A central bank strategy for defending a currency peg
AA central bank strategy for defending a currency peg
Eyal Neuman ∗ Alexander Schied ∗∗ Chengguo Weng †† Xiaole Xue ‡‡ First version: November 29, 2019This version: August 2, 2020
Abstract
We consider a central bank strategy for maintaining a two-sided currency target zone, in which anexchange rate of two currencies is forced to stay between two thresholds. To keep the exchange ratefrom breaking the prescribed barriers, the central bank is generating permanent price impact and therebyaccumulating inventory in the foreign currency. Historical examples of failed target zones illustrate thatthis inventory can become problematic, in particular when there is an adverse macroeconomic trendin the market. We model this situation through a continuous-time market impact model of Almgren–Chriss-type with drift, in which the exchange rate is a diffusion process controlled by the price impactof the central bank’s intervention strategy. The objective of the central bank is to enforce the targetzone through a strategy that minimizes the accumulated inventory. We formulate this objective as astochastic control problem with random time horizon. It is solved by reduction to a singular boundaryvalue problem that was solved by Lasry and Lions (1989). Finally, we provide numerical simulationsof optimally controlled exchange rate processes and the corresponding evolution of the central bankinventory.
Key words . Currency target zone, currency peg, price impact, central bank intervention, singular stochas-tic control, second-order differential equation with infinite boundary conditions
MSC subject classifications.
Currency pegs, also called currency target zones, describe a regime in which a central bank imposes upperand/or lower limits on the exchange rate of the domestic currency against a foreign currency. There is anabundance of current and historical examples for such currency pegs, and in many cases, the terminationof a currency peg had dramatic consequences. A particularly notorious example is the exit of the Britishpound from the European Exchange Rate Mechanism (ERM) on September 16, 1992, a day known as“Black Wednesday”.The goal of this note is to describe mathematically the situation that a central bank is facing whendefending a currency peg against a macroeconomic trend. More precisely, we consider a two-sided currencytarget zone, in which an exchange rate of two currencies is forced to stay between two thresholds. Twocurrent examples of such target zones are shown in Figure 1.To keep the exchange rate from breaking the prescribed barriers, the central bank has two main methodsof intervention:(a) It can change the interest rates for the domestic currency. ∗ Department of Mathematics, Imperial College London, SW7 2AZ London, United Kingdom. E-mail: [email protected] ∗∗ Department of Statistics and Actuarial Science, University of Waterloo, ON, N2L 3G1, Canada. E-mail:[email protected] †† Department of Statistics and Actuarial Science, University of Waterloo, ON, N2L 3G1, Canada. E-mail:[email protected] ‡‡ School of Management, Shandong University, Jinan 250100, China, and Department of Statistics and Actuarial Science,University of Waterloo, ON, N2L 3G1, Canada. E-mail: [email protected] a r X i v : . [ q -f i n . T R ] A ug
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Out[ ! ]= Figure 1: Left-hand panel: Since 1 January 1999, the target zone of the EUR/DKK currency pair is definedthrough the European Exchange Rate Mechanism II (ERM II) as a ± .
25% band around a central rate of7.46038. Right-hand panel: The USD/HKD currency peg has been in place since 1983. Since 18 May 2005,its target zone is defined through the band from 7.75 to 7.85. Note that the EUR/DKK exchange ratestays well away from the boundaries of the target zone, which are indicated by two horizontal gray lines,while the USD/HKD exchange rate spends a significant amount of time in the vicinity of both thresholds.The dotted line indicates Hong Kong’s foreign exchange reserves, which increased from USD 122 billion inMay 2005 to USD 439 billion in September 2019.(b) It can trade domestic versus foreign currency and thereby generate permanent price impact.The instrument (a) is often fraught with political tensions and might have side effects on the economyas a whole. For instance, it was used only once during the entire duration of the EUR/CHF currencypeg. It may also not be very effective, as is illustrated by the fact that the dramatic announcement ofincreasing the British interest rates to the level of 15% could not prevent the “Black Wednesday” [6]. Forthis reason, we will ignore the possible use of interest rate changes and focus instead on instrument (b), thegeneration of permanent price impact through trading foreign against domestic currency. This instrumenthas the side effect of accumulating (long or short) inventory in the foreign currency. Historical examplesof failed target zones illustrate that this inventory can become problematic, in particular when there isan adverse macroeconomic trend in the market. For instance, it was estimated in [6] that immediatelybefore the British exit from the ERM, “40 per cent of Britain’s foreign exchange reserves were spent infrenetic trading”. Another example is the more recent EUR/CHF target zone: In its article Why the Swissunpegged the franc , published on January 18, 2015, The Economist wrote:Thanks to this policy, by 2014 the SNB had amassed about $480 billion-worth of foreign cur-rency, a sum equal to about 70% of Swiss GDP.Following the abandonment of the currency peg on January 15, 2015, the inventory of the Swiss Nationalbank (SNB) lost about CHF 78 billion of its value, which is about 12% of the Swiss GDP. These figures aretaken from Lleo and Ziemba [13], where one can also find the following quotation regarding the inventoryof the Danish central bank accumulated through enforcing the EUR/DKK target zone as shown in Figure1: In an effort to defend the peg, the central bank’s currency reserves have soared between two-thirds and more than 100% in recent months. They now amount to more than USD 110 billion,which is about one third of Denmark’s 2014 GDP. On December 18, 2014, and thus less than one month before the termination of the currency peg, the Swiss NationalBank decreased its 3-months Libor target range from [0 . , . − . , − .
2o model permanent price impact in a pegged currency market, we will use a continuous-time marketimpact model of Almgren–Chriss-type, in which the exchange rate is a diffusion process controlled by theprice impact generated by the central bank’s strategy. The objective of the central bank is to enforce thetarget zone through a strategy that minimizes the inventory in foreign currency. Within our model thisobjective is then formulated as a stochastic control problem with random time horizon. We transform thecorresponding Hamilton–Jacobi–Bellman equation into a certain second-order ordinary differential equationwith infinite boundary conditions. This boundary value problem is a special case of a class of elliptic partialdifferential equations with infinite boundary values for which classical solutions were obtained by Lasryand Lions [12]. Our main result then states that, after reverting the transformation, this classical solutionprovides indeed the solution to our optimal control problem. Finally, we provide numerical simulationsof the optimally controlled exchange rate processes and the corresponding evolution of the central bankinventory. We illustrate that, depending on the model parameters, the controlled exchange rate processcan be both of the type of the EUR/DKK and the USD/HKD exchange rates as shown in Figure 1.The approach we exploit to defend the target zone is different from most of the literature. Many papersin economics assume that the target zone is defended with infinitesimal interventions at its edges, and nointerventions take place when the exchange rate is strictly inside the band. This stream of literaturetypically considers equilibrium models and focuses on the study of the exchange rate behavior; see, e.g.,[11], the survey [17], and the references therein. Some literature adopts the quasi-variational inequalitiesmethod to study the optimal impulse control (i.e., the optimal times and the optimal sizes of interventions)to defend a target zone of exchange rate (e.g., [9] and [10]). These papers commonly follow [11] and describethe dynamics of the exchange rates by a “fundamental” process plus a term proportional to the percentagechange in the exchange rate. They study the optimal interventions on the fundamental process to retainthe exchange rates within an exogenously given band at the minimum total transaction cost (including afixed fee and proportional cost for each intervention). Some more recent literature [4, 5, 14] does not takethe target zone as exogenously given, and instead assumes the central bank aims at keeping the exchangerate as close as possible to a given target. In [14], the exchange rate is controlled both by interventionsin the foreign exchange market (via impulse controls) and by the determination of the domestic interestrates (via stochastic controls). The paper establishes a sufficient condition for a strategy to minimize thesum of (i) the cost due to the deviations of the exchange rates and the domestic interest rates from theirtarget levels and (ii) the cost due to transaction on the foreign exchange market. By assuming a quadraticform for the deviation costs and a linear form for the transaction cost in the model of [14], [5] derives aclosed-form solution which explicitly indicates that it is optimal to maintain the exchange rate within afinite band. Finally, in [16], a Stackelberg equilibrium between a speculative investor and the central bankmaintaining a one-sided target zone is established.
Our goal is to set up a model in which a central bank intervenes in a currency market so as to keepthe exchange rate within the domain D = ( β − , β + ), where β − < β + are two finite barriers. To thisend, the exchange rate will be modeled as a controlled diffusion process X x,u = { X x,u ( t ) } t ≥ , with thecontrol u describing the intervention strategy of the central bank and x denoting the starting point. Theintervention of the central bank will occur via the generation of permanent price impact through buying orselling the foreign currency. That is, buying one lot of the foreign currency will increase the exchange rateby a factor γ >
0, whereas selling one lot decreases the exchange rate by the same amount. Specifically,let B = { B ( t ) } t ≥ be a standard Brownian motion on the probability space (Ω , F , P ) and let {F t } t ≥ be right-continuous filtration generated by B . Following the continuous-time version [1] of the standardAlmgren–Chriss model [2], we assume that for a given constant X (0) = x ∈ D , the dynamics of X x,u aregiven by X x,u ( t ) = x + σB ( t ) + (cid:90) t b ( X x,u ( r )) dr + γ (cid:90) t u ( r ) dr, t ≥ . (2.1)3ere, σ > b ( · ) describes a macro-economic trend of the exchange rate, and u is a progressively measurable stochastic control process such that (cid:82) t ( u ( r )) dr < ∞ P -a.s. for all t ≥ X x,u satisfying X x,u ( t ) ∈ D for all t ≥
0. Any such control process u will be called admissible , and U will denote the setof all admissible control processes.The drift b ( · ) describes a macro-economic tendency of investors to buy or sell the domestic currency.For instance, during the European sovereign debt crisis, such a trend resulted from the desire of many euroinvestors to seek a safe haven in the Swiss franc. As a response, the Swiss National Bank introduced alower currency peg of 1.20 on the EUR/CHF exchange rate, which held from 2011 to 2015. During thattime, the Swiss National Bank had to defend the currency peg through purchasing large amounts of foreigncurrency. In the sequel, it will make sense to think of b ( · ) as being negative, although this assumptionis not necessary from a mathematical perspective. On a technical level, we will assume that b belongs to C ( D ) and that both b and its derivative b (cid:48) are bounded. A few additional assumptions will be formulatedlater.We assume that the central bank has a random planning horizon τ . This planning horizon will bereached when the currency peg is terminated, modified, or when the macroeconomic trend b changes. Sucha change in the macroeconomic drift can be observed in the USD/HKD target zone in the right-hand panelof Figure 1: For more than a decade, the Monetary Authority of Hong Kong had only to defend the lowerbarrier of their target zone; in 2018, however, the emergence of the US-China trade war in 2018 changedthat picture, and since then, the upper barrier has to be defended.Maintaining the target zone is associated with certain costs for the central bank. As discussed in theintroduction, one of the main concerns is the accumulation of large inventory in the foreign currency,which presents a large risk in the event of a break-up of the target zone. The discounted total inventoryaccumulated until the random time τ equals (cid:90) τ u ( t ) e − ρt dt, where ρ > η (cid:90) τ ( u ( t )) e − ρt dt, where η > J ( x, u ) = E (cid:20) (cid:90) τ (cid:0) u ( t ) + η ( u ( t )) (cid:1) e − ρt dt (cid:21) (2.2)over u ∈ U . It is reasonable to assume that τ is an independent exponential time with parameter θ > J ( x, u ) = E (cid:20) (cid:90) ∞ (cid:0) u ( t ) + η ( u ( t )) (cid:1) e − ρt { τ>t } dt (cid:21) = E (cid:20) (cid:90) ∞ (cid:0) u ( t ) + η ( u ( t )) (cid:1) e − λt dt (cid:21) , (2.3)where λ := ρ + θ . Finally, the value function is denoted by V ( x ) = inf u ∈U J ( x, u ) , x ∈ D. (2.4) Remark 2.1.
In the financial interpretation of this cost functional, we use the above-mentioned idea thatthe macro-economic trend b ( · ) should be negative, so that it is primarily the lower barrier, β − , that must4e defended by the central bank. Therefore, it is the long inventory in the foreign currency, which isproblematic. If, as during the British exit from the ERM, the primary focus is on the upper barrier, β + ,then the short inventory in the foreign currency is a concern, and we can simply replace the term u ( t ) with − u ( t ). Clearly, both scenarios are mathematically equivalent.The usual heuristic arguments suggest that V should satisfy the following Hamilton–Jacobi–Bellman(HJB) equation: inf u ∈ R (cid:26) V (cid:48) ( x )( b ( x ) + γu ) + 12 σ V (cid:48)(cid:48) ( x ) + u + ηu − λV ( x ) (cid:27) = 0 , x ∈ D. (2.5)By computing, at least informally, the infimum over u , we obtain the following reduced form of the HJBequation: 12 σ V (cid:48)(cid:48) ( x ) + V (cid:48) ( x ) b ( x ) − λV ( x ) − ( γV (cid:48) ( x ) + 1) η = 0 , x ∈ D. (2.6)Moreover, the minimizer provides a candidate for the optimal Markovian strategy, (cid:98) u ( x ) = − γV (cid:48) ( x ) + 12 η , x ∈ D. (2.7)In addition to (2.6), we need to specify boundary values of V at ∂D = { β − , β + } . Since the goal of thecentral bank is to keep the exchange rate within D , any value outside D should receive an infinite penalty,which amounts to the singular boundary condition V ( x ) −→ + ∞ as x → ∂D . (2.8)Since D is a bounded domain, the boundary condition (2.8) will imply that the candidate strategy (cid:98) u from(2.7) will satisfy (cid:98) u ( x ) → + ∞ as x ↓ β − and (cid:98) u ( x ) → −∞ as x ↑ β + . Since D is a bounded domain, theboundary condition (2.8) implies that V (cid:48) ( x ) → −∞ as x ↓ β − and V (cid:48) ( x ) → + ∞ as x ↑ β + , which yieldsthat the candidate strategy (cid:98) u from (2.7) satisfies (cid:98) u ( x ) → + ∞ as x ↓ β − and (cid:98) u ( x ) → −∞ as x ↑ β + . Thisproperty of (cid:98) u is natural, because only unbounded controls u will be able to keep the controlled diffusion X x,u ( t ) inside the target zone D for all t ≥
0. Now we can formulate our main result.
Theorem 2.2.
Suppose that b ( · ) is bounded, twice continuously differentiable, and Lipschitz continuous.Then: (a) The singular boundary value problem (2.6) , (2.8) admits a unique classical solution V . Moreover, atthe boundary, the solution blows up like the logarithmic distance to ∂D , i.e., lim ε ↓ V ( β ± ∓ ε ) − log ε = 2 σ ηγ . (2.9)(b) The function V from part (a) is equal to the value function (2.4) . Moreover, the optimal control isMarkovian and given in feedback form by u ∗ ( t ) = − γV (cid:48) ( X x,u ∗ ( t )) + 12 η . (2.10)Part (a) of Theorem 2.2 is based on a transformation of the singular boundary value problem (2.6),(2.8) into a related equation for which existence and uniqueness of classical solutions were studied by Lasryand Lions [12]. This transformation is summarized in the following proposition. Proposition 2.3.
Under the same assumptions as Theorem 2.2, fix some x ∈ D . A function V solves (2.6) if and only if W ( x ) := γ ησ (cid:16) γV ( x ) + x − ηγ (cid:90) xx b ( y ) dy (cid:17) (2.11) solves − W (cid:48)(cid:48) ( x ) + (cid:0) W (cid:48) ( x ) (cid:1) + 2 λσ W ( x ) = f ( x ) , x ∈ D, (2.12) for f ( x ) = − γσ η b ( x ) + 1 σ b (cid:48) ( x ) + 1 σ ( b ( x )) + γλσ η x − λσ (cid:90) xx b ( y ) dy. (2.13)5 Numerical experiments
A first initial guess for solving the singular boundary value problem (2.6), (2.8) is to replace the singularboundary condition (2.8) with large but finite boundary values. This is indeed possible, as it was shownin [12] that the boundary value problem defined through (2.12) and W ( β ± ) = R admits at least a weaksolution for each R >
0. By Proposition 2.3, this solution can then be transformed into a solution of ouroriginal differential equation (2.6). We found, however, that this method is numerically unstable for large R . Moreover, it does not accurately reflect the blow-up of solutions at the boundary, which is importantfor the requirement that the controlled diffusion stays within the domain D = ( β − , β + ). To overcomethese disadvantages, we used the following method, which provides a numerical solution that retains theboundary behavior of V .Let d ( · ) be any function in C ( D ) that satisfies d ( x ) > x ∈ D , d ( β ± ) = 0, d (cid:48) ( β − ) = +1, and d (cid:48) ( β + ) = −
1. For instance, one can take d ( x ) = 1 β + − β − (cid:0) − β − β + + ( β − + β + ) x − x (cid:1) (3.1)or d ( x ) = β + − β − π sin (cid:16) π ( x − β − ) β + − β − (cid:17) . Then it follows from (2.9) that, when approaching the boundary ∂D , the value function behaves like aconstant times ( − log d ( x )). Hence, if we define U ( x ) := V ( x ) − log d ( x ) , then the equation (2.6) is equivalent to the following equation for U ,4 ηb (cid:0) U dd (cid:48) + U (cid:48) d log d (cid:1) + (cid:0) γU d (cid:48) + γU (cid:48) d log d − d (cid:1) + 2 ησ (cid:0) dd (cid:48) U (cid:48) + U (cid:48)(cid:48) d log d + U (cid:0) dd (cid:48)(cid:48) − ( d (cid:48) ) (cid:1)(cid:1) − ηλU d log d = 0 , (3.2)and the asymptotic boundary behavior (2.9) translates into the following two-point boundary condition, U ( β ± ) = 2 σ ηγ . (3.3)After solving the boundary value problem (3.2) and (3.3) numerically, a numerical solution for the boundaryvalue problem (2.6), (2.8) is then obtained by setting V ( x ) := − U ( x ) log d ( x ) (3.4)and the optimal Markovian control is given by (cid:98) u ( x ) = γ η (cid:16) U ( x ) d (cid:48) ( x ) d ( x ) + U (cid:48) ( x ) log d ( x ) − γ (cid:17) This method clearly retains the blow-up rate (2.9) of V .For solving the auxiliary boundary value problem (3.2) and (3.3) numerically, we chose the function d as in (3.1). To avoid the singularity of the coefficients of the differential equation (3.2) at the boundary,however, we replaced d ( x ) with d ε ( x ) := ε + (1 − ε ) d ( x ) in (3.2), but not in (3.4); in our simulations wetook ε = 0 . NDSolve to numerically solve the correspondingboundary value problem. The corresponding exchange rate and inventory processes can then be simulatedby means of a standard Euler scheme. When doing so, it is important to choose a small step size, so as toavoid that increments can jump over the barriers β − and β + . In our simulations, we took a step size of1 . × − . 6he results of our simulations are shown in Figures 2 and 3. These two figures illustrate that, dependingon the model parameters, our central bank strategies can exhibit the features of both the EUR/DKK andthe USD/HKK exchange rates as shown in Figure 1. That is, in Figure 2, the controlled diffusion stayswell away from the boundary ∂D = { β − , β + } , whereas in Figure 3, the process spends a significant amountin the vicinity of the lower bound β − . Moreover, in Figure 3, the inventory process appears to be closeto the local time of a diffusion reflecting at β − , with essentially linear decay during excursions from thereflecting barrier. This can justify the common use of reflecting diffusions in target zone models; see, e.g.,[3, 15] and the references therein. We mention finally that by changing the model parameters it is alsopossible to obtain less regularly shaped solutions V and in particular non-monotone Markovian controls (cid:98) u .Our choice of the plots in Figures 2 and 3 was motivated by their similarity to the historical EUR/DKKand USD/HKD in Figure 1.Figure 2: Solution V (top left), optimal Markovian control (cid:98) u (bottom left), as well as a sample path ofthe optimally controlled exchange rate, X . ,u ∗ ( t ) (top right), and the corresponding inventory (cid:82) t u ∗ ( s ) ds (bottom right), for 0 ≤ t ≤
1. The parameters are β − = 0, β + = 1, σ = 0 . η = 6, γ = 1, λ = 0 .
5, and b ( x ) = − (1 − x ) / Proof of Proposition 2.3.
By completing the squares, we see that (2.6) is equivalent to12 σ V (cid:48)(cid:48) ( x ) − η (cid:16) γV (cid:48) ( x ) + 1 − ηγ b ( x ) (cid:17) − γ b ( x ) + ηγ ( b ( x )) − λV ( x ) = 0 . (4.1)From the given relation (2.11) between V and W , we get V ( x ) = 1 γ (cid:16) ησ γ W ( x ) − x + 2 ηγ (cid:90) xx b ( y ) dy (cid:17) . Taking the first two derivatives of the right-hand side of the preceding equation and plugging them into(4.1) yields that V satisfies (4.1) if and only if W satisfies σ ηγ W (cid:48)(cid:48) ( x ) − σ ηγ (cid:0) W (cid:48) ( x ) (cid:1) − σ ηλγ W ( x ) = 1 γ b ( x ) − ησ γ b (cid:48) ( x ) − ηγ ( b ( x )) − λγ x + 2 ληγ (cid:90) xx b ( y ) dy. Dividing by − σ η/γ shows that the preceding equation is the same as (2.12).7igure 3: Solution V (top left), optimal Markovian control (cid:98) u (bottom left), as well as a sample path ofthe optimally controlled exchange rate, X . ,u ∗ ( t ) (top right), and the corresponding inventory (cid:82) t u ∗ ( s ) ds (bottom right), for 0 ≤ t ≤
1. The parameters are β − = 0, β + = 1, σ = 0 . η = 0 . γ = 2, λ = 1, and b ( x ) = − (1 − x ). Proof of Theorem 2.2.
Our assumptions imply that the function f defined in (2.13) is bounded and con-tinuously differentiable. Under this condition, the existence of a unique classical solution W to (2.12) withboundary conditions lim ε ↓ W ( β ± ∓ ε )log ε = − x ∈ D . Let V denote the solution of the singular boundary value problem(2.6), (2.8) as provided by part (a) and (cid:98) u ( x ) = − η ( γV (cid:48) ( x ) + 1) so that u ∗ ( t ) = (cid:98) u ( X x,u ∗ ( t )) by (2.10). Weshow first that u ∗ is indeed an admissible control, i.e., u ∗ ∈ U . For δ >
0, let D δ = ( β − + δ, β + − δ ) and τ δ be the first exit time of X x,u ∗ from D δ , where δ should be small enough so that x ∈ D δ . For the easeof notation, we will also write X for X x,u ∗ . Note that V and its derivatives are bounded on D δ . Hence,applying Itˆo’s formula to e − λt V ( X ( t )) and taking expectations gives V ( x ) = E (cid:20) (cid:90) τ δ e − λt (cid:16) λV ( X ( t )) − V (cid:48) ( X ( t ))[ b ( X ( t )) + γ (cid:98) u ( X ( t ))] − σ V (cid:48)(cid:48) ( X ( t )) (cid:17) dt (cid:21) + E [ e − λτ δ V ( X ( τ δ ))] . (4.3)Using the HJB equation (2.5) and the fact that (cid:98) u attains the infimum, we obtain V ( x ) = E (cid:20) (cid:90) τ δ e − λt (cid:0)(cid:98) u ( X ( t )) + η ( (cid:98) u ( X ( t ))) (cid:1) dt (cid:21) + E [ e − λτ δ V ( X ( τ δ ))] . (4.4)The first term on the right-hand side can be estimated as follows, E (cid:20) (cid:90) τ δ e − λt (cid:0)(cid:98) u ( X ( t )) + η ( (cid:98) u ( X ( t ))) (cid:1) dt (cid:21) = E (cid:20) (cid:90) τ δ e − λt η (cid:18)(cid:16) ( (cid:98) u ( X ( t )) + 12 η (cid:17) − η (cid:19) dt (cid:21) ≥ − η E (cid:20) (cid:90) τ δ e − λt dt (cid:21) ≥ − ηλ . E [ e − λτ δ V ( X ( τ δ ))] ≥ (cid:0) V ( β − + δ ) ∧ V ( β + − δ ) (cid:1) E [ e λτ δ ] . Therefore, V ( x ) ≥ − ηλ + (cid:0) V ( β − + δ ) ∧ V ( β + − δ ) (cid:1) E [ e λτ δ ] . Since V ( x ) is finite in D δ and V ( β − + δ ) ∧ V ( β + − δ ) → + ∞ as δ ↓
0, we must have0 = lim δ ↓ E [ e − λτ δ ] = E [ e − λτ ] , where τ := lim δ ↓ τ δ is equal to the exit time of X from D . Therefore, we must have τ = ∞ P -a.s., andso u ∗ is admissible.Now we show that V ( x ) ≥ J ( x, u ∗ ) ≥ inf u ∈U J ( x, u ) . (4.5)The first inequality will yield in particular that J ( x, u ∗ ) is finite. For sufficiently small δ , we have V ( x ) ≥ D \ D δ , and thus, by (4.4), V ( x ) ≥ E (cid:20) (cid:90) τ δ e − λt (cid:0) u ∗ ( t ) + η ( u ∗ ( t )) (cid:1) dt (cid:21) . Since τ δ → + ∞ as δ ↓
0, we obtain V ( x ) ≥ J ( x, u ∗ ) and thus (4.5).In the remainder of the proof, we show V ( x ) ≤ J ( x, u ) , for all u ∈ U , (4.6)which in turn gives the optimality of u ∗ . To this end, we use the fact, established in Step 2 of the proof of[12, Theorem II.1], that for each n ∈ N , the following two-point boundary value problem admits a solution W n in the Sobolev space W , ( D ), − W (cid:48)(cid:48) n ( x ) + (cid:0) W (cid:48) n ( x ) (cid:1) + 2 λσ W n ( x ) = f ( x ) for x ∈ D,W n ( β ± ) = n. (4.7)In our one-dimensional situation, functions in W , ( D ) are almost everywhere differentiable with a square-integrable derivative and hence extend to continuous functions on D = [ β − , β + ]. The same is true oftheir derivatives, so that W , ( D ) ⊂ C ( D ). In [12], it is shown moreover that the functions W n increasepointwise to the continuous function W , so that W n → W uniformly on compact subsets of D .When defining V n ( x ) := 2 ησ γ W n ( x ) − xγ + 2 ηγ (cid:90) xx b ( y ) dy, Proposition 2.3 thus yields a sequence of solutions to (2.6) that belong to W , ( D ) ∩ C ( D ) and thatconverge to V uniformly on compact subsets of D . Moreover, Itˆo’s formula also applies to such functions(see, e.g., [7, Remark 3.2]).We now take u ∈ U such that J ( x, u ) < ∞ , let X ( t ) := X x,u ( t ), apply Itˆo’s formula to e − λt V n ( X ( t )),and take expectations to get V n ( x ) = E (cid:20) e − λτ δ V n ( X ( τ δ )) − (cid:90) τ δ e − λt (cid:16)(cid:0) b ( X ( t )) + γu ( t ) (cid:1) V (cid:48) n ( X ( t )) − λV n ( X ( t )) + σ V (cid:48)(cid:48) n ( X ( t )) (cid:17) dt (cid:21) ≤ E (cid:2) e − λτ δ V n ( X ( τ δ )) (cid:3) + E (cid:20) (cid:90) τ δ e − λt (cid:16) u ( t ) + η (cid:0) u ( t ) (cid:1) (cid:17) dt (cid:21) , P -a.s. X ( t ) ∈ D for all t ≥
0, we must have τ δ → ∞ as δ ↓
0, and so (cid:12)(cid:12) E [ e − λτ δ V n ( X ( τ δ ))] (cid:12)(cid:12) ≤ sup y ∈ D | V n ( y ) | · E [ e − λτ δ ] −→ δ ↓ . Consequently, V n ( x ) ≤ E (cid:20) (cid:90) ∞ e − λt (cid:16) u ( t ) + η (cid:0) u ( t ) (cid:1) (cid:17) dt (cid:21) = J ( x, u ) . Sending n ↑ ∞ , we obtain (4.6). Acknowledgement . E.N. would like to thank the CFM – Imperial Institute of Quantitative Finance whichsupported this research. A.S. and X.X. gratefully acknowledge financial support from the Natural Sciencesand Engineering Research Council of Canada (NSERC) through grant RGPIN-2017-04054. C.W. andX.X. would like to thank the financial support from NSERC through grant RGPIN-2016-04001. X.X. isgrateful to Peng Liu, Peng Luo and Wei Wei for helpful discussions. All authors thank Abel Cadenillas forkindly pointing out the references [4, 5, 9, 10, 14].
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