𝑥,𝑣,𝑡)−𝑢(𝑥,0,𝑡)=𝑢(𝑥,𝑣,𝑡)−ℎ(𝑥)<𝜖 and 0≤𝑢(𝑥,𝑣,𝑡)−𝑢(𝑥,0,𝑡)=𝑢(𝑥,𝑣,𝑡)−ℎ(𝑥)<𝜖 for 𝑣≤𝛿. In particular, |𝑢(𝑥,𝑣,𝑡)−𝑢(𝑥,𝑣,𝑡)|<𝜖 for all 𝑥, all 𝑡 and 𝑣≤𝛿. Furthermore, by Theorem 5.3, we obtain that 𝑢(𝑥,𝛿,𝑡)≤𝑢(𝑥,𝛿,𝑡)≤𝑢(𝑥,𝛿,𝑡)+𝜖, and 𝑢(𝑥,𝑣,𝑇)=𝑢(𝑥,𝑣,𝑇) by definition. By the comparison principle of Theorem 5.2, we find that 𝑢(𝑥,𝑣,𝑡)≤𝑢(𝑥,𝑣,𝑡)≤𝑢(𝑥,𝑣,𝑡)+𝜖𝑒^{𝑟(𝑇−𝑡)} for all (𝑥,𝑣,𝑡)∈𝒪^𝛿. Hence, 𝑢(𝑥,𝑣,𝑡)≤𝑢(𝑥,𝑣,𝑡)≤𝑢(𝑥,𝑣,𝑡)+𝜖𝑒^𝑟𝑇 for all (𝑥,𝑣,𝑡)∈𝒪. Since 𝜖 is arbitrary, we obtain the desired result.

To prove the uniqueness of the solution, we first modified the optimal stopping problem by defining it on 𝒪^𝛿 in order to avoid the degeneracy of the infinitesimal generator. This suggests that one may be able to use known numerical schemes for nondegenerate IPDEs for Lévy processes to design a numerical scheme for the value function 𝑢^𝛿. When 𝛿 is close to 0, we obtain an approximation for the value function 𝑢. The reader is referred to the work of Levendorskiĭ et al. [6] and references therein for possible numerical implementation techniques and difficulties associated to them. On the other hand, it is likely that the method of proof of the uniqueness property could be used for a larger class of stochastic volatility models, including the models of Heston [12] and Hull and White [13]. Indeed, all these models have the non-Lipschitz term 𝑉_𝑡 in the differential equation for the log-returns. There remain open problems which we leave to future research.

Acknowledgment

Some parts of this paper were done at HEC Montréal, and the author would like to thank Bruno Rémillard and an anonymous referee for their help. This work was supported in part by the NSERC, the IFM2 and the Fonds québécois de la recherche sur la nature et les technologies.