Optimal resource allocation for maintaining system solvency

1 Optimal resource allo cation for main taining system solv ency Gao yue Guo 1 , W enpin T ang 2 , and Nizar T ouzi 3 1 Univ ersit´ e P aris-Saclay Cen traleSup ´ elec, Lab oratoire MICS and CNRS FR-3487 ⋆⋆⋆ gaoyue.guo@centralesupelec.fr 2 Colum bia Universit y , IEOR wt2319@columbia.edu 3 New Y ork Universit y , T andon School of Engineering † nt2635@nyu.edu Summary . W e study an optimal allo cation problem for a system of indep enden t Bro wnian agents whose states ev olve under a limited shared con trol. At eac h time, a unit of resource can b e divided and allocated across comp o- nen ts to increase their drifts, with the ob jective of maximizing either (i) the probability that all comp onen ts av oid ruin, or (ii) the exp ected num b er of comp onen ts that av oid ruin. W e derive the associated Hamilton–Jacobi–Bellman equations on the p ositiv e orthant with mixed b oundary conditions at the absorbing b oundary and at infinity , and w e identify drift thresholds separating trivial and nontrivial regimes. F or the all-survive criterion, w e establish exis- tence, uniqueness, and smo othness of a b ounded classical solution and a v erification theorem linking the PDE to the sto c hastic-con trol v alue function. W e then inv estigate the conjectured optimality of the push-the-laggard allo cation rule: in the nonnegative-drift regime, w e pro ve that it is optimal for the all-survive v alue function, while it is not optimal for the count-surviv ors criterion, by exhibiting a tw o-dimensional coun terexample and then lifting it to all dimensions. Key words : sto chastic control, allocation problem, HJB equation, push-the-laggard p olicy 1.1 In tro duction: scarce resources, surviv al, and fairness W e study a sto chastic allo cation problem for a system of indep enden t Bro wnian agents whose states evolv e under a shared control budget. At each time, the controller may distribute one unit of effort across the co or- dinates in order to increase their drifts, while each co ordinate is absorb ed up on hitting zero. The ob jective is to allo cate the resource dynamically so as to maximize long-run surviv al. Two natural criteria lead to quali- tativ ely differen t con trol problems. The first maximizes the probability that all co ordinates survive forev er; the second maximizes the exp ected n umber of co ordinates that survive forev er. These tw o ob jectives capture, resp ectiv ely , a system-wide robustness criterion and an aggregate surviv al criterion. A cen tral question is whether an optimal p olicy should alw ays allo cate the av ailable effort to the currently w eakest co ordinate. This question is closely related to an interpretation prop osed by McKean and Shepp, who con trasted comp eting allocation principles in a sto chastic model of supp ort across firms. In our setting, it leads to a precise push-the-laggard conjecture: is it optimal to allocate the en tire budget to the smallest positive co ordinate? W e analyze this problem through its Hamilton–Jacobi–Bellman equation on the positive orthant, sup- plemen ted with absorbing b oundary conditions at zero and recursive asymptotic conditions at infinity . Our first result identifies drift thresholds separating trivial and nontrivial regimes. In the nonnegative-drift case, w e pro ve existence, uniqueness, and regularity of b ounded classical solutions, and w e establish a verification theorem linking the PDE to the v alue function. Our second result reveals a sharp dichotom y in the structure of optimal controls: the push-the-laggard policy is optimal for the all-survive criterion, but fails in general for the surviv or-count criterion. Man y real-w orld allocation problems share a common tension: a planner m ust distribute a limited resource among several units whose situations evolv e randomly , and the decision maker’s ob jectiv e ma y reflect either efficiency , meaning maximizing the n um b er of units that do w ell, or robustness, meaning a voiding catastrophic failure of the system as a whole. A vivid example comes from emergency departments: clinicians must allocate scarce resources such as physician time, ICU b eds, ven tilators, and diagnostic capacity to a finite n umber of patien ts, each with uncertain health tra jectories. Op erational researc h mo dels of emergency departmen ts emphasize that decisions must b e made in real time under uncertaint y , and that different prioritization rules ⋆⋆⋆ This work was supp orted by the ANR pro ject MA TH-SP A. † T ouzi is partially supported by NSF grant No. DMS-2508581. 2 Guo, T ang and T ouzi can strongly affect system p erformance and patien t outcomes. F rom a normative viewp oin t, triage guidelines balance sev eral ethical principles, such as “sa ve the most lives”, “protect the worst-off ”, and “treat p eople equally”, and these principles can conflict in scarce-resource settings. A second viewp oin t comes from so cial welfare design: should a planner concentrate effort on those close st to failure, follo wing a maximin or Rawlsian intuition, or instead allo cate effort where it yields the greatest aggregate b enefit, following a more utilitarian logic? McKean and Shepp compared tw o distinct gov ernmen t tax p olicies tow ards companies in [13]: the republican p olicy giv es tax breaks to the ric her companies, while the demo cratic p olicy would p erhaps giv e tax breaks to the weak er companies in the hop e of keeping them aliv e and thereby reducing unemploymen t. Dep ending on the optimization criterion, they used a sto c hastic mo del and provided a mathematical formulation of the question. W e extend their mo del, and the conjecture studied later can b e interpreted precisely as follows: Do es allo cating to the w eakest co ordinate yield the highest marginal b enefit? In the emergency-department analogy , this resembles a sick est-first rule; in a social-welfare analogy , it re- sem bles a prioritize-the-least-adv an taged rule. The pap er shows that the answer depends sharply on the ob jective: prioritizing the w eakest can b e optimal for a stringent system-surviv al criterion, but can fail for a maximize-the-n umber-of-survivors criterion. 1.1.1 Problem formulation W e consider t w o allo cation problems formulated on some filtered probability space ( Ω , F , F = ( F t ) t ≥ 0 , P ) satisfying the usual conditions and supp orting infinitely many independent F -Bro wnian motions W 1 , W 2 , . . . . F or every n ≥ 1, denote b y A n the collection of F -progressively measurable pro cesses Φ = ( φ 1 t , . . . , φ n t ) t ≥ 0 taking v alues in [0 , 1] n and satisfying n X j =1 φ j t ≤ 1 , ∀ t ≥ 0 . F or eac h Φ ∈ A n , define the controlled pro cess X Φ = ( X Φ, 1 t , . . . , X Φ,n t ) t ≥ 0 b y d X Φ,i t := b d t + φ i t d t + d W i t , ∀ t ≥ 0 , and the hitting time τ Φ i := inf  t ≥ 0 : X Φ,i t ≤ 0  . W e in terpret X Φ,i t as the health, wealth, or reserve level of unit i : absorption at 0 represents failure, such as death, default, or bankruptcy . The control φ i t represen ts instantaneous assistance allo cated to unit i , sub ject to a unit total budget 4 . W e consider the follo wing sto c hastic control problems. F or x = ( x 1 , . . . , x n ) ∈ R n + , define V n ( x ) := sup Φ ∈A n E " n Y i =1 1 { τ Φ i = ∞}    X Φ 0 = x # , (1.1) U n ( x ) := sup Φ ∈A n E " n X i =1 1 { τ Φ i = ∞}    X Φ 0 = x # . (1.2) The functional V n corresp onds to a system-reliability ob jective: maximize the probability that every unit surviv es forev er. The functional U n corresp onds to a surviv or-count ob jective: maximize the expected n umber of units that surviv e forev er. These t w o ob jectives mirror, respectively , a no-one-left-behind policy and a sav e- as-man y-as-p ossible p olicy in triage ethics [15, 17]. Another issue concerns the structure of optimal allo cation. A natural conjecture is that, whenever one co ordinate is strictly smaller than another, the v alue function is steep er in that co ordinate, meaning that the marginal v alue of assistance is larger for the weak est unit. Equiv alently , an optimal feedback would allo cate effort to the smallest co ordinate. This conjecture formalizes the question whether a prioritize-the-worst-off p olicy is optimal. Define the push-the-laggard strategy Φ by φ i t := 1 { i = I t } , where I t := min { 1 ≤ j ≤ n : X j t = Z t } and Z t := min { X j t : X j t > 0 , j = 1 , . . . , n } . 4 All the results presented b elo w remain v alid if the unit budget is replaced by any p ositiv e budget. 1 Optimal resource allocation for maintaining system solv ency 3 This system of SDEs has rank-dependent drifts and corresp onds to an A tlas-type model whose existence and uniqueness are guaranteed b y standard results on rank-based diffusions. This motiv ates the following question. Conjecture* (push-the-laggard conjecture) : Is Φ optimal for V n ( x ) and for U n ( x ) ? 1.1.2 Main results Our answer is summarized in the theorem b elow. Theorem 1. L et b ≥ 0 and n ≥ 2 . Then Conje ctur e* holds for V n and do es not hold for U n . F or n = 1, the unique optimizer is trivially given by φ 1 ≡ 1, and V 1 ( z ) = U 1 ( z ) = P [ z + (1 + b ) t + W 1 t > 0 , ∀ t ≥ 0] = H (( b + 1) + z ) , ∀ z ≥ 0 , (1.3) where H : R + → R is defined by H ( z ) := 1 − e − 2 z . Our analysis is built on the PDE counterpart of (1.1) and (1.2). It is worth noting that the parameter b pla ys a substan tial role in the analysis of the corresp onding HJB equation and in the study of the conjecture. Before turning to the PDE form ulation, w e record some basic properties of (1.1) and (1.2). Their pro ofs are p ostponed to the appendix. Lemma 1. As functions on R n + , V n and U n satisfy the fol lowing pr op erties. 1. V n and U n ar e symmetric on R n + . F or every x ∈ R n + , 0 ≤ V n ( x ) ≤ n Y i =1 H (( b + 1) + x i ) , 0 ≤ U n ( x ) ≤ n X i =1 H (( b + 1) + x i ) . In p articular, V n = 0 on ∂ E n . 2. The maps R + ∋ x i 7− → V n ( x 1 , . . . , x n ) ∈ R + and R + ∋ x i 7− → U n ( x 1 , . . . , x n ) ∈ R + ar e nonde cr e asing. Lemma 2. F or every x ∈ E n , one has V n ( x ) > 0 if and only if b > − 1 n , U n ( x ) > 0 if and only if b > − 1 . In view of Lemma 2, the allo cation problem is trivial when b is to o negativ e, and it suffices to consider b > − 1 /n for V n and b > − 1 for U n . The next lemma provides the recursive relation b et ween V n and V n − 1 , and b et ween U n and U n − 1 . F or later use, we adopt the notation that for x ∈ R n and i = 1 , . . . , n , x − i := ( x 1 , . . . , x i − 1 , x i +1 , . . . , x n ) ∈ R n − 1 , ( x − i , z ) := ( x 1 , . . . , x i − 1 , z , x i +1 , . . . , x n ) ∈ R n , max( x ) := max { x 1 , . . . , x n } , min( x ) := min { x 1 , . . . , x n } . Lemma 3. Fix an arbitr ary i ∈ { 1 , . . . , n } . 1. If b ≥ 0 or b ≤ − 1 / ( n − 1) , then V n ( x − i , ∞ ) = V n − 1 ( x − i ) . 2. If b ≥ 0 , then U n ( x − i , ∞ ) = U n − 1 ( x − i ) + 1 . Her e we write f ( ∞ ) := lim z →∞ f ( z ) whenever the limit exists. Both V n ( x − i , ∞ ) and U n ( x − i , ∞ ) ar e wel l define d by L emma 1. This pap er s tudies the push-the-laggard conjecture by combining tec hniques from sto c hastic control and PDE analysis. Denote b y E n := (0 , ∞ ) n the interior of R n + and by ∂ E n its b oundary , namely x ∈ ∂ E n :=  x ∈ R n + : min( x ) = 0  . F or n ≥ 2, we write the corresp onding HJB equation, with 1 n := (1 , . . . , 1) ∈ R n , 1 2 ∆p + b 1 n · ∇ p + max( ∇ p ) = 0 in E n , (1.4) 4 Guo, T ang and T ouzi together with suitable b oundary conditions on ∂ E n : p ( x − i , 0) = 0 in ∂ E n , (1.5) p ( x − i , 0) = U n − 1 ( x − i ) in ∂ E n . (1.6) The following prop osition relates the sto c hastic control problems to the HJB equation. Prop osition 1. Our r esults ar e state d sep ar ately for V n and U n , with n ≥ 2 . (i) If b ≤ − 1 /n , then (1.4) and (1.5) ar e wel l-p ose d, and V n ≡ 0 is their unique b ounde d classic al solution. (ii) If b ≤ − 1 , then (1.4) and (1.6) ar e wel l-p ose d, and U n ≡ 0 is their unique b ounde d classic al solution. (iii) If b ≥ 0 , then (1.4) , (1.5) and (1.7) ar e wel l-p ose d, and V n ∈ C 2 ( E n ) ∩ C ( R n + ) is their unique b ounde d classic al solution, wher e p ( x − i , ∞ ) = V n − 1 ( x − i ) . (1.7) (iv) If b ≥ 0 , then (1.4) , (1.6) and (1.8) ar e wel l-p ose d, and U n ∈ C 2 ( E n ) ∩ C ( R n + ) is their unique b ounde d classic al solution, wher e p ( x − i , ∞ ) = U n − 1 ( x − i ) + 1 . (1.8) Lemmas 2 and 3, together with Proposition 1, explain from both stochastic and PDE viewpoints the natural splitting of the drift parameter in to three regimes: • a trivial regime, where the v alue function v anishes identically; • a nonnegative-drift regime, in whic h the recursion at infinity closes on the same ( n − 1)-dimensional problem; • an intermediate negative-drift regime , − 1 n < b < 0 for V n , − 1 < b < 0 for U n , in which the v alue functions are nonzero, but the recursiv e b oundary conditions at infinity no longer close on the same ( n − 1)-dimensional problem. Conjecture (gradien t form of the push-the-laggard conjecture) : F or ev ery x = ( x 1 , . . . , x n ) ∈ E n , x i = min( x ) = ⇒ ( ∂ x i V n ( x ) = max  ∇ V n ( x )  , ∂ x i U n ( x ) = max  ∇ U n ( x )  . It is kno wn that: • F or b = 0 and n = 2, McKean and Shepp gav e in [13] the explicit expression V 2 ( x ) = 1 − e − 2 min( x ) − 2 min( x )e − 1 2 · x , whic h pro ves the conjecture by direct verification. Moreov er, they presented numerical evidence that the conjecture fails for U 2 . • F or b = 0 and n − → ∞ , this conjecture is “asymptotically true” for U n , as established by T ang and Tsai [16]. W e ha ve the following result. Theorem 2. L et b ≥ 0 and n ≥ 2 . (i) Conje ctur e holds for V n . (ii) Conje ctur e do es not hold for U n . Theorem 1 follows from Theorem 2, Prop osition 1, and the verification argumen t giv en in App endix A. In particular, the feedback control that allo cates the full budget to a co ordinate attaining the minim um lev el realizes the Hamiltonian p oin twise for V n , whereas Theorem 2 (ii) sho ws that the same feedbac k cannot be optimal for U n . F or simplicit y , we write X Φ,i ≡ X i , τ Φ i ≡ τ i , and so on, and also V n ≡ V , U n ≡ U , and so on, whenever the con text is clear. 1 Optimal resource allocation for maintaining system solv ency 5 1.1.3 On the in termediate negative-drift regime In this subsection, we briefly discuss the intermediate negative-drift regimes − 1 n < b < 0 for V n , − 1 < b < 0 for U n . By Lemma 2, the v alue functions are strictly p ositiv e in these regimes. How ever, the recursive b oundary con- ditions at infinity from Lemma 3 are no longer exp ected to hold in their present form, and the corresp onding PDE analysis b ecomes more delicate. First, the univ ersal upp er b ounds V n ( x − i , z ) ≤ V n − 1 ( x − i ) , U n ( x − i , z ) ≤ 1 + U n − 1 ( x − i ) , ∀ z ≥ 0 , (1.9) follo w directly from the definitions and hold for all b ∈ R . The difficulty is to identify the correct matching lo wer b ounds, and hence the correct asymptotic trace at infinity . Indeed, if the i -th co ordinate is started from a large level z , then in order for it to survive with probabilit y tending to one, one must assign to it a p ositiv e effective drift. If one freezes a constant con trol η ∈ [0 , 1] on the i -th co ordinate, then P  τ i = ∞   X i 0 = z  = H (( b + η ) + z ) . Hence, when b < 0, asymptotic surviv al of the remote co ordinate requires η > − b , so a nonv anishing fraction of the total budget m ust b e reserv ed for it. Consequently , the remaining n − 1 co ordinates do not asymptotically inherit the full unit budget. T o formalize this observ ation, for a ∈ [0 , 1], let V n − 1 ,a and U n − 1 ,a denote the ( n − 1)-dimensional v alue functions asso ciated with the same dynamics, but with total budget a in place of 1, namely V n − 1 ,a ( y ) := sup Ψ E h n − 1 Y j =1 1 { τ Ψ j = ∞}    X Ψ 0 = y i , and U n − 1 ,a ( y ) := sup Ψ E h n − 1 X j =1 1 { τ Ψ j = ∞}    X Ψ 0 = y i , where the supremum is tak en ov er all progressively measurable con trols Ψ = ( ψ 1 , . . . , ψ n − 1 ) satisfying ψ j t ≥ 0 and n − 1 X j =1 ψ j t ≤ a, t ≥ 0 . Then, for ev ery η ∈ ( − b, 1], freezing φ i ≡ η on the remote coordinate yields lim inf z →∞ V n ( x − i , z ) ≥ V n − 1 , 1 − η ( x − i ) , (1.10) and lim inf z →∞ U n ( x − i , z ) ≥ 1 + U n − 1 , 1 − η ( x − i ) . (1.11) Since the maps a 7→ V n − 1 ,a and a 7→ U n − 1 ,a are nondecreasing, these estimates suggest that the correct asymptotic traces at infinity should b e gov erned by the reduced budget 1 + b . More precisely , one is naturally led to conjecture that lim z →∞ V n ( x − i , z ) = V n − 1 , 1+ b ( x − i ) , (1.12) and lim z →∞ U n ( x − i , z ) = 1 + U n − 1 , 1+ b ( x − i ) . (1.13) A t present, ho wev er, we do not ha ve a matc hing upp er-bound argument pro ving these identities. Establishing suc h an upp er b ound app ears to require a gen uinely new argumen t showing that an y asymptotically optimal strategy must devote at least − b units of budget to the remote co ordinate. This also clarifies the PDE situation. Consider first the case of V n with − 1 /n < b < 0. The pair consisting of the interior HJB equation (1.4) and the finite-b oundary condition (1.5) is not w ell-p osed in the class of b ounded solutions: the zero function is a b ounded classical solution, while Lemma 2 implies that the sto c hastic con trol v alue function V n is strictly p ositiv e on E n . Hence uniqueness fails if one prescrib es only the b oundary condition on ∂ E n and do es not sp ecify the b eha viour at infinity . The same remark applies to 6 Guo, T ang and T ouzi U n in the regime − 1 < b < 0, where again the corresp onding HJB problem with only the b oundary condition (1.6) is not exp ected to b e w ell-p osed. Although the asymptotic b oundary condition at infinity is unclear in the intermediate negative-drift regime, the stochastic con trol problems themselv es are unc hanged, and the same w eak dynamic-programming argumen t still leads formally to the HJB equation (1.4) in E n . F or instance, in the standard viscosity sense, the upp er semicon tinuous en velope of V n is a viscosity subsolution of (1.4) on E n , while the lo wer semicon tinuous en velope of V n is a viscosit y supersolution of (1.4) on E n . The main obstacle is differen t: without an identified b oundary condition at infinity , the present metho d do es not provide contin uity of V n or U n on E n , although their contin uit y on ∂ E n is clear from the con trol problem. Without contin uity in E n , the later regularit y argumen t cannot b e applied in the same wa y . Finally , let us emphasize that the missing ingredien t is global, not lo cal. If contin uity of V n or U n in E n could b e established by another argument, then the lo cal elliptic regularity part of the pro of, as sho wn in Section 1.2, should still apply to obtain V n , U n ∈ C 2 ( E n ). In other w ords, the principal obstruction in the in termediate negative-drift regime is the lack of an identified trace at infinity and the resulting failure of global comparison and uniqueness, rather than the lo cal regularity theory itself. 1.2 Pro of of Prop osition 1 This section is devoted to the pro of of Prop osition 1. W e b egin by recalling some basic prop erties of the v alue functions, and then distinguish t wo cases according to the sign of the drift parameter b . F or n ≥ 2, we write the HJB equation in the form F ( ∇ p, ∇ 2 p ) = 0 in E n , (1.14) where the Hamiltonian F : R n × S n → R is defined by F ( q , Γ ) := 1 2 tr( Γ ) + b 1 n · q + max( q ) , with S n := { Γ = ( γ ij ) 1 ≤ i,j ≤ n : γ ij = γ j i ∈ R } . 1.2.1 The case of sufficien tly negative drift By Lemma 2, one has V n ≡ 0 when b ≤ − 1 /n and U n ≡ 0 when b ≤ − 1. So it remains to show the uniqueness, summarized in the prop osition b elo w. Prop osition 2. L et v n ∈ C 2 ( E n ) ∩ C ( R n + ) b e a b ounde d solution of (1.4) and (1.5) . If b ≤ − 1 /n , then v n ≡ 0 . L et u n ∈ C 2 ( E n ) ∩ C ( R n + ) b e a b ounde d solution of (1.4) and (1.6) . If b ≤ − 1 , then u n ≡ 0 . Prop osition 2 yields parts (i) and (ii) of Prop osition 1. Its pro of, similar to that of Theorem 2, also follows from the v erification argumen t and is p ostponed to the appendix. 1.2.2 The case b ≥ 0 W e assume b ≥ 0 throughout this subsection. W e first prov e the result for V n , and the proof for U n is iden tical up to the corresp onding b oundary data. W e pro ceed by induction on n . F or n = 1, the claim is immediate, since (1.4) reduces to an ODE and V 1 ( z ) = H (( b + 1) z ) ∈ C 2 ( E 1 ) ∩ C ( R + ) is the unique b ounded classical solution. Assume now that the statement has b een pro ved up to dimension n − 1. F or the sake of presentation, we write V ≡ V n without any danger of confusion. Recall that V ( x ) = sup Φ ∈A n E h n Y i =1 1 { τ Φ i = ∞}    X Φ 0 = x i , x ∈ R n + , and denote by V ∗ and V ∗ its upp er and low er semicontin uous env elop es on E n . By the weak dynamic- programming principle of Bouchard and T ouzi [2], which applies to the present con trolled diffusion setting, V ∗ is a viscosit y subsolution and V ∗ is a viscosit y sup ersolution of (1.14) on E n . The follo wing prop osition shows the contin uity of V , whic h is the key to establish higher regularit y of V . Prop osition 3. One has V ∗ ≤ V ∗ on E n . Conse quently, V = V ∗ = V ∗ is c ontinuous on R n + and is a visc osity solution of (1.14) on E n . 1 Optimal resource allocation for maintaining system solv ency 7 The idea is to transform the comparison argument to a b ounded domain. Namely , w e introduce a co ordi- natewise compactification. Let f : (0 , 1) → (0 , ∞ ) b e defined b y f ( z ) := z 1 − z , z ∈ (0 , 1) , and extend it comp onen twise to a homeomorphism, still denoted by f : J n := (0 , 1) n − → E n , f ( y ) :=  y 1 1 − y 1 , . . . , y n 1 − y n  . Its inv erse is the comp onent wise map g ( x ) := x 1 + x , x ∈ (0 , ∞ ) . Define e V := V ◦ f , e V ∗ := V ∗ ◦ f and e V ∗ := V ∗ ◦ f , and set for y ∈ J n : a i ( y ) := (1 − y i ) 4 , β i ( y ) := (1 − y i ) 2  b − (1 − y i )  , γ i ( y ) := (1 − y i ) 2 . Define the alternativ e Hamiltonian H : J n × R n × S n → R by H ( y , q , Γ ) := 1 2 n X i =1 a i ( y ) γ ii + n X i =1 β i ( y ) q i + max 1 ≤ i ≤ n γ i ( y ) q i . (1.15) Lemma 4. The function e V ∗ is a b ounde d upp er semic ontinuous visc osity subsolution of H ( y , ∇ u, ∇ 2 u ) = 0 in J n , and e V ∗ is a b ounde d lower semic ontinuous visc osity sup ersolution of the same e quation. Pr o of. This is the standard inv ariance of viscosity inequalities under C 2 diffeomorphisms, applied to the map f : J n → E n . The sp ecific form of (1.15) is obtained b y the chain rule. ⊓ ⊔ W e next iden tify the b oundary v alues induced by the absorbing b oundary and the recursion at infinity . F or y ∈ ∂ J n , set I 0 ( y ) := { i : y i = 0 } , I 1 ( y ) := { i : y i = 1 } , m ( y ) := | I 1 ( y ) | . If m ( y ) < n , let y | ( I 1 ( y )) c denote the ( n − m ( y ))-tuple obtained by deleting the co ordinates equal to 1. With the conv ention V 0 ≡ 1, define G : ∂ J n → [0 , 1] b y G ( y ) := ( 0 , if I 0 ( y )  = ∅ , V n − m ( y )  f ( y | ( I 1 ( y )) c )  , if I 0 ( y ) = ∅ . (1.16) Lemma 5. The b oundary function G is c ontinuous on ∂ J n . Pr o of. Fix y ∈ ∂ J n and let y k → y in ∂ J n . If I 0 ( y )  = ∅ , then at least one co ordinate of y equals 0. Along every sequence y k → y , at least one co ordinate of f ( y k ) conv erges to 0, and by Lemma 1 we hav e 0 ≤ V  f ( y k )  ≤ n Y i =1 H  ( b + 1) f ( y k i )  − → 0 . Hence G ( y k ) → 0 = G ( y ). Assume now that I 0 ( y ) = ∅ . Then only co ordinates equal to 1 may lie on the b oundary . If I 1 ( y ) = { i 1 , . . . , i m } , then along every sequence y k → y with y k ∈ ∂ J n , the co ordinates f ( y k i r ) tend to ∞ , whereas the remaining co ordinates con verge to f ( y | ( I 1 ( y )) c ). Rep eated application of Lemma 3 therefore yields G ( y k ) − → V n − m  f ( y | ( I 1 ( y )) c )  = G ( y ) . This prov es contin uity of G on ∂ J n . ⊓ ⊔ Lemma 6. One has e V ∗ = G = e V ∗ on ∂ J n . 8 Guo, T ang and T ouzi Pr o of. If I 0 ( y )  = ∅ , then f ( y ) approac hes ∂ E n , and Lemma 1 implies e V ∗ ( y ) = e V ∗ ( y ) = 0 = G ( y ) . Assume no w that I 0 ( y ) = ∅ . Then m ( y ) ≥ 1, and along every sequence y k → y with y k ∈ J n , one has f ( y k ) i → ∞ for i ∈ I 1 ( y ) while the remaining co ordinates conv erge to f ( y | ( I 1 ( y )) c ). By Lemma 3, the corresp onding limit of V is precisely V n − m ( y )  f ( y | ( I 1 ( y )) c )  = G ( y ) . This gives e V ∗ ( y ) ≤ G ( y ) ≤ e V ∗ ( y ). Since alwa ys e V ∗ ≤ e V ∗ , the conclusion follows. ⊓ ⊔ No w w e are ready to prov e Prop osition 3. Pr o of (of Pr op osition 3). By Lemma 4, e V ∗ is a b ounded upp er semicontin uous viscosity subsolution and e V ∗ is a b ounded lo wer semicontin uous viscosity sup ersolution of H ( y , ∇ u, ∇ 2 u ) = 0 in J n . Set W := e V ∗ − e V ∗ on J n . By Lemma 6, for ev ery y ∈ ∂ J n one has lim sup J n ∋ z → y W ( z ) ≤ 0 . (1.17) W e claim that W ≤ 0 on J n . Assume, for con tradiction, that M := sup J n W > 0 . Since W is upp er semicontin uous, (1.17) implies that W cannot attain a p ositive maximum at the b oundary . Hence there exists ¯ y ∈ J n suc h that W ( ¯ y ) = M > 0 . Cho ose ε ∈ (0 , 1 / 2) so small that ¯ y ∈ J ε n := ( ε, 1 − ε ) n . Since ¯ y is an in terior maximizer and W is upp er semicontin uous, after p ossibly decreasing ε we may also assume sup J n \ J ε n W ≤ M 2 . (1.18) W e no w w ork on the compact set J ε n . F or δ > 0, define Ψ δ ( y , z ) := e V ∗ ( y ) − e V ∗ ( z ) − | y − z | 2 2 δ , ( y , z ) ∈ J ε n × J ε n . Since Ψ δ is upp er semicontin uous on the compact set J ε n × J ε n , it attains its maxim um at some p oin t ( y δ , z δ ). Step 1: lo calization of the maximizers. Because Ψ δ ( ¯ y , ¯ y ) = W ( ¯ y ) = M , w e ha ve Ψ δ ( y δ , z δ ) ≥ M . On the other hand, Ψ δ ( y δ , z δ ) ≤ W ( y δ ) . Hence W ( y δ ) ≥ M . In particular, b y (1.18), one must ha ve y δ ∈ J ε n . Similarly , Ψ δ ( y δ , z δ ) ≤ sup J n e V ∗ − e V ∗ ( z δ ) , 1 Optimal resource allocation for maintaining system solv ency 9 and the standard doubling-of-v ariables argument shows that, up to extracting a subsequence, y δ → ¯ y , z δ → ¯ y , | y δ − z δ | 2 δ → 0 as δ ↓ 0 . In particular, for all sufficien tly small δ , both y δ and z δ b elong to J ε n . Step 2: uniform ellipticity on J ε n . F or every y ∈ J ε n and every i = 1 , . . . , n , a i ( y ) = (1 − y i ) 4 ≥ ε 4 , γ i ( y ) = (1 − y i ) 2 ≥ ε 2 . Therefore, on J ε n , the op erator H ( y , q , Γ ) = 1 2 n X i =1 a i ( y ) γ ii + n X i =1 β i ( y ) q i + max 1 ≤ i ≤ n γ i ( y ) q i is contin uous in all v ariables, proper, and uniformly elliptic in the matrix v ariable. Step 3: application of Ishii’s lemma. Set p δ := y δ − z δ δ . By Ishii’s lemma, there exist symmetric matrices X δ , Y δ ∈ S n suc h that ( p δ , X δ ) ∈ J 2 , + e V ∗ ( y δ ) , ( p δ , Y δ ) ∈ J 2 , − e V ∗ ( z δ ) , and  X δ 0 0 − Y δ  ≤ 3 δ  I − I − I I  . In particular, X δ ≤ Y δ . (1.19) Since e V ∗ is a viscosit y subsolution and e V ∗ is a viscosit y sup ersolution, H ( y δ , p δ , X δ ) ≤ 0 , H ( z δ , p δ , Y δ ) ≥ 0 . Subtracting gives 0 ≤ H ( z δ , p δ , Y δ ) − H ( y δ , p δ , X δ ) . (1.20) Step 4: estimate of the difference. W e decomp ose the right-hand side of (1.20) in to the second-order part, the drift part, and the Hamiltonian part. F or the second-order part, 1 2 n X i =1 a i ( z δ )( Y δ ) ii − 1 2 n X i =1 a i ( y δ )( X δ ) ii ≤ 1 2 n X i =1  a i ( z δ ) − a i ( y δ )  ( Y δ ) ii , where we used (1.19) together with a i ( y δ ) ≥ 0. F or the first-order drift part, n X i =1 β i ( z δ )( p δ ) i − n X i =1 β i ( y δ )( p δ ) i = n X i =1  β i ( z δ ) − β i ( y δ )  ( p δ ) i . F or the nonlinear term, since the map ( y , q ) 7− → max 1 ≤ i ≤ n γ i ( y ) q i is contin uous on J ε n × R n , we hav e max i γ i ( z δ )( p δ ) i − max i γ i ( y δ )( p δ ) i − → 0 as δ ↓ 0 , b ecause y δ − z δ → 0 and ( y δ − z δ ) / √ δ → 0 imply that p δ remains controlled at the scale needed for the standard comparison argumen t. More explicitly , since the co efficients a i , β i , γ i are Lipschitz on J ε n , there exists a constan t C ε > 0 such that | β i ( z δ ) − β i ( y δ ) | + | γ i ( z δ ) − γ i ( y δ ) | ≤ C ε | y δ − z δ | . 10 Guo, T ang and T ouzi Hence      n X i =1  β i ( z δ ) − β i ( y δ )  ( p δ ) i      ≤ C ε | y δ − z δ | | p δ | = C ε | y δ − z δ | 2 δ − → 0 , and similarly    max i γ i ( z δ )( p δ ) i − max i γ i ( y δ )( p δ ) i    ≤ C ε | y δ − z δ | 2 δ − → 0 . F or the second-order term, we use the standard trace estimate coming from Ishii’s lemma. Since the co efficien ts a i are Lipschitz on J ε n , there exists C ε > 0 such that | a i ( z δ ) − a i ( y δ ) | ≤ C ε | y δ − z δ | , i = 1 , . . . , n. Moreo ver, the matrix inequality from Ishii’s lemma implies the standard b ound tr   A ( z δ ) − A ( y δ )  Y δ  ≤ C ε | y δ − z δ | 2 δ , where A ( y ) := diag  a 1 ( y ) , . . . , a n ( y )  . Equiv alently ,      n X i =1  a i ( z δ ) − a i ( y δ )  ( Y δ ) ii      ≤ C ε | y δ − z δ | 2 δ − → 0 . Therefore the second-order contribution also tends to 0 as δ ↓ 0. Therefore the righ t-hand side of (1.20) tends to 0 as δ ↓ 0. This contradicts the assumption that W has a strict p ositiv e interior maximum, and hence M ≤ 0. Hence sup J n ( e V ∗ − e V ∗ ) ≤ 0 , that is, e V ∗ ≤ e V ∗ on J n . Since alwa ys e V ∗ ≤ e V ∗ , it follo ws that e V = e V ∗ = e V ∗ on J n . Comp osing with g = f − 1 yields V = V ∗ = V ∗ on E n . Th us V is contin uous on E n , and then on R n + b y Lemma 1. In particular, V is a viscosit y solution of (1.14) on E n . ⊓ ⊔ W e now deriv e lo cal regularity . The argument itself is standard in interior elliptic regularity theory . Ho wev er, since our equation is p osed on the unbounded domain E n and we did not find a reference stated in exactly this form. F or completeness, we include the pro of. Lemma 7. L et K ⋐ E n b e c omp act. Then ther e exists C K > 0 such that | V ( x ) − V ( y ) | ≤ C K | x − y | , ∀ x, y ∈ K. Pr o of. The Hamiltonian F is degenerate elliptic and has no explicit dependence on the space v ariable or on the unknown itself. Moreov er, it satisfies the structural assumptions of Ishii–Lions [12, Sec. VI I.1]. Since 0 ≤ V ≤ 1 on E n , the interior Lipschitz regularity theorem of Ishii–Lions [12, Thm. VI I.2] applies and yields the desired estimate. ⊓ ⊔ W e no w turn to higher interior regularit y . W e briefly indicate the strategy and p ostpone the details to the app endix. With the interior Lipschitz contin uity of V ensured by Lemma 7, V is differentiable almost everywhere b y Rademac her’s theorem. So the nonlinear term G ( ∇ V ) := − b 1 n · ∇ V − max( ∇ V ) is well defined almost ev erywhere in R n + . Since G is globally Lipsc hitz, one may rewrite the HJB equation in the form ∆V = 2 G ( ∇ V ) . 1 Optimal resource allocation for maintaining system solv ency 11 The pro of then pro ceeds in three steps. First, one shows that the identit y ∆V = 2 G ( ∇ V ) holds in the distributional sense on compact subsets of E n . Second, interior Calder´ on–Zygm und estimates imply that V ∈ W 2 ,p loc ( E n ) for every p ∈ (1 , ∞ ) . Third, Sob olev embedding and interior Schauder estimates yield V ∈ C 2 ,α loc ( E n ) for some α ∈ (0 , 1) . F or completeness, we recall the relev ant lo cal function spaces and giv e a detailed proof in the app endix. Lemma 8. F or every p ∈ (1 , ∞ ) , one has V ∈ W 2 ,p loc ( E n ) . Conse quently, ther e exists α ∈ (0 , 1) such that V ∈ C 2 ,α loc ( E n ) , and in p articular V ∈ C 2 ( E n ) ∩ C ( R n + ) . Finally , w e conclude part (iii) of Prop osition 1 by the lemma b elo w. Lemma 9. L et v ∈ C 2 ( E n ) ∩ C ( R n + ) b e a b ounde d classic al solution of (1.4) , (1.5) , and (1.7) . Then v = V . Its pro of is the same as that of Prop osition 2, and is therefore omitted. R emark 1. The pro of for U n is identical in spirit, but the b oundary function on ∂ J n m ust account sim ulta- neously for co ordinates that hit the absorbing b oundary and for co ordinates that are sen t to infinity under the compactification. F or y ∈ ∂ J n , recall I 0 ( y ) := { i : y i = 0 } , I 1 ( y ) := { i : y i = 1 } , m 0 ( y ) := | I 0 ( y ) | , m 1 ( y ) := | I 1 ( y ) | . If m 0 ( y ) + m 1 ( y ) < n , let y | ( I 0 ( y ) ∪ I 1 ( y )) c denote the tuple obtained by deleting all co ordinates equal to 0 or 1. With the conv ention U 0 ≡ 0, define b G : ∂ J n → [0 , n ] b y b G ( y ) := m 1 ( y ) + U n − m 0 ( y ) − m 1 ( y )  f  y | ( I 0 ( y ) ∪ I 1 ( y )) c   . In words, each co ordinate sent to infinity contributes 1, while each co ordinate that hits 0 is remov ed from the system and contributes nothing further. With this definition, the analogue of Lemmas 5 and 6 holds for U n , and the same comparison and regularit y argumen ts apply . This prov es part (iv) of Prop osition 1. 1.3 Pro of of Theorem 2: Conjecture fails for U n This section is dev oted to the pro of of Theorem 2 (ii), namely the failure of Conjecture for U n . W e pro ceed in tw o steps. • First, we treat the tw o-dimensional case by deriving a con tradiction from a mixed Dirichlet–Neumann corner estimate near the origin. • Then w e lift the t wo-dimensional failure to all dimensions b y com bining the boundary condition at infinit y (1.8) with a compactness argumen t. 1.3.1 The tw o-dimensional counterexample Theorem 3. Write u ≡ U 2 . Then ther e exists a nonempty op en set O ⊂ { ( x 1 , x 2 ) ∈ E 2 : x 1 < x 2 } such that ∂ x 2 u ( x 1 , x 2 ) > ∂ x 1 u ( x 1 , x 2 ) , ∀ ( x 1 , x 2 ) ∈ O. 12 Guo, T ang and T ouzi Pr o of. Let D := { ( x 1 , x 2 ) ∈ E 2 : x 1 < x 2 } . W e argue by contradiction and assume that ∂ x 1 u ( x 1 , x 2 ) ≥ ∂ x 2 u ( x 1 , x 2 ) , ∀ ( x 1 , x 2 ) ∈ D. (1.21) Step 1. Under (1.21), one has max { ∂ x 1 u, ∂ x 2 u } = ∂ x 1 u on D . Hence u solv es the linear equation 1 2 ∆u + ( b + 1) ∂ x 1 u + b∂ x 2 u = 0 in D . (1.22) Step 2. The b oundary of D inside R 2 + consists of the tw o rays Γ D := { (0 , x 2 ) : x 2 > 0 } , Γ N := { ( z , z ) : z > 0 } . On the Diric hlet part Γ D , the b oundary condition (1.6) giv es u (0 , x 2 ) = U 1 ( x 2 ) = 1 − e − 2( b +1) x 2 , x 2 > 0 . (1.23) On the diagonal Γ N , the symmetry of u implies ∂ x 1 u ( z , z ) = ∂ x 2 u ( z , z ) , z > 0 . If n denotes the in ward unit normal to D along Γ N , prop ortional to ( − 1 , 1), then ∂ n u = 0 on Γ N . (1.24) Therefore u solv es the mixed b oundary v alue problem        1 2 ∆u + ( b + 1) ∂ x 1 u + b∂ x 2 u = 0 in D , u = U 1 on Γ D , ∂ n u = 0 on Γ N . (1.25) Step 3. Since U 1 ( z ) = 1 − e − 2( b +1) z = 2( b + 1) z + O ( z 2 ) as z ↓ 0 , w e in tro duce the affine function ¯ u ( x 1 , x 2 ) := − 2 b x 1 + 2( b + 1) x 2 . (1.26) Then ∆ ¯ u = 0, and 1 2 ∆ ¯ u + ( b + 1) ∂ x 1 ¯ u + b∂ x 2 ¯ u = ( b + 1)( − 2 b ) + b · 2( b + 1) = 0 . So ¯ u solves the same interior equation as u . Moreov er, ¯ u (0 , x 2 ) = 2( b + 1) x 2 , whic h matc hes the b oundary datum U 1 ( x 2 ) to first order at the origin. Set H := u − ¯ u. Then H solves 1 2 ∆H + ( b + 1) ∂ x 1 H + b∂ x 2 H = 0 in D, (1.27) with b oundary conditions H (0 , x 2 ) = u (0 , x 2 ) − 2( b + 1) x 2 = O ( x 2 2 ) as x 2 ↓ 0 , (1.28) ∂ n H = − ∂ n ¯ u on Γ N . (1.29) Since ¯ u is affine, the Neumann datum in (1.29) is constant. Step 4. A lo cal mixed-b oundary estimate near the corner. 1 Optimal resource allocation for maintaining system solv ency 13 W e no w justify that ∇ H ( x ) − → 0 as x → 0 , x ∈ D. (1.30) Recall that D = { ( x 1 , x 2 ) ∈ R 2 + : x 1 < x 2 } , so that the origin is the unique corner p oin t where the Dirichlet side Γ D = { (0 , x 2 ) : x 2 > 0 } and the Neumann side Γ N = { ( z , z ) : z > 0 } meet. The function H = u − ¯ u satisfies 1 2 ∆H + ( b + 1) ∂ x 1 H + b∂ x 2 H = 0 in D, together with H (0 , x 2 ) = O ( x 2 2 ) as x 2 ↓ 0 , and ∂ n H = − ∂ n ¯ u on Γ N . W e first reduce to homogeneous b oundary conditions. Cho ose r 1 > 0 small, and let ψ ∈ C ∞ ( D ∩ B (0 , r 1 )) b e suc h that ψ (0 , x 2 ) = 0 for (0 , x 2 ) ∈ Γ D ∩ B (0 , r 1 ) , and ∂ n ψ = − ∂ n ¯ u for ( x 1 , x 2 ) ∈ Γ N ∩ B (0 , r 1 ) . Suc h a function may b e constructed by prescribing the Neumann trace on Γ N and multiplying by a cutoff supp orted near the origin. Define K := H − ψ . Then K satisfies, in D ∩ B (0 , r 1 ), 1 2 ∆K + ( b + 1) ∂ x 1 K + b∂ x 2 K = f , (1.31) where f := −  1 2 ∆ψ + ( b + 1) ∂ x 1 ψ + b∂ x 2 ψ  ∈ C ∞ ( D ∩ B (0 , r 1 )) , and the b oundary conditions b ecome K (0 , x 2 ) = O ( x 2 2 ) on Γ D ∩ B (0 , r 1 ) , ∂ n K = 0 on Γ N ∩ B (0 , r 1 ) . (1.32) W e next straighten the geometry . In tro duce the linear change of v ariables y 1 := x 1 , y 2 := x 2 − x 1 . Under this map, the domain D is transformed in to the first quadran t Q := { ( y 1 , y 2 ) : y 1 > 0 , y 2 > 0 } , the Dirichlet side Γ D is mapp ed to { y 1 = 0 , y 2 > 0 } , and the Neumann side Γ N is mapp ed to { y 2 = 0 , y 1 > 0 } . Let e K ( y ) := K ( x ( y )) . Since the change of v ariables is linear, e K satisfies in Q ∩ B (0 , r 2 ), for some r 2 > 0, a linear uniformly elliptic equation of the form 2 X i,j =1 a ij ∂ y i y j e K + 2 X i =1 e b i ∂ y i e K = e f , (1.33) 14 Guo, T ang and T ouzi where ( a ij ) is a constant p ositiv e-definite matrix, e b i are constants, and e f ∈ C ∞ ( Q ∩ B (0 , r 2 )) . Moreo ver, (1.32) b ecomes e K (0 , y 2 ) = O ( y 2 2 ) on { y 1 = 0 } , ∂ y 2 e K = 0 on { y 2 = 0 } , (1.34) after multiplication of the Neumann condition by a nonzero constant if needed. W e no w reflect evenly across the Neumann b oundary . Define, for y 2 < 0, e K ( y 1 , y 2 ) := e K ( y 1 , − y 2 ) . Because the Neumann condition in (1.34) is homogeneous, the reflected function, still denoted by e K , b elongs to C 1 across { y 2 = 0 } and solves, in the weak sense, a linear uniformly elliptic equation with b ounded measurable co efficien ts in the half-ball Q + := { ( y 1 , y 2 ) : y 1 > 0 , | y | < r 2 } . The Dirichlet condition remains e K (0 , y 2 ) = O ( | y 2 | 2 ) on { y 1 = 0 } . A t this p oint one may in vok e the standard b oundary regularit y theory for uniformly elliptic equations near a flat Dirichlet b oundary . Since the boundary datum v anishes to order at least tw o at the corner and the righ t-hand side is smo oth, there exist α ∈ (0 , 1), ρ ∈ (0 , r 2 ), and C > 0 such that |∇ e K ( y ) | ≤ C | y | α , y ∈ Q, | y | < ρ. (1.35) F or instance, this follows from the standard Schauder theory after subtracting a smo oth function matching the Dirichlet trace, or from the regularity results for mixed b oundary v alue problems on corner domains; see, e.g., Grisv ard [10] or Dauge [5]. Returning to the original v ariables, (1.35) implies that there exist α ∈ (0 , 1), ρ ′ > 0, and C ′ > 0 such that |∇ K ( x ) | ≤ C ′ | x | α , x ∈ D ∩ B (0 , ρ ′ ) . (1.36) Since ψ is smooth, w e also hav e ∇ ψ ( x ) → ∇ ψ (0) as x → 0 . By construction, ψ v anishes on Γ D and has constant normal deriv ativ e on Γ N ; in particular, after subtracting an affine correction if necessary , we may assume ∇ ψ (0) = 0 without changing the previous prop erties. Therefore (1.36) implies ∇ H ( x ) = ∇ K ( x ) + ∇ ψ ( x ) − → 0 as x → 0 , x ∈ D , whic h pro ves (1.30). Step 5. Since u = ¯ u + H , b y (1.26) w e ha ve ∂ x 1 u = − 2 b + ∂ x 1 H , ∂ x 2 u = 2( b + 1) + ∂ x 2 H . Hence, by (1.30), ∂ x 1 u ( x ) − → − 2 b, ∂ x 2 u ( x ) − → 2( b + 1) as x → 0 , x ∈ D . Since 2( b + 1) − ( − 2 b ) = 2 + 4 b > 0 , there exists r 0 ∈ (0 , ρ ′ ) such that, for O := { x ∈ D : | x | < r 0 } , one has ∂ x 2 u ( x ) > ∂ x 1 u ( x ) , ∀ x ∈ O. This contradicts (1.21). Therefore (1.21) cannot hold on all of D , and the theorem follows. ⊓ ⊔ 1 Optimal resource allocation for maintaining system solv ency 15 1.3.2 A dimension-lifting argumen t The next lemma pro vides the k ey step for lifting the tw o-dimensional counterexample to all higher dimensions. Lemma 10. Fix m ≥ 2 . F or R > 0 , define the function w R ( x ) := U m +1 ( x, R ) − 1 , x ∈ R m + . Then w R − → U m in C 2 loc ( E m ) as R → ∞ . Pr o of. Fix a compact set K ⋐ E m , and c ho ose another compact set K ′ ⋐ E m suc h that K ⋐ in t( K ′ ) . Set δ := dist( K, ∂ K ′ ) > 0 . Since K ′ ⋐ E m , there exists δ 0 > 0 such that every p oin t of K ′ lies at distance at least δ 0 from ∂ E m . F or R ≥ 1, define the lifted compact sets e K R := { ( x, R ) : x ∈ K } , e K ′ R := { ( x, R ) : x ∈ K ′ } . Then every p oin t of e K ′ R lies at distance at least min( δ 0 , R ) ≥ min( δ 0 , 1) > 0 from ∂ E m +1 . In particular, the distance from e K R to the b oundary of E m +1 is b ounded from b elo w uniformly in R ≥ 1. No w U m +1 is a b ounded viscosity solution of the HJB equation in E m +1 and, by the ( m + 1)-dimensional analogue of Lemma 8, b elongs to C 2 ,α loc ( E m +1 ) for some α ∈ (0 , 1). More precisely , the pro of of Lemma 8 gives in terior W 2 ,p and C 2 ,α estimates on compact subsets, with constants dep ending only on the L ∞ b ound, the ellipticit y constan ts, and the distance to the b oundary . Since 0 ≤ U m +1 ≤ m + 1 on E m +1 , and the distance from e K ′ R to ∂ E m +1 is b ounded below uniformly in R , there exists a constan t C K > 0, indep enden t of R , such that ∥ U m +1 ∥ C 2 ,α ( e K R ) ≤ C K , ∀ R ≥ 1 . Since w R ( x ) = U m +1 ( x, R ) − 1 , x ∈ E m , this implies ∥ w R ∥ C 2 ,α ( K ) ≤ C K , ∀ R ≥ 1 . By the compact embedding C 2 ,α ( K )  → C 2 ( K ) , the family ( w R ) R ≥ 1 is precompact in C 2 ( K ). Since K ⋐ E m w as arbitrary , a diagonal argument sho ws that from every sequence R ℓ → ∞ one ma y extract a subsequence, still denoted by ( R ℓ ), and a function w ∈ C 2 ( E m ) such that w R ℓ → w in C 2 loc ( E m ) . It remains to identify the limit. By the b oundary condition at infinit y (1.8) in dimension m + 1, for ev ery fixed x ∈ E m , lim R →∞ U m +1 ( x, R ) = 1 + U m ( x ) . Equiv alently , lim R →∞ w R ( x ) = U m ( x ) , x ∈ E m . Hence every subsequential limit w must coincide point wise with U m . Therefore w = U m on E m . Since every subsequence has the same limit, the whole family conv erges: w R → U m in C 2 loc ( E m ) as R → ∞ . This completes the pro of. ⊓ ⊔ 16 Guo, T ang and T ouzi 1.3.3 F ailure for all n ≥ 2 Theorem 4. Fix b ≥ 0 and n ≥ 2 . Then Conje ctur e fails for U n : ther e exists x ∈ E n such that x 1 = min( x ) and ∂ x 2 U n ( x ) > ∂ x 1 U n ( x ) . Mor e over, the strict ine quality holds on a nonempty op en subset of { x ∈ E n : x 1 < x 2 } . Pr o of. W e argue by induction on n . F or n = 2, the result is exactly Theorem 3. Assume now that the statement holds in dimension m ≥ 2. Then there exists x ∗ ∈ E m suc h that x ∗ 1 = min( x ∗ ) and ∂ x 2 U m ( x ∗ ) − ∂ x 1 U m ( x ∗ ) ≥ 2 ε (1.37) for some ε > 0. W e show that the statement then holds in dimension m + 1. Cho ose a compact set K ⋐ E m con taining x ∗ in its in terior. By Lemma 10, U m +1 ( · , R ) − 1 − → U m in C 2 ( K ) as R → ∞ . In particular, for i = 1 , 2, ∂ x i U m +1 ( x ∗ , R ) − → ∂ x i U m ( x ∗ ) as R → ∞ . Hence, by (1.37), for R large enough one has ∂ x 2 U m +1 ( x ∗ , R ) − ∂ x 1 U m +1 ( x ∗ , R ) ≥ ε. Cho ose suc h an R moreov er satisfying R > m ax( x ∗ ), and define b x := ( x ∗ , R ) ∈ E m +1 . Then b x 1 = min( b x ) and ∂ x 2 U m +1 ( b x ) > ∂ x 1 U m +1 ( b x ) . This prov es that Conjecture fails in dimension m + 1. The conclusion for all n ≥ 2 follows by induction. Finally , since U n ∈ C 2 ( E n ), the strict inequality at one p oin t implies the same inequalit y on a sufficien tly small neighborho o d of that p oin t. Therefore it holds on a nonempt y op en subset of { x ∈ E n : x 1 < x 2 } . ⊓ ⊔ 1.4 Pro of of Theorem 2: Conjecture holds for V n F or con venience, we recall tw o basic prop erties of V n : • V n is symmetric in its co ordinates; • V n is comp onen twise nondecreasing on R n + . T o pro ve Conjecture , it is enough to establish the following pairwise comparison. Theorem 5. L et b ≥ 0 . Then for al l i  = j and al l x ∈ E n such that x i < x j , ∂ x i V n ( x ) ≥ ∂ x j V n ( x ) . Clearly , Theorem 5 implies Theorem 2. By symmetry , it is enough to prov e the statement for the pair ( i, j ) = (1 , 2), namely x 1 < x 2 = ⇒ ∂ x 1 V n ( x ) ≥ ∂ x 2 V n ( x ) . 1 Optimal resource allocation for maintaining system solv ency 17 1.4.1 Difference quotient on the w edge Define D := { x ∈ E n : x 1 < x 2 } . Fix h > 0, and for x ∈ D define u ( x ) := V n ( x 1 + h, x 2 , x 3 , . . . , x n ) , e u ( x ) := V n ( x 1 , x 2 + h, x 3 , . . . , x n ) , and w h ( x ) := u ( x ) − e u ( x ) . Since V n ∈ C 2 ( E n ) solv es (1.4), both u and e u solve the same HJB equation on D . Subtracting their equations yields 1 2 ∆w h + b n X i =1 ∂ x i w h +  max( ∇ u ) − max( ∇ e u )  = 0 in D . (1.38) Since the map q 7→ max( q ) is conv ex, for each x ∈ D there exists λ ( x ) = ( λ 1 ( x ) , . . . , λ n ( x )) ∈ [0 , 1] n with P k λ k ( x ) = 1 such that max( ∇ u ( x )) − max( ∇ e u ( x )) = n X k =1 λ k ( x ) ∂ x k w h ( x ) . Hence w h solv es the linear equation 1 2 ∆w h + n X k =1 c k ( x ) ∂ x k w h = 0 in D , (1.39) where c k ( x ) := b + λ k ( x ) ∈ [ b, b + 1] . In particular, (1.39) is a linear uniformly elliptic equation with b ounded measurable drift coefficients. 1.4.2 Boundary v alues of w h W e next identify the sign of w h on the finite b oundary of D . (i) The diagonal x 1 = x 2 . If x 1 = x 2 = z > 0, then w h ( z , z , x 3 , . . . , x n ) = V n ( z + h, z , x 3 , . . . , x n ) − V n ( z , z + h, x 3 , . . . , x n ) = 0 b y symmetry of V n in the first tw o co ordinates. (ii) The face x 1 = 0 . If x 1 = 0, then w h (0 , x 2 , x 3 , . . . , x n ) = V n ( h, x 2 , x 3 , . . . , x n ) − V n (0 , x 2 + h, x 3 , . . . , x n ) . The second term v anishes b ecause (0 , x 2 + h, . . . ) ∈ ∂ E n , while the first term is nonnegativ e. Th us w h (0 , x 2 , x 3 , . . . , x n ) ≥ 0 . (iii) The faces x k = 0 for k ≥ 3 . If x k = 0 for some k ∈ { 3 , . . . , n } , then b oth arguments of V n lie on ∂ E n , and therefore w h ( x ) = 0 . Hence w h ≥ 0 on the finite b oundary of D . (1.40) 18 Guo, T ang and T ouzi 1.4.3 Barrier argument and conclusion Define the linear op erator Lφ := 1 2 ∆φ + n X k =1 c k ( x ) ∂ x k φ. Then (1.39) b ecomes Lw h = 0 in D . No w in tro duce Φ ( x ) := x 1 + · · · + x n , u δ ( x ) := w h ( x ) + δ Φ ( x ) , δ > 0 . Since ∆Φ = 0 and ∂ x k Φ = 1, we ha ve LΦ = n X k =1 c k ( x ) = nb + 1 ≥ 1 . Therefore Lu δ = Lw h + δ LΦ = δ ( nb + 1) ≥ δ > 0 in D . On the finite b oundary of D , (1.40) implies u δ = w h + δ Φ ≥ 0 . Moreo ver, since 0 ≤ V n ≤ 1, one has | w h | ≤ 1, so u δ ( x ) ≥ − 1 + δ Φ ( x ) . As | x | → ∞ in D , one has Φ ( x ) → ∞ , hence u δ ( x ) → ∞ . It follows that u δ attains its minim um at some p oin t x ∗ ∈ D . W e claim that this minimum is nonnegative. Indeed, if x ∗ ∈ ∂ D ∩ R n + , then u δ ( x ∗ ) ≥ 0 b y the b oundary condition ab o ve. If x ∗ ∈ D , then u δ ∈ C 2 ( D ) satisfies Lu δ ≥ 0 and attains an in terior minim um. By the strong maxim um principle for linear uniformly elliptic equations with b ounded co efficien ts, u δ m ust b e constant on D . Since u δ ≥ 0 on the finite b oundary , this constant must b e nonnegativ e. Hence in all cases u δ ( x ) ≥ 0 , ∀ x ∈ D . That is, w h ( x ) ≥ − δ Φ ( x ) , ∀ x ∈ D . Letting δ ↓ 0 gives w h ( x ) ≥ 0 , ∀ x ∈ D . By definition of w h , this means V n ( x 1 + h, x 2 , x 3 , . . . , x n ) ≥ V n ( x 1 , x 2 + h, x 3 , . . . , x n ) , ∀ x ∈ D, ∀ h > 0 . Subtracting V n ( x ) from b oth sides, dividing b y h , and letting h ↓ 0, we obtain ∂ x 1 V n ( x ) ≥ ∂ x 2 V n ( x ) , ∀ x ∈ D. This prov es Theorem 5, and therefore Theorem 2 (i). A App endix of related proofs Pr o of (of L emma 1). W e pro ve the stated properties in the order symmetry , bounds, and monotonicity . Symmetry . Let σ b e a p erm utation of { 1 , . . . , n } , and define P x := ( x σ (1) , . . . , x σ ( n ) ) . F or ev ery admissible control Φ = ( φ 1 , . . . , φ n ), define the p erm uted con trol e Φ = ( e φ 1 , . . . , e φ n ) by 1 Optimal resource allocation for maintaining system solv ency 19 e φ i t := φ σ ( i ) t , t ≥ 0 . Since the Brownian motions are indep enden t and identically distributed, the law of the con trolled pro cess started from P x under e Φ is the same as that of the p erm uted process ( X Φ,σ (1) , . . . , X Φ,σ ( n ) ) started from x under Φ . Therefore, V n ( P x ) = V n ( x ) , U n ( P x ) = U n ( x ) . Bounds. F or every admissible con trol Φ ∈ A n and every i = 1 , . . . , n , one has X Φ,i t = x i + Z t 0 ( b + φ i s ) d s + W i t ≤ x i + ( b + 1) t + W i t , t ≥ 0 , since 0 ≤ φ i t ≤ 1. Hence, { τ Φ i = ∞} ⊂ n x i + ( b + 1) t + W i t > 0 , ∀ t ≥ 0 o . Therefore P [ τ Φ i = ∞ | X Φ 0 = x ] ≤ H (( b + 1) + x i ) . By indep endence of the Brownian motions, E h n Y i =1 1 { τ Φ i = ∞}    X Φ 0 = x i ≤ n Y i =1 H (( b + 1) + x i ) , E h n X i =1 1 { τ Φ i = ∞}    X Φ 0 = x i ≤ n X i =1 H (( b + 1) + x i ) . T aking the supremum ov er Φ ∈ A n yields 0 ≤ V n ( x ) ≤ n Y i =1 H (( b + 1) + x i ) , 0 ≤ U n ( x ) ≤ n X i =1 H (( b + 1) + x i ) . Monotonicit y . Fix i ∈ { 1 , . . . , n } and let x, y ∈ R n + b e such that x j = y j for j  = i and x i ≤ y i . F or any admissible control Φ , the corresp onding controlled pro cesses satisfy X Φ,j,y t = X Φ,j,x t , j  = i, X Φ,i,y t = X Φ,i,x t + ( y i − x i ) , t ≥ 0 . Th us τ Φ,y j = τ Φ,x j for j  = i , while τ Φ,y i ≥ τ Φ,x i . Hence n Y j =1 1 { τ Φ,x j = ∞} ≤ n Y j =1 1 { τ Φ,y j = ∞} , n X j =1 1 { τ Φ,x j = ∞} ≤ n X j =1 1 { τ Φ,y j = ∞} . T aking exp ectations and then the supremum ov er Φ ∈ A n pro ves the monotonicity . ⊓ ⊔ Pr o of (of L emma 2). W e pro ve the “if ” and “only if ” parts separately . The “if ” part. Assume first that b > − 1 /n . Consider the constant con trol Φ ∈ A n giv en b y φ i t := 1 n , i = 1 , . . . , n, t ≥ 0 . Then the co ordinates are indep enden t and satisfy X Φ,i t = x i +  b + 1 n  t + W i t , i = 1 , . . . , n. Hence V n ( x ) ≥ n Y i =1 P h x i +  b + 1 n  t + W i t > 0 , ∀ t ≥ 0 i = n Y i =1 H  b + 1 n  + x i  > 0 . Assume next that b > − 1. Consider the control Φ ∈ A n defined by φ 1 t := 1 , φ i t := 0 , i = 2 , . . . , n, t ≥ 0 . Then X Φ, 1 t = x 1 + ( b + 1) t + W 1 t , and therefore U n ( x ) ≥ P  x 1 + ( b + 1) t + W 1 t > 0 , ∀ t ≥ 0  = H (( b + 1) + x 1 ) > 0 . 20 Guo, T ang and T ouzi The “only if ” part for U n . By Lemma 1, 0 ≤ U n ( x ) ≤ n X i =1 H (( b + 1) + x i ) . If b ≤ − 1, then ( b + 1) + = 0, hence eac h term on the right-hand side v anishes and therefore U n ( x ) = 0. The “only if ” part for V n . Fix an arbitrary admissible control Φ ∈ A n and define the sum pro cess S t := n X i =1 X Φ,i t . Then d S t =  nb + n X i =1 φ i t  d t + n X i =1 d W i t . Since P n i =1 φ i t ≤ 1, there exis ts a one-dimensional Bro wnian motion B such that S t = S 0 + Z t 0  nb + n X i =1 φ i s  d s + √ n B t ≤ S 0 + ( nb + 1) t + √ n B t . On the ev ent that all co ordinates survive forever one has S t > 0 for all t ≥ 0. Therefore n τ Φ 1 = · · · = τ Φ n = ∞ o ⊆ n S t > 0 , ∀ t ≥ 0 o . If b ≤ − 1 /n , then the comparison pro cess S 0 + ( nb + 1) t + √ n B t has nonp ositiv e drift, and hence its probabilit y of staying p ositiv e for all time is 0. Consequently , P x  τ Φ 1 = · · · = τ Φ n = ∞  = 0 for every Φ ∈ A n . T aking the supremum ov er Φ gives V n ( x ) = 0 whenever b ≤ − 1 /n . This prov es the lemma. ⊓ ⊔ Pr o of (of L emma 3). The case b ≤ − 1 / ( n − 1) in (i) is immediate since V n − 1 ≡ 0 ≡ V n b y Lemma 2. W e th us assume b ≥ 0 in the sequel. (i) F or every admissible con trol Φ ∈ A n and every k , n Y j =1 1 { τ Φ j = ∞} ≤ Y j  = i 1 { τ Φ j = ∞} , n X j =1 1 { τ Φ j = ∞} ≤ 1 + X j  = i 1 { τ Φ j = ∞} . T aking exp ectations and then the suprem um ov er Φ ∈ A n , and observing that by restricting to con trols with φ i t ≡ 0 the remaining co ordinates are controlled by an element of A n − 1 , we obtain for all z ≥ 0, V n ( x − i , z ) ≤ V n − 1 ( x − i ) , U n ( x − i , z ) ≤ 1 + U n − 1 ( x − i ) . Hence, the monotonicit y yields V n ( x − i , ∞ ) ≤ V n − 1 ( x − i ) , U n ( x − i , ∞ ) ≤ 1 + U n − 1 ( x − i ) . (1.41) (ii) Fix ε > 0. Cho ose Ψ ∈ A n − 1 suc h that E h Y j  = i 1 { τ Ψ j = ∞}    X Ψ 0 = x − i i ≥ V n − 1 ( x − i ) − ε. Let ( x k i ) k ≥ 1 b e an y sequence such that x k i → ∞ as k → ∞ . F or η ∈ (0 , 1), define Φ η ∈ A n b y φ η ,i t := η , ( φ η , 1 t , . . . , φ η ,i − 1 t , φ η ,i +1 t , . . . , φ η ,n t ) := (1 − η ) Ψ t . Then Φ η ∈ A n since P n j =1 φ η ,j t ≤ η + (1 − η ) = 1. Let X η b e the controlled pro cess under Φ η started from ( x − i , x k i ). The i th co ordinate satisfies 1 Optimal resource allocation for maintaining system solv ency 21 d X η ,i t = ( b + η ) d t + d W i t , X η ,i 0 = x k i , and, since b + η > 0, P h τ Φ η i = ∞   X η ,i 0 = x k i i = 1 − e − 2( b + η ) x k i − − − − → k →∞ 1 . (1.42) F or the remaining co ordinates j  = i , the dynamics coincide with those of the ( n − 1)-dimensional system driv en b y (1 − η ) Ψ started from x − i . Denote b y τ ( n − 1) ,η the minim um ruin time among these n − 1 co ordinates. Since b ≥ 0 and Ψ ≥ 0 comp onen twise, decreasing the control in tensity from 1 to 1 − η decreases drifts and therefore decreases surviv al probabilities; in particular, E h Y j  = i 1 { τ (1 − η ) Ψ j = ∞}    X (1 − η ) Ψ 0 = x − i i ↑ E h Y j  = i 1 { τ Ψ j = ∞}    X Ψ 0 = x − i i as η ↓ 0 . Hence, choosing η > 0 small enough, we may ensure E h Y j  = i 1 { τ (1 − η ) Ψ j = ∞}    X (1 − η ) Ψ 0 = x − i i ≥ V n − 1 ( x − i ) − 2 ε. (1.43) Since the Brownian motions are indep enden t and φ η ,i t ≡ η is deterministic, conditioning on the σ -field generated by ( W j ) j  = i yields E h n Y j =1 1 { τ Φ η j = ∞}    X Φ η 0 = ( x − i , x k i ) i = E h Y j  = i 1 { τ (1 − η ) Ψ j = ∞}    X (1 − η ) Ψ 0 = x − i i P h τ Φ η i = ∞   X η ,i 0 = x k i i . Com bining (1.42)–(1.43) and letting k → ∞ giv es V n ( x − i , ∞ ) = lim inf k →∞ V n ( x − i , x k i ) ≥ V n − 1 ( x − i ) − 2 ε. Since ε > 0 is arbitrary , V n ( x − i , ∞ ) ≥ V n − 1 ( x − i ) . (1.44) T ogether with (1.41), this prov es (i) for b ≥ 0. (iii) Fix ε > 0 and choose Ψ ∈ A n − 1 suc h that E h X j  = i 1 { τ Ψ j = ∞}    X Ψ 0 = x − i i ≥ U n − 1 ( x − i ) − ε. Let Φ η b e defined as ab o v e. Then E h n X j =1 1 { τ Φ η j = ∞}    X Φ η 0 = ( x − i , x k i ) i = E h 1 { τ Φ η i = ∞}   X η ,i 0 = x k i i + E h X j  = i 1 { τ (1 − η ) Ψ j = ∞}    X (1 − η ) Ψ 0 = x − i i . The first term conv erges to 1 as k → ∞ b y (1.42). F or the second term, the same monotonicity-in- η argument yields E h X j  = i 1 { τ (1 − η ) Ψ j = ∞}    X (1 − η ) Ψ 0 = x − i i ↑ E h X j  = i 1 { τ Ψ j = ∞}    X Ψ 0 = x − i i as η ↓ 0 , so choosing η > 0 small enough we can ensure E h X j  = i 1 { τ (1 − η ) Ψ j = ∞}    X (1 − η ) Ψ 0 = x − i i ≥ U n − 1 ( x − i ) − 2 ε. Letting k → ∞ then giv es U n ( x − i , ∞ ) = lim inf k →∞ U n ( x − i , x k i ) ≥ 1 + U n − 1 ( x − i ) − 2 ε. Since ε is arbitrary , together with (1.41) we obtain (ii). ⊓ ⊔ Pr o of (of The or em 1). W e prov e s eparately the statements for V n and U n . (i) Optimality of the push-the-laggard strategy for V n when b ≥ 0. By Proposition 1, the v alue function V n b elongs to C 2 ( E n ) ∩ C ( R n + ) and is the unique b ounded classical solution of (1.4), (1.5), and (1.7). 22 Guo, T ang and T ouzi Let Φ ∈ A n b e arbitrary , and let X = ( X 1 , . . . , X n ) b e the corresp onding controlled pro cess started from x ∈ E n . Define τ := min 1 ≤ i ≤ n τ i , σ K := inf { t ≥ 0 : max( X t ) ≥ K } , K > 0 . Then τ ∧ σ K < ∞ a.s. Since V n ∈ C 2 ( E n ) and X t ∈ E n on [0 , τ ∧ σ K ), Itˆ o’s formula gives V n ( X τ ∧ σ K ) = V n ( x ) + n X j =1 Z τ ∧ σ K 0 ∂ x j V n ( X s ) d W j s + Z τ ∧ σ K 0  1 2 ∆V n + b 1 n · ∇ V n + n X j =1 φ j s ∂ x j V n  ( X s ) d s. (1.45) No w, for every y ∈ E n , n X j =1 φ j s ∂ x j V n ( y ) ≤ max( ∇ V n ( y )) , b ecause φ j s ≥ 0 and P n j =1 φ j s ≤ 1. Since V n solv es (1.4), we obtain 1 2 ∆V n + b 1 n · ∇ V n + n X j =1 φ j s ∂ x j V n ≤ 0 on E n . T aking exp ectations in (1.45) therefore yields V n ( x ) ≥ E x  V n ( X τ ∧ σ K )  . (1.46) W e next let K → ∞ . Since 0 ≤ V n ≤ 1, bounded conv ergence may b e applied once we identify the limit of V n ( X τ ∧ σ K ). On { τ < ∞} , one has τ ∧ σ K → τ and X τ ∈ ∂ E n , hence b y (1.5), V n ( X τ ∧ σ K ) − → V n ( X τ ) = 0 . On { τ = ∞} , the pro cess nev er hits the absorbing b oundary , and either σ K < ∞ for all K, or σ K = ∞ for all sufficiently large K . The second alternative cannot o ccur on { τ = ∞} , b ecause if all co ordinates stay ed strictly p ositiv e forev er while remaining b ounded, then the pro cess w ould remain in a compact subset of E n , whic h is incompatible with the Bro wnian fluctuations of the controlled diffusion. Hence, on { τ = ∞} , neces sarily σ K < ∞ for every K and τ ∧ σ K = σ K − → ∞ . Moreo ver, along { τ = ∞} one has max( X σ K ) = K, so at least one co ordinate tends to infinit y along the sequence K → ∞ . Th us, after extracting a subsequence if necessary , for every sample p oin t in { τ = ∞} there exists a nonempt y subset I ⊂ { 1 , . . . , n } such that X i σ K → ∞ for i ∈ I , while the remaining co ordinates either conv erge in [0 , ∞ ] or admit further subsequen tial limits. Rep eated use of the recursive b oundary condition at infinity (1.7), together with the b oundary condition (1.5) on ∂ E n and the contin uity of V n on R n + , implies that along ev ery suc h subsequence, V n ( X σ K ) − → n Y i =1 1 { τ i = ∞} on { τ = ∞} . Indeed, on { τ = ∞} all co ordinates remain strictly p ositiv e, so ev ery co ordinate that sta ys finite contributes through the lo wer-dimensional v alue function, while every co ordinate sent to infinit y is remov ed recursively; after all suc h co ordinates are remov ed, the limiting v alue is V 0 = 1. Com bining the cases { τ < ∞} and { τ = ∞} , w e conclude that 1 Optimal resource allocation for maintaining system solv ency 23 V n ( X τ ∧ σ K ) − → n Y i =1 1 { τ i = ∞} a.s. Since the left-hand side is b ounded by 1, b ounded conv ergence in (1.46) yields V n ( x ) ≥ E x h n Y i =1 1 { τ i = ∞} i . (1.47) As Φ ∈ A n w as arbitrary , this prov es that V n dominates the pay off under every admissible control, as exp ected. W e now sp ecialize to the push-the-laggard feedbac k Φ . By Theorem 2 (i), for ev ery y ∈ E n and ev ery index i suc h that y i = min( y ), ∂ x i V n ( y ) = max( ∇ V n ( y )) . Since Φ allo cates the whole budget to one co ordinate attaining the minimum, we hav e n X j =1 φ j ( y ) ∂ x j V n ( y ) = max( ∇ V n ( y )) , y ∈ E n . Therefore, under Φ , the drift term in (1.45) v anishes iden tically , and V n ( x ) = E x  V n ( X τ ∧ σ K )  for every K > 0 . P assing to the limit K → ∞ exactly as ab ov e yields V n ( x ) = E x h n Y i =1 1 { τ i = ∞} i under Φ. Comparing with (1.47), we conclude that Φ is optimal for V n . (ii) Non-optimalit y of the push-the-laggard strategy for U n . By Theorem 2 (ii), there exists a p oin t x ∈ E n suc h that x 1 < x j for every j ≥ 2 , and ∂ x 2 U n ( x ) > ∂ x 1 U n ( x ) . By contin uit y of ∇ U n and of the co ordinate ordering, after shrinking to a sufficiently small ball B ( x, r ) w e ma y assume that for every y ∈ B ( x, r ), y 1 < y j for every j ≥ 2 , so that the push-the-laggard p olicy allo cates the full budget to co ordinate 1, and moreo ver ∂ x 2 U n ( y ) > ∂ x 1 U n ( y ) . Hence on B ( x, r ) one has n X j =1 φ j ( y ) ∂ x j U n ( y ) = ∂ x 1 U n ( y ) < max( ∇ U n ( y )) . Since U n satisfies (1.4), it follows that 1 2 ∆U n ( y ) + b 1 n · ∇ U n ( y ) + n X j =1 φ j ( y ) ∂ x j U n ( y ) < 0 , y ∈ B ( x, r ) . By compactness there exists ε > 0 such that 1 2 ∆U n ( y ) + b 1 n · ∇ U n ( y ) + n X j =1 φ j ( y ) ∂ x j U n ( y ) ≤ − ε, y ∈ B ( x, r ) . Let X = ( X 1 , . . . , X n ) b e the con trolled pro cess under Φ , started from x , and let θ := inf { t ≥ 0 : X t / ∈ B ( x, r ) } . 24 Guo, T ang and T ouzi Applying Itˆ o’s formula to U n ( X t ) on [0 , θ ] and taking exp ectations gives E x [ U n ( X θ )] ≤ U n ( x ) − ε E x [ θ ] . Since θ > 0 a.s., this yields E x [ U n ( X θ )] < U n ( x ) . (1.48) If Φ were optimal for U n at x , then b y the strong Marko v prop erty the contin uation of Φ after θ would still b e optimal from X θ , implying U n ( x ) = E x [ U n ( X θ )] , whic h con tradicts (1.48). Hence Φ is not optimal for U n . ⊓ ⊔ Pr o of (of Pr op osition 2). The proof is the same verification argument as in the pro of of Theorem 1. Assume first that b ≤ − 1 /n , and let v n ∈ C 2 ( E n ) ∩ C ( R n + ) b e a b ounded solution of (1.4) and (1.5). Cho ose a measurable selector I ( x ) ∈ arg max 1 ≤ j ≤ n ∂ x j v n ( x ) , x ∈ E n , for instance the smallest maximizing index, and define the feedback control by φ i ( x ) := 1 { i = I ( x ) } , i = 1 , . . . , n. Then φ = ( φ 1 , . . . , φ n ) is admissible and realizes the Hamiltonian p oin t wise. Exactly as in the pro of of Theorem 1, Itˆ o’s form ula giv es v n ( x ) = E x  v n ( X τ ∧ σ K )  . No w, setting Y t := P n j =1 X j t , one has d Y t =  nb + n X j =1 φ j t  d t + n X j =1 d W j t , and since nb + 1 ≤ 0, it follo ws that τ < ∞ a.s. Letting K → ∞ and using b oundedness of v n , we obtain v n ( x ) = E x [ v n ( X τ )] . Because X τ ∈ ∂ E n and v n = 0 on ∂ E n , w e conclude that v n ( x ) = 0 for all x ∈ E n . Hence v n ≡ 0 on R n + b y con tinuit y . The pro of for u n is identical. If b ≤ − 1 and u n ∈ C 2 ( E n ) ∩ C ( R n + ) is a b ounded solution of (1.4) and (1.6), define the same maximizing feedback control. Then Itˆ o’s formula giv es u n ( x ) = E x  u n ( X τ ∧ σ K )  . Since now each c oordinate has drift b ounded ab ov e by 0, one has τ < ∞ a.s. Letting K → ∞ yields u n ( x ) = E x [ u n ( X τ )] . Using the boundary condition (1.6) and arguing by induction on n , exactly as in the original pro of, giv es u n ( X τ ) = 0, hence u n ≡ 0. ⊓ ⊔ Pr o of (of L emma 8). W e begin b y recalling the lo cal function spaces used b elo w. F or an op en set O ⊂ R n and p ∈ (1 , ∞ ), the Sob olev space W 2 ,p ( O ) consists of all functions u ∈ L p ( O ) whose weak deriv atives up to order t wo b elong to L p ( O ), endo wed with the norm ∥ u ∥ W 2 ,p ( O ) := X | β |≤ 2 ∥ D β u ∥ L p ( O ) . W e write W 2 ,p loc ( E n ) :=  u : E n → R : u ∈ W 2 ,p ( K ) for ev ery compact K ⋐ E n  . Similarly , for α ∈ (0 , 1) and an open set O ⊂ R n , the H¨ older space C 2 ,α ( O ) consists of all functions u ∈ C 2 ( O ) suc h that ∥ u ∥ C 2 ,α ( O ) := X | β |≤ 2 ∥ D β u ∥ L ∞ ( O ) + X | β | =2 [ D β u ] C 0 ,α ( O ) < ∞ , 1 Optimal resource allocation for maintaining system solv ency 25 where [ w ] C 0 ,α ( O ) := sup x,y ∈ O x  = y | w ( x ) − w ( y ) | | x − y | α . W e write C 2 ,α loc ( E n ) :=  u : E n → R : u ∈ C 2 ,α ( K ) for ev ery compact K ⋐ E n  . Fix compact sets K ⋐ K ′ ⋐ E n . By Lemma 7, V is Lipsc hitz on K ′ . Hence, b y Rademacher’s theorem, V is differen tiable almost everywhere on K ′ and ∇ V ∈ L ∞ ( K ′ ) . Recall that G ( q ) := − b 1 n · q − max( q ) , q ∈ R n . Since G is globally Lipsc hitz, the function f := G ( ∇ V ) is well defined almost everywhere on K ′ and satisfies f ∈ L ∞ ( K ′ ) . Because V is a viscosity solution of (1.14), the equation may b e rewritten as ∆V = 2 f in the viscosit y sense on K ′ . Since the Laplacian is uniformly elliptic and f ∈ L ∞ ( K ′ ), the equiv alence b et w een viscosity and distributional solutions for linear uniformly elliptic equations with b ounded right-hand side implies that V is a distributional solution of ∆V = 2 f in K ′ . A conv enient reference is Caffarelli–Cabr´ e [3, Prop. 2.9]. Let p ∈ (1 , ∞ ) b e arbitrary . By the interior Calder´ on–Zygm und estimate for Poisson’s equation, see Gilbarg–T rudinger [7, Thm. 9.11], there exists a constan t C K,K ′ ,p > 0 such that ∥ V ∥ W 2 ,p ( K ) ≤ C K,K ′ ,p  ∥ V ∥ L p ( K ′ ) + ∥ f ∥ L p ( K ′ )  . Since V is b ounded and f ∈ L ∞ ( K ′ ), the righ t-hand side is finite. Therefore V ∈ W 2 ,p ( K ) . As K ⋐ E n is arbitrary , we conclude that V ∈ W 2 ,p loc ( E n ) for every p ∈ (1 , ∞ ) . Cho ose no w p > n and set α := 1 − n p ∈ (0 , 1) . By the Sob olev em b edding theorem, see Gilbarg–T rudinger [7, Thm. 7.26], W 2 ,p ( K )  → C 1 ,α ( K ) . Hence V ∈ C 1 ,α ( K ) . Since G is globally Lipsc hitz, it follows that f = G ( ∇ V ) ∈ C 0 ,α ( K ) . Finally , applying the in terior Sc hauder estimate for P oisson’s equation, see Gilbarg–T rudinger [7, Thm. 6.2 and Thm. 6.6], we obtain V ∈ C 2 ,α ( K ) . 26 Guo, T ang and T ouzi As K ⋐ E n is arbitrary , this shows that V ∈ C 2 ,α loc ( E n ) . In particular, V ∈ C 2 ( E n ). 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