Appl Math Optim 41:25–50 (2000) DOI: 10.1007/s002459910003

© 2000 Springer-Verlag New York Inc.

Controlling Inflation: The Infinite Horizon Case∗ M. B. Chiarolla1 and U. G. Haussmann2 1 Facolt´ a

di Economia, Universit`a degli Studi di Roma “La Sapienza”, via del Castro Laurenziano n. 9, 00161 Roma, Italy

2 Department

of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2

Communicated by I. Karatzas

Abstract. This paper studies the two-dimensional singular stochastic control problem over an infinite time-interval arising when the Central Bank tries to contain the inflation by acting on the nominal interest rate. It is shown that this problem admits a variational formulation which can be differentiated (in some sense) to lead to a stochastic differential game with stopping times between the conservative and the expansionist tendencies of the Bank. Substantial regularity of the free boundary associated to the differential game is obtained. Existence of an optimal policy is established when the regularity of the free boundary is strengthened slightly, and it is shown that the optimal process is a diffusion reflected at the boundary. Key Words. Central Bank, Inflation, Singular stochastic control, Variational inequality, Stochastic differential game, Free boundary, Two-obstacle problem, Reflected diffusion. AMS Classification.

1.

90A70, 93E20, 93E05, 49J40, 35R35.

Introduction

The issue of modeling the dynamics of the interest rate and the inflation rate due to monetary decision is still fresh in the literature, although it is undeniable that some sort ∗ This work was supported by the Natural Sciences and Engineering Research Council of Canada under Grant 88051.

26

M. B. Chiarolla and U. G. Haussmann

of a relationship between these rates exists. The work of Mishkin [18] contains several general observations about correlations between interest rates, inflation, and money supply growth. Moreover, the Central Banks exploit such a relationship in an effort to strike a balance between raising interest rates to limit inflation and lowering them to stimulate the economy. For example, in 1988 the newly appointed Governor of the Bank of Canada stated in a public lecture [2]: Monetary policy should be conducted so as to achieve a pace of monetary expansion that promotes stability in the value of money. This means pursuing a policy aimed at achieving and maintaining stable prices. In Canada the Bank controls the demand for money by influencing interest rates through intervention in the weekly auction of 91 day T-bills and by trading in the secondary market for T-bills. The Bank rate is then set slightly above the T-bill rate which itself is set at the weekly auction. Reserves are provided to the commercial banks to support this action; hence the monetary base is only controlled passively. This paper proposes a two-dimensional, infinite horizon, stochastic model for the changes operated by the Central Bank on the interest rate X 1 with the aim of containing the inflation rate X 2 . The dynamics of the jointly dependent variables (X 1 , X 2 ) are consistent with some equilibrium asset-pricing models already existing in the literature, e.g., [19]. A control process k ∈ V of locally finite variation entering the dynamics of the interest rate additively represents the action of the Central Bank. Inflation targeting is achieved by minimizing over V a cost functional involving not only the inflation rate X 2 , but also X 1 and k to reflect simultaneously the desire to provide stable interest rates (so large changes are penalized) as well as the desire to stimulate the economy (so interest rates must be kept low). As suggested by the fact that the control is explicitly exerted only in the x1 -direction, we show that this problem can be almost characterized in terms of the partial derivative vx1 of the value function v of the minimization problem. In fact, we prove that vx1 is the value of a two-players, zero-sum stochastic differential game with stopping times but with running cost depending on vx2 . The game is fictitious, between conservative forces aiming to contain inflation and expansionist forces primarily interested in stimulating the economy. This is in agreement with some economists’ conviction (see [21]) that in financial markets with imperfections there is a conflict (a game) between the Central Bank and its Government because of the real effects of monetary policy. We show that the variational formulation of the stochastic differential game corresponds to a “two-obstacle” problem; i.e., a free boundary problem with two boundaries. By methods similar to those in [6] we study the regularity of the free boundary, we prove that it is locally Lipschitz, and that the (x1 , x2 )-plane splits into three regions: the inflation region I, where interest rates are so low as to produce significant inflation, and hence it is optimal for the Bank to increase interest rates; the inflation–output tradeoff region T, where inflation and output growth are in equilibrium, more or less, so it is optimal for the Bank not to act; the deflation region D, where interest rates are so high as to prevent economic expansion, hence the Bank intervenes (possibly at the Government’s request) to lower interest rates. Moreover, the optimal stopping times θˆ1 , θˆ2 are precisely the times of intervention of the Bank to raise and to lower the rates, respectively. (In [8] we have

Controlling Inflation: The Infinite Horizon Case

27

obtained a similar result for the finite horizon case. Work concerning the regularity of the corresponding “moving” (i.e., time dependent) free boundary is in progress.) From the mathematical point of view the problem under study is a Bounded Variation Follower Problem with Infinite Horizon, hence a singular stochastic control problem; in fact, it allows even control processes which (as functions of time) are singular with respect to the Lebesgue measure. Usually the variational formulation of singular control problems gives rise to a free boundary problem, and the optimal process is a diffusion reflected at the free boundary. In the case of control of Brownian motion, a one-dimensional follower problem with infinite horizon was studied by Karatzas [13] (the free boundary being given by two points), whereas a two-dimensional follower problem was completely solved by Soner and Shreve [20]. A d-dimensional follower problem with infinite horizon for the control of a diffusion with constant coefficients was studied by Menaldi and Taksar [17]; they proved the existence of an optimal control by means of compactness, but were unable to obtain the regularity of the corresponding free boundary and hence could not construct the optimal reflected diffusion. The regularity issue was considered by Williams et al. [22] and the authors [5], [6] for the case of constant coefficient dynamics. We extend the results to the case of an affine drift coefficient. Early results on singular stochastic control problems, considered as limits of impulse control problems, can be found in [16]. For the present problem, we do obtain Lipschitz regularity of the free boundary, but we still have unbounded coefficients and some other complications to circumvent in order to construct the optimal diffusion process. This is achieved by means of Girsanov’s well known technique of transformation of probability measures. This paper is organized as follows. In Section 2 we state the control problem, we set the notation, and we recall some properties of the value function (derived utility function) v. In Section 3 a variational inequality satisfied by vx1 is derived by means of penalization and localization. Moreover, this problem is interpreted as a stochastic differential game with stopping times with value vx1 and saddle point (θˆ1 , θˆ2 ), where θˆ1 is the optimal time to increase interest rates in order to contain the inflation and θˆ2 is the optimal time to lower interest rates in order to stimulate the economy. Section 4 contains the analysis of the regularity of the free boundary following a method of Friedman [10], as extended in [6] and [22]. Finally, in Section 5 the optimal control of the original control problem is constructed using ideas to be found in [7]. In [8] we investigated the finite horizon version of this problem. The time dependence of the resulting value function makes this a more difficult problem requiring some different techniques from those used here, see Remark 3.7. In fact we were only able to obtain results equivalent to those of the first three sections of this work.

2.

Statement of the Problem

Consider an economy with a single capital-consumption good and a single technology to transform capital into output as in [19]. Let X 1,t and X 2,t be the nominal (spot) interest rate and expected rate of inflation, respectively, then X t = (X 1,t , X 2,t ) satisfies d X t = (a + bX t ) dt + σ dWt ,

t > 0.

(2.1)

28

M. B. Chiarolla and U. G. Haussmann

This model does not take into consideration the direct influence of the Central Bank on the interest rate. It also allows these rates, in particular the nominal interest rate, to go negative! Nevertheless, as a first step we adopt this model except that we add a control term which reflects the actions of the Bank. In [12] it is shown that the model, when applied to Canadian data from 1983 to 1988, is not unreasonable, i.e. statistical tests checking for nonnormality of the residuals corresponding to 1W are not significant. To be fair, we must add that for other time segments of the data this is not the case. If the inflation rate is ignored, i.e., X 2 = 0, then the model for the interest rate has been used in derivative pricing, see [4]. ¡¢ Let c be a positive constant, let a be a constant vector in R2 and e1 = 10 , let ¡b11 b12 ¢ b = b21 b22 , and let σ be constant 2 × 2 matrices such that α := 12 σ σ ∗ is positive ¡ ¢ definite (∗ denotes transpose). Then X s = XX 1,s is the process starting at time 0 from 2,s 2 x ∈ R and governed by the stochastic differential equation ½ t > 0, d X t = (a + bX t ) dt + σ d Wt + ce1 dkt , (2.2) X 0 = x + ce1 k0 , on some filtered probability space (Ä, F, Ft , P) with the filtration {Ft , t ≥ 0} satisfying the usual conditions, where {Wt , t ≥ 0} is a standard two-dimensional Brownian motion and the control {kt , t ≥ 0} is a real-valued cadlag (i.e., right-continuous with left-limits), Ft -adapted process, almost surely of locally finite variation with k0− := 0. We denote by V the set of all such control processes; we refer to x as “the initial condition.” Remark 2.1. We set X¯ = −b−1 a, i.e., the steady state for the average rates in the absence of intervention by the Bank. For some Canadian data of the 1980s, this turns out to be approximately (0.1, 0.04), see [12]. Now if X˜ = X − X¯ denotes the fluctuations about X¯ , then it satisfies d X˜ t = b X˜ t dt + σ dWt + ce1 dkt , so we see that b11 and b22 are the natural growth rates of the interest and inflation rates (they are usually negative yielding reversion of the mean to X¯ ). b12 represents the influence of the inflation rate on the growth of the interest rate. We expect it to be positive. Similarly b21 represents the natural influence of the interest rate on the growth of the inflation rate. In the long run, this ought to be negative, but at least initially an interest rates rise may have an inflationary effect. This effect is neglected in the model, and certainly the policy of the Central Bank is based (in this model) on the premise that this parameter is negative. Let f (x) = 12 [νx12 + x22 ] for x = (x1 , x2 ) ∈ R2 and ν ≥ 0, then the function f is convex. Let ρ > 0 be a given discount factor such that ρ > 2λb ,

(2.3)

where λb := max{Re(λ(b)): λ(b) eigenvalue of the matrix b} and let δ > 0. To any control process k is associated a cost, namely, ½Z ∞ ¾ Z ∞ −ρt −ρt e f (X t ) dt + δ e d|k|t + δ|k0 | , (2.4) Jx (k) = E 0

0

Controlling Inflation: The Infinite Horizon Case

29

where |k| is the total variation process of k. The term involving X 1 reflects the desire to keep interest rates low so as to stimulate the economy; the term involving X 2 reflects the desire to keep the inflation rate at zero (we could have chosen any constant), and the terms involving k reflect the Bank’s reluctance to make large changes in the rate, i.e., it prefers to provide stable interest rates. The coefficient ν is determined by the Bank to reflect the weight it wishes to give to fighting inflation compared with stimulating the economy. By rescaling c and k we can take δ = 1. The problem is to minimize J and to find the value function (P) v(x) = inf{Jx (k): k ∈ V }. Note that (2.3) implies that Jx (0) < ∞, so v(x) < ∞. ¯ Let p > 1, m ∈ IN, N ∈ IN, and let Q denote an open set in R2 with closure Q, then we set • B N = {x ∈ R2 : |x| < N }; 2 (Q) = the set of all functions u ∈ C 2 (Q) which satisfy a polynomial growth • Cpol condition on Q; i.e., for some constants C > 0 and m ∈ IN, |u(x)| ≤ C(1 + |x|m ),

∀x ∈ Q;

• W (Q) = the space of all functions g which have weak derivatives D α g in L p (Q) for all |α| ≤ m; m; p • Wloc (R2 ) = the space of all functions g belonging to W m; p (Q) for all bounded Q ⊂ R2 ; • Lg(x) = 12 trace[σ σ ∗ D2 g(x)] + (a + bx) · ∇g(x), where D2 g(x) and ∇g(x) are the Hessian matrix and the gradient of g(x), respectively; ¯ = the space of all functions g continuous on Q¯ with continuous partial • H m+µ ( Q) j derivatives Dx g for j ≤ m, whose derivatives of order m, Dxm g, are H¨older continuous with respect to x (exponent µ ∈ (0, 1)); ¯ for all Ä such that • H m+µ (Q) = the set of all functions g belonging to H m+µ (Ä) ¯ ⊂ Q; Ä m+µ ¯ for all bounded • Hloc (R2 ) is the set of all functions g belonging to H m+µ ( Q) Q ⊂ R2 . m; p

The following simple lemma holds: Lemma 2.2. If X t0 is the uncontrolled process starting at time 0 from x, then there exists a positive constant C > 0 independent of x such that ¡ ¢ ª © (2.5) E |X t0 |2 ≤ C(1 + |x|2 ) e2λb t + 1 for every t > 0. y For every k ∈ V and x, y ∈ R2 , if X tx and X t denote the processes controlled by k and starting from x and y, respectively, then E{|X t − X tx |} ≤ |y − x| eλb t , y

(2.6)

y

(2.7)

E{|X t − X tx |2 } ≤ |y − x|2 e2λb t , for every t > 0.

30

M. B. Chiarolla and U. G. Haussmann

Notice that f (x) − f (y) ≤ C(|x| + |y|)|x − y|

(2.8)

for all x, y in R2 . Then the main properties of v(x, t) are obtained as in Theorem 3.2 of [8], but using Lemma 2.2 and (2.3) in place of Lemma 3.1 of [8] to take care of the present infinite horizon setting. Theorem 2.3. There exists a positive constant C such that for every λ ∈ (0, 1) and for all x, x 0 in R2 , |x 0 | ≤ 1, one has 0 ≤ v(x) ≤ C(1 + |x|2 );

(2.9)

|v(x) − v(x + x 0 )| ≤ C(1 + |x| + |x 0 |)|x 0 |;

(2.10)

0 ≤ v(x + λx 0 ) + v(x − λx 0 ) − 2v(x) ≤ Cλ2 .

(2.11)

2;∞ (R2 ), and Hence v is convex, v ∈ Wloc

|vxi (x)| ≤ C(1 + |x|),

|vxi x j (x)| ≤ C,

i, j = 1, 2,

(2.12)

1+µ

almost surely in R2 . In particular, for any µ ∈ (0, 1), v ∈ Hloc (R2 ). Note that the H¨older continuity of v and vxi follows from an embedding theorem concerning the space W 2; p (B N ) with p > 2 (see Theorem 7.26, p. 171, of [11]).

3.

Variational Formulation

We now wish to derive equations which characterize first v and then vx1 . The following heuristic discussion motivates the results. Itˆo’s formula applied to v(X (t))e−ρt for any control k gives, formally, Z t (L − ρ)v(X (s))e−ρs ds Ev(X (t))e−ρt = v(x) + E 0 Z t X [v(X (s)) − v(X (s−))], + E cvx1 (X (s−))e−ρs dk c (s) + E 0

s≥0

where k is the continuous part of k. Assume that Ev(X (t))e−ρt → 0 as t → ∞, so Z ∞ Z ∞ −ρs (−L + ρ)v(X (s))e ds − E cvx1 (X (s−))e−ρs dk c (s) v(x) = E 0 0 X −E [v(X (s)) − v(X (s−))] c

s≥0 ∞

Z

(−L + ρ)v(X (s))e−ρs ds Z ∞ cvx1 (X (s−))e−ρs [d(k c )+ (s) − d(k c )− (s)] −E

=E

0

0

Controlling Inflation: The Infinite Horizon Case

−E

X s≥0 ∞

Z

¢ ¡ cvx1 ( X¯ (s))e−ρs [k + (s) − k + (s−)] − [k − (s) − k − (s−)]

f (X (s)e

≤E

31

−ρs

Z ds + E

0

∞

e−ρs d|k|(s)

0

= Jx (k)

if (−L + ρ)v ≤ f and c|vx1 | ≤ 1. The superscripts ± denote positive and negative variations. Moreover, for the optimal control we must have equality, hence ª © a.e. in R2 . max [−L + ρ]v − f, |vx1 | − c−1 = 0 This is the required equality for v, see (3.14). The following decomposition of R2 plays a major role:   I := {x ∈ R2 : cvx1 (x) = −1}, T := {x ∈ R2 : |cvx1 (x)| < 1},  D := {x ∈ R2 : cvx1 (x) = 1}.

(3.1)

The convexity of v implies that I is to the left of T which again is to the left of D. Moreover, the above implies that [−L+ρ]v− f = 0 on T, hence {[−L+ρ]v− f }x1 = 0 on T. Direct computation shows that on D ∪ I, {[−L + ρ]v − f }x1 x1 = −ν, i.e., {[−L + ρ]v − f }x1 is decreasing along horizontal lines. It follows that {[−L + ρ]v − f }x1 ≥ 0 on I and ≤ 0 on D. This gives  x ∈ T, [−L + (ρ − b11 )]vx1 = b21 vx2 + νx1 , x ∈ D, [−L + (ρ − b11 )]vx1 ≤ b21 vx2 + νx1 ,  [−L + (ρ − b11 )]vx1 ≥ b21 vx2 + νx1 , x ∈ I, i.e., (3.25). The above three inequalities can be written in variational form as A(vx1 , g − vx1 ) ≥ (b21 vx2 + νx1 , g − vx1 ),

∀g ∈ K,

(3.2)

1;2 (R2 ): |g(x)| ≤ c−1 a.e.}. The operator A corresponds to the where K = {g ∈ Wloc variational form of −L + (ρ − b11 ), i.e.,

A(g, h) =

2 X i, j=1

αi j (gxi , h x j ) −

2 X ([a + bx]i gxi , h) + (ρ − b11 )(g, h),

(3.3)

i=1

where (·, ·) represents the inner product in L 2 (R2 ). The global equation (3.2) fails due to lack of integrability of vx1 on R2 , but the local version (3.22) is correct. In order to obtain these results rigorously, we approximate our singular control problem by a classical one, where we know that the Hamilton–Jacobi–Bellman equation characterizes the value function, see [9]. Then we pass to the limit to obtain the equation for v. The obvious choice R t is to replace the bounded variation control k(t) by the absolutely continuous control 0 η(s) ds, and restrict |η(s)| < 1/ε to obtain compactness. Then let ε ↓ 0. An easy calculation shows that the equation satisfied by the value function, vε , of this new problem is 1 ε ˜ − Lv ε + ρv ε + β(|cv x1 | − 1) = f ; ε

(3.4)

32

M. B. Chiarolla and U. G. Haussmann

˜ ) = r + , the positive part of r . Unfortunately, to obtain the equation for vxε we where β(r 1 ˜ We replace it by a require extra differentiability of v ε , hence of the penalty function β. smooth β, see below, but this requires a more complicated approximate control problem, the problem (P ε ) below. Nevertheless, we still obtain (3.4) with β˜ replaced, i.e., (3.13). We now proceed with the rigorous derivation of the results. We use penalization as in [8] and [22]. Let β be a C ∞ (R), convex, nondecreasing function such that β(r ) = 0 for r ≤ 0, β(r ) > 0 for r ∈ (0, 1), and β(r ) = 2r − 1 for r ≥ 1. For every ε > 0 the set ¾ ½ 1 1 U ε = (η, ξ ) ∈ R × [0, ∞): |η|θ − β(θ(θ + 2)) ≤ ξ ≤ for all θ ≥ 0 (3.5) ε ε is convex and compact. Let V ε be the set of all measurable, Ft -adapted processes (η, ξ ): [0, +∞) → U ε , and define the penalized problem © ª (P ε ) v ε (x) = inf Jxε (η, ξ ): (η, ξ ) ∈ V ε , where Jxε (η, ξ )

½Z =E

∞

e

−ρt

¾ [ f (X t ) + |ηt | + ξt ] dt

(3.6)

0

and X t is the diffusion governed by the stochastic differential equation ½ t > 0, d X t = (a + bX t + e1 cηt ) dt + σ d Wt , X 0 = x.

(3.7)

The value function v ε (x) is convex in x since X is affine in η and x and the cost is simultaneously convex in (η, ξ ) and x, i.e.,

Jxε (η, ξ )

ε (λ(η, ξ ) + (1 − λ)(η0 , ξ 0 )) ≤ λJxε (η, ξ ) + (1 − λ)Jyε (η0 , ξ 0 ) Jλx+(1−λ)y

for λ ∈ [0, 1]. As expected (see Theorem 4.1 of [8]), estimates analogous to those of Theorem 2.3 hold for v ε uniformly in ε; that is, Theorem 3.1. There exists a positive constant C such that for every ε > 0, for every λ ∈ (0, 1), and for all x, x 0 in R2 , |x 0 | ≤ 1, one has 0 ≤ v ε (x) ≤ C(1 + |x|2 );

(3.8)

|v ε (x) − v ε (x + x 0 )| ≤ C(1 + |x| + |x 0 |)|x 0 |;

(3.9)

0 ≤ v ε (x + λx 0 ) + v ε (x − λx 0 ) − 2v ε (x) ≤ Cλ2 .

(3.10)

2;∞ (R2 ), and Hence v ε is convex, v ε ∈ Wloc

|vxεi (x)| ≤ C(1 + |x|),

|vxεi x j (x)| ≤ C,

i, j = 1, 2,

(3.11)

almost surely in R2 . In particular v ε ∈ C 1 (R2 ). Moreover, for each initial condition x ∈ R2 , lim v ε (x) = v(x).

ε→0+

(3.12)

Controlling Inflation: The Infinite Horizon Case

33

Proposition 3.2. For every 2 < p < +∞ there exists a sequence of positive numbers {εn }n∈IN with εn → 0 as n → ∞ such that for every N ∈ IN one has uniformly on B¯ N , v εn → v as n → ∞ εn uniformly on B¯ N , vxi → vxi as n → ∞ weakly in L p (B N ). vxεinx j → vxi x j as n → ∞ Proof. This follows from (3.11), the reflexivity of W 2; p (B N ), (3.12), and the fact that the embedding W 2; p (B N ) ,→ C 1 (B N ) is compact if p > 2 (2 being the dimension of our state space). The Hamilton–Jacobi–Bellman equation for the (P ε )-problem is ¢ 1 ¡ − Lv ε + ρv ε + β [cvxε1 ]2 − 1 = f ; ε

(3.13)

then by using Theorem VI-6.1a , Corollary VI-4.1, and arguments as in the proofs of Theorem VI-6.2 and Theorem VI-6.3 of [9], the following result is obtained. Proposition 3.3. The value function v ε of the penalized problem (P ε ) is continuous 2 on R2 and is the unique solution in Cpol (R2 ) of the Hamilton–Jacobi–Bellman equation (3.13). In fact, v ε ∈ Hloc (R2 ) as shown in the proof of Theorem VI-6.2 of [9]. Now standard elliptic bootstrapping arguments allow us to improve the regularity of v ε . 2+µ

Proposition 3.4.

For every ε > 0 and for some µ ∈ (0, 1) one has

v ε ∈ Hloc (R2 ). 4+µ

Proof.

We write (3.13) as

¢ 1 ¡ −Lv ε + ρv ε = f − β [cvxε1 ]2 − 1 , ε 1+µ

then the right-hand side above is in Hloc (R2 ). Therefore by Theorem 6.17 of [11] we obtain v ε ∈ Hloc (R2 ) 3+µ

2+µ

and hence the right-hand side above is in Hloc (R2 ). Another application of Theorem 6.17 of [11] completes the proof.

34

M. B. Chiarolla and U. G. Haussmann

Remark 3.5. Notice that by taking limits in (3.13) (in the sense of Proposition 3.2) we can characterize the value function v by the Hamilton–Jacobi–Bellman equation ( 2;∞ (R2 ); v convex; v ∈ Wloc (3.14) ª © a.e. in R2 . max [−L + ρ]v − f, |vx1 | − c−1 = 0 The proof is similar to that of Theorem 2.11, p. 31, of [5]. Clearly (3.14) is a variational problem in terms of both v and vx1 , and hence not very easy to handle. Instead, the fact that the control problem under study allows the control to be explicitly exerted only in the x1 -direction suggests the possibility of a complete characterization of the problem in terms of the partial derivative vx1 . We then study the properties of vx1 . Lemma 3.6. The function vx1 is in K. Proof. Because of the estimates (2.12) we need only show that a.e. |vx1 | ≤ c−1 . In fact, from (3.13) and (3.11) it follows that ° ° °β([cv ε ]2 − 1)° 2 ≤ Cε, ∀ε > 0, (3.15) x1 L (B ) N

with C independent of ε. Also β is continuous and vxε1 → vx1 pointwise (along a subsequence if necessary) by Proposition 3.2, hence lim β

ε→0

³£

cvxε1

¤2

´ ³£ ´ ¤2 − 1 = β cvx1 − 1 .

Finally, (3.11) implies 0≤β

³£

cvxε1

¤2

´ ³£ ´ ¡ ¤2 ¢ − 1 ≤ 2 cvxε1 − 1 ≤ C 1 + |x|2 ,

so we can pass to the limit in (3.15) to obtain kβ([cvx1 (. )]2 − 1)k2L 2 (B N ) ≤ 0, and hence |vx1 | ≤ c−1 a.e. in B N and hence in R2 . Because of Proposition 3.4, v ε is regular enough to allow us to differentiate (3.13) with respect to x1 ; we then have ´ ¤2 2 ³£ (3.16) − Lvxε1 + (ρ − b11 )vxε1 + β 0 cvxε1 − 1 cvxε1 vxε1 x1 = uˆ ε , ε where we have set uˆ ε := b21 vxε2 + νx1 .

(3.17)

Remark 3.7. As pointed out previously, the finite horizon version of this problem has been studied in [8]. That problem is more difficult due to the time dependence of the value function. Of course there are also similarities. In particular, the approximating or penalized problem (P ε ) is the same except that the horizon is T . Here we have been

Controlling Inflation: The Infinite Horizon Case

35

able to derive fairly easily the convergence of v ε and its derivatives of order one and two to the corresponding expressions for v. In Section 4 of [8], we had to use more complicated methods and could only obtain weaker convergence—the convergence of vxε1 t in particular is the problem. In the remainder of this section we obtain results which are equivalent to those of Section 5 of [8] but again in [8] we had to surmount greater difficulties. Specifically, we had to work with weak variational inequalities in weighted Sobolev spaces, i.e., an integration over time (and integration by parts) was used to eliminate the time derivative, and integrability as |x| → ∞ was achieved by using a suitable weight. In the present paper we are able to work with a strong variational inequality directly, Theorem 3.9, using localization to overcome the integrability problem. Both Theorem 3.9 here and Theorem 5.2 of [8] are based on ideas in Theorem 4.5 of [22]. Whereas in the infinite horizon case we can easily convert the strong local variational inequality into a pointwise inequality, see Theorem 3.11, much technical analysis [8, Sections 7 and 8] is required in the finite horizon case to prove uniqueness of the solution, and the equivalence of the weak, the strong, and the pointwise variational inequalities. We continue now with the analysis of the infinite horizon problem. Integrating by parts in the equation for vxε1 and passing to the limit will give a variational inequality for 1;2 (R2 ) and the operator defined in vx1 involving the operator A. Unfortunately vx1 ∈ Wloc (3.3) is not well defined on that set, so we proceed locally. Let Q be a bounded, open set in R2 with smooth boundary. Now restrict attention to L 2 (Q), W 1;2 (Q), and W◦1;2 (Q), the elements of W 1,2 (Q) whose trace on the boundary of Q is zero. The inner product in L 2 (Q) is denoted by (·, ·) Q . Now define the analog of A on W 1;2 (Q) by A Q (g, h) =

2 X

αi j (gxi , h x j ) Q

i, j=1

−

2 X ([a + bx]i gxi , h) Q + (ρ − b11 )(g, h) Q .

(3.18)

i=1

Definition 3.8. An operator A: W 1;2 (Q) × W 1;2 (Q) → R is weakly coercive if there exists η > 0 and λ ≥ 0 such that A(g, g) ≥ ηkgk2W 1;2 (Q) − λkgk2L 2 (Q) ,

∀g ∈ V.

(3.19)

If λ = 0, then A is coercive. Since σ σ ∗ is positive definite, it is easy to see that there exists η > 0 such that 2 X i, j=1

αi j ξi ξ j ≥ η

2 X

ξi2 ,

∀ξ1 , ξ2 ∈ R,

(3.20)

i=1

and hence A Q is weakly coercive. Let ψ ∈ C◦∞ be a nonnegative smooth function with compact support in Q and let Kψ := ψK. Let Vψε := vxε1 ψ and uˆ ε := b21 vxε2 + νx1 ,

36

M. B. Chiarolla and U. G. Haussmann

Uˆ ε := uˆ ε ψ − vxε1 Lψ − ∇vxε1 · α∇ψ, Uˆ := uψ ˆ − vx1 Lψ − ∇vx1 · α∇ψ. Then Vψε ∈ Kψ := ψK and

´ ¤2 2 ³£ − LVψε + (ρ − b11 )Vψε + β 0 cvxε1 − 1 cVψε vxε1 x1 = Uˆ ε , ε

∀x ∈ R2 .

(3.21)

Theorem 3.9. The function u = vx1 ψ solves the variational inequality A Q (u, g − u) ≥ (Uˆ , g − u),

∀g ∈ Kψ .

(3.22)

Moreover, if ρ > b11 then the solution is unique. Proof. The proof of the first part is similar to the proof of Theorem 5.2 of [8]. (It was originally suggested by one in [22].) Let g ∈ Kψ and take the inner product of g − Vψε with (3.21) in L 2 (Q); then an integration by parts gives ³£ ¤2 ´ 2 A Q (Vψε , g−Vψε )+ (β 0 cvxε1 −1 cVψε vxε1 x1 , g−Vψε ) Q = (Uˆ ε , g−Vψε ) Q . (3.23) ε Now we proceed to take limits in (3.23) as ε → 0; by Proposition 3.2, as ε → 0 (along a subsequence) we have A Q (Vψε , g) → A Q (u, g), (Uˆ ε , g − Vψε ) Q → (Uˆ , g − u) Q , and [A Q − A◦ ](Vψε , Vψε ) → [A Q − A◦ ](u, u), P where A◦ (g, g) = i, j αi j (gxi , gx j ) Q . Finally, since (A◦ (g, g))1/2 defines a norm on W◦1,2 (Q) and vxε1 → vx1 (hence Vψε → u) pointwise, we have lim inf A◦ (Vψε , Vψε ) ≥ A◦ (u, u), ε→0

hence lim inf A Q (Vψε , Vψε ) ≥ A Q (u, u). ε→0

It remains only to show that the penalty term in (3.23) is nonpositive. Write g = gψ ˆ with gˆ ∈ K. We examine two cases. On {x: gˆ 2 (x) ≥ vxε1 (x)2 } we have [cvxε1 (x)]2 ≤ 1 since gˆ ∈ K, hence

Case 1. β0

³£

¤2

cvxε1 (x)

´ − 1 = 0.

Controlling Inflation: The Infinite Horizon Case

Case 2.

37

On {x: gˆ 2 (x) < (vxε1 (x))2 } it follows that

 ³ ´ ³ ´2 ³ ´2  2V ε (x) g(x) − V ε (x) ≤ V ε (x) + g 2 (x) − 2 V ε (x) < 0, ψ ψ ψ ψ ´ ³  β 0 £cv ε (x)¤2 − 1 ≥ 0; x1 but vxε1 x1 ≥ 0 by convexity, hence ´ ´ 2 ³ 0 ³£ ε ¤2 β cvx1 − 1 cVψε vxε1 x1 , g − Vψε ≤ 0, ε and (3.22) follows. If ρ > b11 , then the uniqueness is obtained from Chapter 3, Theorem 1.5, p. 199 of [1]. Proposition 3.10. kψvx1 kW 2; p (Q)

For every p ≥ 2 there exists a constant C Q such that ¡ ¢ ≤ C Q 1 + kvx1 k L p (Q) + kvx2 k L p (Q) .

(3.24)

2; p

Hence vx1 ∈ Wloc (R2 ). Proof. For λ > 0 we define A Q,λ (g, h) := A Q (g, h) + λ(g, h) Q , and we choose λ so large as to make the operator A Q,λ coercive on W 1; p (Q). Since u = ψvx1 solves A Q,λ (u, g − u) ≥ (Uˆ + λu, g − u),

∀g ∈ Kψ ,

then (3.24) follows from Corollary I.1, p. 15, of [3] and the boundedness of ∇vx1 . Since ψ and Q are arbitrary, the last conclusion follows from (3.24) and (2.12). The local W 2; p -regularity of vx1 allows us to write (3.22) in the form ¤ £ ¤ ¢ ¡£ ∀g ∈ K, [−L + (ρ − b11 )]vx1 − uˆ ψ, g − vx1 ψ Q ≥ 0, and in fact this holds for all g ∈ Kˆ := {g ∈ L 2loc (R2 ): |cg(x)| ≤ 1}. We can now obtain a pointwise variational inequality for vx1 . Recall the definition of T, I, D from (3.1). Theorem 3.11. Assume that ρ > b11 . Then the function vx1 is the unique solution of the pointwise variational inequality  2; p a.e. in R2 ; v ∈ Wloc (R2 ) and − 1 ≤ cvx1 ≤ +1   x1 a.e. in T; [−L + (ρ − b11 )]vx1 = uˆ (3.25) a.e. in D;  [−L + (ρ − b11 )]vx1 ≤ uˆ  [−L + (ρ − b11 )]vx1 ≥ uˆ a.e. in I.

38

M. B. Chiarolla and U. G. Haussmann

Proof. We take g = vx1 + ψˆ with suitable ψˆ and write Lˆ := [−L + (ρ − b11 )]vx1 − u. ˆ Now let ε > 0 and Tε := {x ∈ R2 : c|vx1 | < 1 − ε}. With N > 0 and 0 < η < ε/(cN ), 1 S is the indicator of the set S. Then g ∈ Kˆ and set ψˆ := −η Lˆ 11Tε 11{| L|
ˆ − vx1 )ψ 2 d x = −η L(g

0≤ Q

Z R2

Lˆ 2 ψ 2 11Tε 11{| L|
i.e. Z R2

Lˆ 2 ψ 2 11Tε 11{| L|
p Since Lˆ ∈ L loc (Q) then as N ↑ ∞, 11{| L|
Z

Lˆ 2 ψ 2 d x = 0, T

i.e., Lˆ = 0 a.e. and we have (3.25) on T. ˆ ˆ+ Now let ψˆ := −η Lˆ + 11D 11{| L|
Proof. We prove the required regularity in {vx1 < c−1 }; similar arguments apply to {vx1 > −c−1 }. Let Q 0 and Q be bounded open sets in {vx1 < c−1 } with Q 0 ⊂ Q; then it follows from Theorem 3.11 that vx1 + c−1 solves  a.e. in Q; vx1 +c−1 ∈ W 2; p (Q) and vx1 +c−1 ≥ 0 a.e. in [−L+(ρ −b11 )](vx1 +c−1 ) = uˆ +(ρ −b11 )c−1  a.e. in [−L+(ρ −b11 )](vx1 +c−1 ) ≥ uˆ +(ρ −b11 )c−1

Q ∩ {vx1 +c−1 > 0}; Q ∩ {vx1 +c−1 = 0}.

Therefore, if ψ ∈ C◦∞ is a smooth function with compact support in Q such that 0 ≤ ψ ≤ 1 in Q and ψ = 1 in Q 0 , the function ψ(vx1 + c−1 ) solves  2; p ψ(vx1 + c−1 ) ∈ W0 (Q) and    2  X   − αi j (ψ(vx1 + c−1 ))xi x j = f ∗

ψ(vx1 + c−1 ) ≥ 0 a.e. in

a.e. in Q;

Q ∩ {ψ(vx1 + c−1 ) > 0};

i, j=1

  2 X     αi j (ψ(vx1 + c−1 ))xi x j ≥ f ∗ −  i, j=1

a.e. in

Q ∩ {ψ(vx1 + c−1 ) = 0};

(3.26)

Controlling Inflation: The Infinite Horizon Case

39

with f ∗ := −

2 X

αi j ψxi x j (vx1 + c−1 ) − 2

i, j=1

+

2 X

αi j ψxi vx1 x j

i, j=1

2 X [a + bx]i vx1 xi ψ − [(ρ − b11 )vx1 − u]ψ; ˆ i=1

1; p ¯ for p > 2 by Sobolev embedding (see so f ∗ ∈ W0 (Q), and hence f ∗ ∈ H 1−2/ p ( Q) p. 164 of [11]). Now problem (3.26) verifies all the conditions of Theorem 4.1, p. 31, of [10] (with ϕ = 0, f = f ∗ , and g = 0 in the notation of [10]), from which it follows that 2;∞ ({vx1 < ψ(vx1 + c−1 ) ∈ W 2;∞ (Q) and hence vx1 + c−1 ∈ W 2;∞ (Q 0 ). Thus vx1 is in Wloc −1 c }).

We now observe that (3.25) is also satisfied by the value of a two-player zero-sum stochastic differential game. In fact, for every x ∈ R2 consider the control-free diffusion process starting at time 0 from x; i.e., Z t Z t 0 0 (a + bX s ) ds + σ d Ws , ∀t ≥ 0, Xt = x + 0

0

and denote by S the collection of all stopping times relative to the underlying filtration {Ft }t≥0 . Let θ1 , θ2 be stopping times in S; if θ1 and θ2 play the role of strategies, then we define the evaluation function of the game by setting ½Z θ1 ∧θ2 u(X ˆ t0 )e−(ρ−b11 )t dt − c−1 e−(ρ−b11 )θ1 11 θ1 ≤θ2 G x (θ1 , θ2 ) = E 0

¾

+ c−1 e−(ρ−b11 )θ2 11θ2 <θ1 .

θ1 <∞

(3.27)

We prove that the differential game has a solution. Theorem 3.13. Assume that ρ > b11 and define ½ θˆ1 = inf{t ≥ 0: X t0 ∈ I}, θˆ2 = inf{t ≥ 0: X t0 ∈ D}.

(3.28)

Then for every initial condition x, the evaluation function G x has a saddle point at (θˆ1 , θˆ2 ) with value vx1 (x); i.e., ½ ∀θ1 , θ2 ∈ S, G x (θ1 , θˆ2 ) ≤ G x (θˆ1 , θˆ2 ) ≤ G x (θˆ1 , θ2 ), (3.29) vx1 (x) = G x (θˆ1 , θˆ2 ), and vx1 (x) = inf sup G x (θ1 , θ2 ) = sup inf G x (θ1 , θ2 ). θ2 ∈S θ1 ∈S

θ1 ∈S θ2 ∈S

(3.30)

40

M. B. Chiarolla and U. G. Haussmann

Proof. Clearly (3.30) follows directly from (3.29). On the other hand, since vx1 is in 2; p Wloc (R2 ), (3.29) can be proved as in Theorem 2.3 of [8], by applying a generalized Itˆo formula (see Theorem 8.5, p. 185, of Chapter 2 of [1]) to e−(ρ−b11 )t vx1 (X t0 ). In essence we have moved from the control problem (P) to the variational inequality satisfied by its value function v, we have differentiated to find a a variational inequality for vx1 , and then we have found the corresponding control problem, which happens to be a differential game. Remark 3.14. The result has an interesting interpretation. First, observe that b11 is the natural growth rate of the interest rate (usually it is negative), so that ρ − b11 represents a discount rate rescaled to factor out this natural growth. From (3.27) we see that the game has a running cost which is a measure of the change of f with respect to the interest rate ( f x1 ) plus the indirect change in v produced by the interest rate acting through the inflation rate (vx2 b21 —if the interest rate did not affect the inflation rate, then this term would be zero), and a penalty of c−1 to whichever player acts to terminate the game. In economic terms, this terminal cost reflects the cost of exercising control. The solution then represents a dynamic equilibrium between the need to contain inflation and the desire to expand the economy. In the inflation region I, it is optimal for the conservative forces of the Bank to intervene by increasing interest rates; in the inflation–output tradeoff region T, the conservative and the expansionist forces of the Bank find an equilibrium and hence it is optimal for both to do nothing; finally, in the deflation region D it is optimal for the expansionist forces to intervene by lowering interest rates.

4.

The Free Boundary

For any set G we denote by ∂G its boundary, and we define • ∂ I := ∂I, • ∂ D := ∂D. The pointwise variational problem (3.25) given in Theorem 3.11 is in fact the variational formulation of a “two-obstacle problem,” the two obstacles being the functions h I (x) = −c−1 and h D (x) = +c−1 ; i.e., (3.25) is a free boundary problem with two boundaries, ∂ I and ∂ D . As expected in singular stochastic control, the optimal process of the original control problem should be a diffusion reflected at the boundaries if these are regular enough to guarantee the existence of such a diffusion. We aim to prove that ∂ I and ∂ D are locally Lipschitz. Proposition 4.1. Assume that ρ > b11 and ν > 0. Then ½ on ∂ I , cuˆ < −(ρ − b11 ) on ∂ D . cuˆ > +(ρ − b11 )

(4.1)

Controlling Inflation: The Infinite Horizon Case

41

Proof. We establish the first inequality; the other follows similarly. Let Ä be a bounded ˆ If w = cvx1 , subset of I ∪ T which intersects ∂ I . Let L = −L + (ρ − b11 ) and F = cu. then it follows from (3.25) that Lw − F ≥ 0,

w ≥ −1,

(Lw − F)(w + 1) = 0

a.e. on Ä.

Recall that vx1 ∈ H 1+1 (Ä) by Corollary 3.12. Since vx1 attains its minimum value in I, it follows that ∇vx1 = 0 on I, hence also on ∂ I . Thus, for xo ∈ ∂ I , ∇[−F + L(−1)](xo ) = (−ν, −vx2 x2 (xo )) 6= 0. Now the proof of Theorem 4.8 of [22] gives the result since −F + L(−1) = −cuˆ − (ρ − b11 ). The only divergence of our case from that in [22] is the presence of the term bx in the drift coefficient. This only plays a role in (4.7) of [22], but it is readily seen that any locally bounded coefficient will preserve (4.7) of [22]. Theorem 4.2. Assume that ρ > b11 and ν > 0. Then the boundaries ∂ I and ∂ D are locally Lipschitz; that is, • for every x I = (x1I , x2I ) ∈ ∂ I there exists a neighborhood U = U1 × U2 of x I such that ∂ I ∩ U is given by x1 = ψ I (x2 ),

∀x2 ∈ U2 ,

with ψ I Lipschitz continuous; • for every x D = (x1D , x2D ) ∈ ∂ D there exists a neighborhood U = U1 × U2 of x D such that ∂ D ∩ U is given by x1 = ψ D (x2 ),

∀x2 ∈ U2 ,

with ψ D Lipschitz continuous. Proof. The proof is similar to that of Theorem 6.2, p. 716, of [6] but we must allow for the dependence of uˆ on vx2 and not only on f ; however, the method still works thanks to the boundedness of vxi x j and the condition ν > 0. Notice that T is open since vx1 is continuous. From (3.14) it follows that [−L + ρ]v = f a.e. in T, so by elliptic regularity (see Theorem 8.10, p. 186, of [11]) we have 4;2 (T), i.e., v ∈ C 2 (T) by Sobolev’s embedding theorem (see (7.30), p. 158 of v ∈ Wloc [11]). Hence 4+µ

v ∈ Hloc (T)

(4.2)

by the interior regularity theorem [11, Proposition 6.17, p. 109], and so also 3+µ

vx1 ∈ Hloc (T). Thus (3.25) may be differentiated with respect to x1 in T. ˆ the continuity Now let x I = (x1I , x2I ) ∈ ∂ I . By Proposition 4.1, the continuity of u, of vx1 xi , ν > 0, and ∇vx1 = 0 on ∂ I , we can find an open rectangle U with center at x I

42

M. B. Chiarolla and U. G. Haussmann

and some ε¯ > 0 such that ½ ν + b11 vx1 x1 + 2b21 vx1 x2 > ν/2 cuˆ + ρ − b11 < −¯ε

in U.

Let R > 0 be such that, for some r > 0, © ª O R := (x1 , x2 ): |x2 − x2I | < R, |x1 − x1I | < r ⊂ U, ª © O2R := (x1 , x2 ): |x2 − x2I | < 2R, |x1 − x1I | < r ⊂ U. Let D R = O R ∩ T. For K > 0, F > 0 to be determined later, and |H | ≤ 1, define w := K vx1 x1 + H vx1 x2 − F(cvx1 + 1). Then (

[−L + (ρ − b11 )]w = −F(cuˆ + ρ − b11 ) + K (ν + b11 vx1 x1 + 2b21 vx1 x2 ) a.e. in D2R , + H (b12 vx1 x1 + b22 vx1 x2 + b21 vx2 x2 ) w=0 in ∂ D R ∩ ∂ I ;

hence for F large enough (independently of K > 0 and H, |H | ≤ 1) and some ε > 0 it follows that ½ a.e. in D2R , [−L + (ρ − b11 )]w > ε > 0 (4.3) w=0 in ∂ D R ∩ ∂ I , since vxi x j is bounded, see (2.12). We also have  a.e. in U ∩ T, [−L + (ρ − b11 )]vx1 x1 > ν/2   in U ∩ ∂ I , vx1 x1 = 0   vx1 x1 ≥ 0 2+µ in U, (U ∩ T) ∩ H 0+1 (U ) vx1 x1 ∈ H (the last inequality above being due to the convexity of v). Since ρ − b11 > 0 then the strong maximum principle [11, Theorem 8.19, p. 198] implies vx1 x1 > 0

in U ∩ T.

(4.4)

For δ ∈ (0, 1) let Sδ := {x ∈ T: dist(x, ∂ I ) ≤ δ}. Then (4.4) implies that for each δ there exists K (δ) such that if K > K (δ), then w>0

on

D2R ∩ T\Sδ .

(4.5)

This is (6.12) of [6]. Now by the same arguments as those employed on pp. 178–179, of [10]) or pp. 718– 719 of [6] we can show that there exists δ > 0 such that w>0

on ∂ D R ∩ Sδ ,

(4.6)

see (6.13) of [6]. Then the weak maximum principle (see Theorem 8.1, p. 179, of [11]) applied to (4.3), (4.5), and (4.6) implies that w ≥ 0 in D R , i.e., for any |H | ≤ 1 we have K vx1 x1 + H vx1 x2 ≥ F(cvx1 + 1) > 0

in D R .

Controlling Inflation: The Infinite Horizon Case

43

Hence vx1 increases along lines of slope H/K in D R , for every |H | ≤ 1. Since vx1 = −c−1 in I, then we can conclude that there exists a cone γ + with vertex at 0, angle 2 arctan(1/K ), and with axis the positive x1 -axis such that (γ + + x I ) ∩ O R ⊂ T,

(−γ + + x I ) ∩ O R ⊂ I.

In fact for any x ∈ ∂ I in a sufficiently small neighborhood of x I we have (γ + + x) ∩ O R ⊂ T,

(−γ + + x) ∩ O R ⊂ I.

This suffices to guarantee not only that the portion of ∂ I near x I may be represented by the graph of a Lipschitz function but also that near x I , I must be the subgraph of ψ I (x2 ) := inf{x1 ∈ R: vx1 (x1 , x2 ) > −c−1 }, with ψ I locally Lipschitz continuous since |ψ I (z 2 ) − ψ I (y2 )| ≤K |z 2 − y2 | for every z 2 , y2 near x2I . Similar arguments show the existence of the Lipschitz function ψ D (x2 ) := sup{x1 ∈ R: vx1 (x1 , x2 ) < +c−1 } representing the boundary ∂ D . Remark 4.3. Notice that the cone property obtained in the proof above implies that the boundaries ∂ I and ∂ D cannot contain horizontal segments; moreover, points of the boundaries are of positive Lebesgue density for I and D, respectively. This and Proposition 4.1 allow us to apply a fundamental result of Caffarelli (see Theorem 3.10 and Corollary 3.11, p. 162, of [10]) to obtain even more regularity. That is, ½

5.

ψ I and ψ D are C 1 -functions, vx1 ∈ C 2 (∂ I ∪ T ∪ ∂ D ).

The Optimal Control

Now that we have obtained some smoothness of the free boundary, we no longer need the variational inequality for vx1 ; instead, we can return to that for v, i.e., (3.14). Usually in singular stochastic control the continuous part of the optimal control acts like the local time of the optimal process at the free boundary, and the control acts only in the complement of the region {x: [−L + ρ]v(x) = f (x)} where the dynamic programming equation holds, and it pushes only enough to keep the state process inside this region. Moreover, when the boundary is represented by a function having jumps in the direction of pushing, then the optimal control has matching jumps.

44

M. B. Chiarolla and U. G. Haussmann

In our case the control is exerted only in the x1 -direction and the boundaries ∂ I and ∂ D do not contain any horizontal segment by Remark 4.3, i.e., the functions ψ I and ψ D representing the boundaries do not have jumps (by Theorem 4.2); therefore we can expect the optimal control to be continuous after possibly an initial jump. Intuitively, if the value function v were a C 2 -function, then by (3.14) we could ¯ Also, if we could construct a control extend [−L + ρ]v = f to hold almost surely in T. ¯ with an initial jump to the boundary ∂ I ∩ ∂ D ˆk able to keep the state process Xˆ inside T, ˆ In fact, if there were if necessary, then it would not be hard to show the optimality of k. + − ± ˆ ˆ ˆ such ˆ ˆ a continuous k = k − k in V (k are the positive and negative variations of k) 2 that for every initial condition x ∈ R and for every t > 0, (i) v(x) − v( Xˆ 0 ) = |kˆ0 |, Z 11{cvx ( Xˆ s )=−1} d kˆs+ = kˆt+ , (ii) Z

1

(0,t]

(iii) (0,t]

11{cvx

1

( Xˆ s )=+1}

d kˆs− = kˆt− ,

¯ t ∈ [0, ∞) almost surely, (iv) Xˆ t ∈ T, then the Itˆo formula for semimartingales applied to v( Xˆ t )e−ρt would give v(x) − E|kˆ0 | = Ev( Xˆ 0 ) ½Z t e−ρs [−L + ρ]v( Xˆ s ) ds =E 0 ¾ Z −ρs −ρt ˆ ˆ ˆ e vx1 ( X s− )c d ks + v( X t )e . − (0,t]

Hence also

½Z

t

v(x) ≥ E

e 0

−ρs

f ( Xˆ s ) ds +

Z e (0,t]

−ρs

¾

ˆ s + |kˆ0 | d|k|

by (i)–(iv) and the positivity of v, and by the monotone convergence theorem we would ˆ conclude that v(x) = Jx (k). Recall that the boundaries ∂ I and ∂ D are respectively represented by the C 1 -functions ½ I ψ (x2 ) := inf{x1 ∈ R: cvx1 (x1 , x2 ) > −1}, ψ D (x2 ) := sup{x1 ∈ R: cvx1 (x1 , x2 ) < +1}. Since the above sets are not necessarily bounded from below, respectively from above, for some x2 ∈ R, we may have ½ I ψ (x2 ) := −∞, ψ D (x2 ) := +∞. We now aim to construct a control kˆ with the properties described above. We begin by treating the case with b = 0 and then we use the well known technique of transformation of probability measure due to Girsanov to introduce b. (See [7] for a similar construction for the case b = 0 in a monotone control problem.) Let B be a standard Brownian motion on the filtered probability space (Ä, G, Gt , Q) where the filtration {Gt }

Controlling Inflation: The Infinite Horizon Case

45

is right continuous and G0 contains the null sets of G. The smallest such filtration is the filtration generated by B augmented by the null sets. We denote this filtration by {Ft }. Let Yt = x + at + σ Bt . Define the first exit time of Yt from T: τ1 = τ1I ∧ τ1D , where τ1I (ω) := inf{t ≥ 0: Yt ∈ I}, τ1D (ω) := inf{t ≥ 0: Yt ∈ D}. ˆ For t ≥ τ1 , if ω ∈ {τ1 = τ1I }, For t < τ1 let kˆt = 0 and define Xˆ t = Yt , i.e., Xˆ = Y +e1 ck. then define kˆ from ( + kˆt = kˆτ+1 − + sup [ψ I (Y2,s ) − (Y1,s + ckˆτ1 − )]+ /c, τ1 ≤s≤t (5.1) kˆt− = 0 for all t ∈ [τ1 (ω), τ2 (ω)), with τ2 (ω) given by τ2 (ω) = τ2D (ω) := inf{t > τ1 (ω): Xˆ t− ∈ ∂ D }, where Xˆ t = Yt + e1 ckˆt . Similarly if ω ∈ {τ1 = τ1D }, then (

kˆt+ kˆt−

= =

0, kˆτ−1 − + sup [(Y1,s + ckˆτ1 − ) − ψ D (Y2,s )]+ /c

(5.2)

τ1 ≤s≤t

for all t ∈ [τ1 (ω), τ2 (ω)), with τ2 (ω) given by τ2 (ω) = τ2I (ω) := inf{t > τ1 (ω): Xˆ t− ∈ ∂ I }. Note that in both cases the sup above is a max since ψ I and ψ D are continuous. Moreover, ¯ since Y0 = x, then kˆ0 = 0 if and only if x ∈ T. Now for ω ∈ {τ2 = τ2D } we set (

kˆt+ = kˆτ+2 − , kˆt− = kˆτ−2 − + max [(Y1,s + ckˆτ2 − ) − ψ D (Y2,s )]+ /c τ2 ≤s≤t

for all t ∈ [τ2 (ω), τ3 (ω)), with τ3 (ω) given by τ3 (ω) = τ3I (ω) := inf{t > τ2 (ω): Xˆ t− ∈ ∂ I }.

46

M. B. Chiarolla and U. G. Haussmann

Observe that the definition (5.1) is also valid at t = τ2 and that kˆ is continuous for t > 0. By induction we can define kˆ by ( + kˆt − kˆτ+n − = max [ψ I (Y2,s ) − (Y1,s + ckˆτn − )]+ /c τn ≤s≤t if τn = τnI , (5.3) kˆt− − kˆτ−n − = 0 ( + kˆt − kˆτ+n − = 0 if τn = τnD (5.4) kˆt− − kˆτ−n − = max [(Y1,s + ckˆτn − ) − ψ D (Y2,s )]+ /c τn ≤s≤t

ˆ Note if τn (ω) = τnI (ω), then for all t ∈ [τn (ω), τn+1 (ω)) and n ∈ IN and Xˆ := Y + e1 ck. D (ω) := inf{t > τn (ω): Xˆ t− ∈ ∂ D } τn+1 (ω) = τn+1 and similarly if τn (ω) = τnD (ω). We observe that the boundedness of vx1 x1 implies that inf y [ψ D (y) − ψ I (y)] > 0, so D and I are separated by a positive distance and hence {τn } cannot accumulate at a finite value. Theorem 5.1. The process kˆ = kˆ + − kˆ − defined by (5.3), (5.4) is Ft -progressively measurable, right continuous at 0, continuous for t > 0, and satisfies the fixed point problem  − + ckˆt+ = max [ψ I (Y2,s ) − (Y1,s − ckˆs− )] , 0≤s≤t t ≥ 0. (5.5) + ckˆt− = max [(Y1,s + ckˆs− ) − ψ D (Y2,s )]+ , 0≤s≤t

Moreover kˆ is constant when Xˆ t ∈ T, and Xˆ t ∈ T¯ a.s. for all t < ∞. Proof. Note that Xˆ 1,t = Y1,t + ckˆt and Xˆ 2,t = Yt . It follows from (5.3), (5.4) that kˆ is progressively measurable and continuous for t > 0. For t < τ1 , kˆ = 0 so (5.5) holds. Since kˆτ±1 − = 0, then (5.1), (5.2) imply (5.5) for t < τ2 . Now (5.5) is obtained by induction on n ∈ IN (as in Corollary 4.8, p. 116, of [7]). If d kˆt+ > 0, then from (5.5) ckˆt+ = ψ I (Y2,t ) − (Y1,t − ckˆt− ) = ψ I ( Xˆ 2,t ) − Xˆ 1,t + ckˆt+ , i.e. Xˆ t ∈ ∂ I , whereas if Xˆ ∈ T, then ψ I ( Xˆ 2,t ) < Xˆ 1,t . Similarly if d kˆt− > 0. Finally if Xˆ t ∈ I\∂ I , then 0 ≤ ckˆt+ < ckˆt+ + ψ I ( Xˆ 2,t ) − Xˆ 1,t = ψ I (Y2,t ) − (Y1,t − ckˆt− ) = [ψ I (Y2,t ) − (Y1,t − ckˆt− )]+ ≤ max [ψ I (Y2,s ) − (Y1,s − ckˆs− )]+ , 0≤s≤t

which contradicts (5.5). The case Xˆ t ∈ D\∂ D is treated similarly.

Controlling Inflation: The Infinite Horizon Case

47

We can now introduce the term bX into the dynamics, but we require one additional assumption: the functions ψ I and ψ D are globally Lipschitz. This strengthening of the local Lipschitz property, which has been proved earlier, seems to hold in practice at least if ν > 0, see [12]. It then follows that, for t ≤ T , Xˆ 1,t ≤ ψ D (Y2,t ) ≤ ψ D (x2 ) + κo |Y2,t − x2 | ≤ κ1 (1 + |Bt |), and with a similar lower bound derived from ψ I , we obtain | Xˆ 1,t | ≤ κ1 (1 + |Bt |).

(5.6)

Set m(t, B(ω)) := σ −1 b Xˆ t (ω). It is Ft -progressively measurable since Xˆ is. Thanks to (5.6), for any T < ∞, there exists a constant κT < ∞ such that ¶ µ t ≤ T. |m(t, B)| ≤ κT 1 + max |Bs | , 0≤s≤t

According to the result of Girsanov, see Sections 3.5A and 3.5D of [14], there exists B , the (uncompleted) σ -algebra generated by B, such that, if a measure P on F B := F∞ Z t m(s, B) ds, Wt := Bt − 0

then W is a standard Brownian motion on (Ä, F B , FtB , P) and for any T < ∞ and any set A ∈ FTB , ·Z T ¸ Z Z 1 T 2 m(s, B) · d Bs − |m(s, B)| ds d Q. P(A) = PT (A) := 11 A exp 2 0 0 It follows that Xˆ and kˆ satisfy P-a.e. Z t (a + b Xˆ s ) ds + σ Wt + e1 ckˆt Xˆ t = x +

(5.7)

0

 + + ckˆt+ = max [ψ I ( Xˆ 2,s ) − ( Xˆ 1,s − ckˆs− )] , 0≤s≤t

− ckˆt− = max [( Xˆ 1,s + ckˆs− ) − ψ D ( Xˆ 2,s )]+ ,

t ≥ 0.

(5.8)

0≤s≤t

Remark 5.2. We have not mentioned uniqueness so far, but certainly Y is unique, and so is kˆ defined by (5.3), (5.4). Since k0− = 0 for k ∈ V and since ψ I and ψ D are ˆ continuous, then it follows that solutions of (5.5) or (5.8) have the same structure as k, ˆ i.e., they satisfy (5.3), (5.4) and hence are unique. Nevertheless, X is constructed as a weak solution of (5.7). We shall show that it is pathwise unique, hence a strong solution. Let X˜ be another solution of (5.7), but with the same process W . Assume that Xˆ s = X˜ s

48

M. B. Chiarolla and U. G. Haussmann

so kˆs+ := ks+ ( Xˆ ) = k˜s+ := ks+ ( X˜ ) for 0 ≤ s ≤ τ = τ I , where the processes are in ∂ I at time τ I . This is true with τ = τ1 . Since Z t b Xˆ s ds + σ Wt , Yˆt = x + at + 0 Z t b X˜ s ds + σ Wt Y˜t = x + at + 0

and ψ I and ψ D are Lipschitz, then (5.3) implies |kˆt+ − k˜t+ | ≤ K o

Z τ

t

| Xˆ s − X˜ s | ds

for some constant K o . Since also Xˆ t − X˜ t =

Z

t

τ

b( Xˆ s − X˜ s ) ds + e1 c(kˆt+ − k˜t+ )

for t ∈ [τ, τ D ), where τ D is the next time that the other boundary is hit, then Xˆ t = X˜ t on that interval, and by induction, Xˆ = X˜ , i.e., pathwise uniqueness holds. It now follows ˆ that Xˆ is adapted to W , and so is k. Finally we need to show that kˆ is optimal. We cannot apply the classical Itˆo formula ¯ whereas since this would require the function v to be of class C 2 in a neighborhood of T, 2;∞ 2 our v is only in Wloc (R ). We then apply Theorem 7.1, p. 125, of [7] which is an extension to semimartingales of an Itˆo formula given by Krylov [15]. Theorem 5.3. Assume (5.6), ρ − b11 > 0, ν > 0. The process kˆ = kˆ + − kˆ − given by (5.3), (5.4) is optimal for the control problem (P), i.e., ˆ v(x) = Jx (k),

∀x ∈ R2 .

Proof. Let m > 0 be such that |x| < m and ckˆ0+ + ckˆ0− < m; let Bm be the open ball centered at the origin with radius m. Define τm0 as the first exit time of Y (which is just X 0 ) ˆ from [−m, m]. Finally let τˆm = min{τm0 , τmk }. from Bm , and τmk as the first exit time of c|k| Then the first exit time of Xˆ from B2m is greater than τˆm . From Theorem 5.1, Theorem 7.5 of [7], and (3.14) we obtain Z τˆm |[−L + ρ]v( Xˆ t ) − f ( Xˆ t )| dt E 0

Z

τˆm

=E 0

ˆ ˆ 11{ Xˆ t ∈T} ¯ |[−L + ρ]v( X t ) − f ( X t )| dt

≤ K 1 k[−L + ρ]v − f k L 2 (T∩B ¯ 2m ) =0 for some constant K 1 . Note that E denotes expectation with respect to P.

Controlling Inflation: The Infinite Horizon Case

49

From Theorem 7.1 of [7] it follows that v(x) − E|kˆ0 | = Ev( Xˆ 0 ) (Z τˆm e−ρs [−L + ρ]v( Xˆ s ) ds =E 0

Z −

e

−ρs

vx1 ( Xˆ s )c d kˆs + e

−ρ τˆm

v( Xˆ τˆm )

0

(Z

τˆm

=E

e (Z

)

τˆm

−ρs

f ( Xˆ s ) ds +

0

)

τˆm

e

−ρs

0 τˆm

≥E

Z

e

−ρs

f ( Xˆ s ) ds +

0

Z

ˆ s + e−ρ τˆm v( Xˆ τˆm ) d|k| )

τˆm

e

−ρs

ˆs . d|k|

(5.9)

0

The usual estimates show that ¾ ½ 0 2 E sup |X s | ≤ (1 + |x|2 )ϕ(t), 0≤s≤t

where ϕ(t) < ∞ for t < ∞. Now Chebychev’s inequality implies that P-limm→∞ τm0 = ∞. Since Z τˆm ˆs e−ρs d|k| v(x) ≥ E 0− ( ) Z τmk −ρs ˆ ≥ E 11{τ k <τ 0 } e d|k|s m

m

0−

o k ≥ (m/c)E 11{τmk <τm0 } e−ρτm , n

then, for any Mo > 0, hn lim P

m→∞

o © ªi k e−ρτm > Mo ∩ τmk < τm0 = 0.

Moreover o © oi hn hn ªi k 0 P e−ρτm > Mo ∩ τmk ≥ τm0 ≤ P e−ρτm > Mo → 0 since P-limm→∞ τm0 = ∞. It follows that P-limm→∞ τmk = ∞, and hence P-limm→∞ τˆm = ∞. Thus when we take the limit on m in (5.9), we obtain ˆ v(x) ≥ Jx (k) and the result follows. Remark 5.4. The condition (5.6) cannot be verified in practice as we can only approximate the control numerically on a compact set, and there we already know that

50

M. B. Chiarolla and U. G. Haussmann

the functions ψ I , ψ D are Lipschitz. Nevertheless, if the process Xˆ is ever observed to leave a sufficiently large set, i.e., the rates exceed say 50%, then it is clear that the model is insufficient and the problem must be re-examined. However, until such time we can take the optimal control to be the one which provides reflection at the boundaries of T without concern about (5.6).

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

22.

A. Bensoussan, J. L. Lions (1982) Applications of Variational Inequalities in Stochastic Control, NorthHolland, Amsterdam. H. H. Binhammer (1993) Money, Banking and the Canadian Financial System, 6th edn., Nelson Canada, Scarborough. H. Br´ezis (1972) Probl`emes unilat´eraux, J. Math. Pures Appl., 51: 1–168. K. Chan, A. Karolyi, F. Longstaff, A. Sanders (1992) An empirical comparison of alternative models of the short-term interest rates, J. Finance, 47: 1209–1227. M. B. Chiarolla (1992) Geometric Approach to Monotone Stochastic Control, Ph.D. Dissertation, The University of British Columbia, Vancouver, BC. M. B. Chiarolla, U. G. Haussmann (1994) The free boundary of the monotone follower, SIAM J. Control Optim., 32: 690–727. M. B. Chiarolla, U. G. Haussmann (1994) The optimal control of the cheap monotone follower, Stochastics Stochastics Rep., 49: 99–128. M. B. Chiarolla, U. G. Haussmann (1998) Optimal control of inflation: a Central Bank problem, SIAM J. Control Optim., 36: 1099–1132. W. H. Fleming, R. W. Rishel (1975) Deterministic and Stochastic Optimal Control, Springer-Verlag, New York. A. Friedman (1982) Variational Principles and Free-Boundary Problems, Wiley, New York. D. Gilbarg, N. S. Trudinger (1983) Elliptic Partial Differential Equations of Second Order, SpringerVerlag, New York. B. Han, U. G. Haussmann (1998) Interest rates and inflation: parameter identification and control in a model using Canadian data, Preprint. I. Karatzas (1983) A class of singular stochastic control problems, Adv. in Appl. Probab., 15: 225–254. I. Karatzas, S. E. Shreve (1988) Brownian Motion and Stochastic Calculus, Springer-Verlag, New York. N. V. Krylov (1980) Controlled Diffusion Processes, Springer-Verlag, New York. J. L. Menaldi, M. Robin (1993) On some cheap control problems for diffusion processes, Trans. Amer. Math. Soc., 278: 771–802. J. L. Menaldi, M. I. Taksar (1989) Optimal correction problem of a multidimensional stochastic system, Automatica, 25: 223–232. F. S. Mishkin (1993) Money, Interest Rates and Inflation, Edward Elgar, Aldershot. G. G. Pennacchi (1991) Identifying the dynamics of real interest rates and inflation: evidence using survey data, Rev. Financial Studies, 4: 53–86. H. M. Soner, S. E. Shreve (1989) Regularity of the value function for a two-dimensional singular stochastic control problem, SIAM J. Control Optim., 27: 876-907. G. Vaciago (1992) Banca Centrale tra governo e mercato, in Il Ruolo della Banca Centrale nella Politica Economica – XXXII Riunione Scientifica Annuale della Societ`a Italiana degli Economisti, M. Arcelli ed., il Mulino, Bologna, pp. 61–70. S. A. Williams, P. L. Chow, J. L. Menaldi (1994) Regularity of the free boundary in singular stochastic control, J. Differential Equations, 111: 175–201.

Accepted 22 May 1998