Existence results for Isaacs equations with local conditions and related semilinear Cauchy problems

∈C × ∩C

×

2 x

x ANNALES

POLONICI MATHEMATICI

Online First version

Existence results for Isaacs equations with local conditions and related semilinear Cauchy problems

Dariusz Zawisza(Kraków)

Abstract.O u r goalistoproveexistenceresultsforclassicalsolutionstosomegeneral nondegenerateCauchyproblemswhicharenaturalgeneralizationsofIsaacsequations.Forthe

latterweare able to extend our resultsbyadmitting local conditions for coefficients. Such equations appear naturally for instance in robust controltheory.Using our general results,wecan solve not only Isaacs equations, but also equations for other sophisticated control problems, for instance models with state dependent constraints on the controlse t .

1. Introduction.Ourmainconcernhereistoprovesomegeneralresults regarding classical solutions (u^2,1(R^N[0, T)) (R^N[0, T])) to the semilinearCauchyproblemofthetype

(1.1) (ut+¹ T

r(a(x,t)D²u)+H(Dxu,u,x,t)=0, (x,t)∈R^N×[0,T),

u(x, T) =β(x), x∈R^N.

Weuseutto denote the derivative with respect tot,Dxuto denotet h e

gradient(ux1,...,u^xN order derivatives.

), andD²uis used to denote the matrix of the second

Our motivation comes from the fact that equation(1.1)canbeused as an excellent starting point to solve many control and dynamic game prob- lems.However,in the existing literature it is usually hard to find sufficiently general and easily verifiable results for classical solutions which canbedi- rectly applied to the HJBtheory.Forinstance, equation(1.1)is a natural generalizationofthefollowingIsaacstypeequation:

2010Mathematics Subject Classification: 35K58, 49J20, 91A15, 91A23.

Keywordsandphrases:Cauchyproblem,Hamilton–Jacobi–Bellman–Isaacsequation,ro- bust control, semilinear parabolic equation, stochasticgame.

Received 5 October 2017; revised 12 June 2018.

Published online 20 September 2018.

DOI:1 0 . 4 0 6 4 / a p 1 7 1 0 0 5 - 2 8 - 6 [1]

0cInstytutMatematycznyPAN,2018

⊂ D. Zawisza

2

(1.2) ut+¹Tr(a(x,t)D₂ ²u)_x + max min(

i(x, t, δ,η)Dxu+h(x, t, δ,η)u+f(x, t, δ,η))

= 0,

δ∈D

η∈Γ (x, t)∈R^N ×[0, T),

with the terminal conditionu(x, T)=0, whereDR^kandΓR^lare fixed compact sets. In stochastic control context, the existence of a classical solution is often crucial to determine the optimal control/saddle point and helpful to establish a convergence rate

for numerical methods.Toexplore

thistopicmore,itisworthtoreadDupuisandJames[8].

Equation(1.2)is very popular in stochastic game theory and has gained a lot of attention recently in robust stochastic optimal control, where it is used to solve optimization problems with model ambiguity (or model misspecifi- cation).Forfinancial aspects of model ambiguity see for example Hernández- HernándezandSchied[16],Schied[27],Tevzadzeetal.[28],Zawisza[33]and referencestherein.Foradiscussionconcerningrobustcontrolinenvironmen- tal economics see Xepapadeas[30],Jasso-Fuentesand López–Barrientos [19]or López-Barrientos et al.[18].In fact they formulate problems in the infinite time horizon setting, but there is no problem in rewriting it in the fixed time framework. The last-mentioned work

provides general existence results for

classicalsolutionstotheassociatedellipticIsaacsequations.

Moreover, equation (1.2) can also be used as the first step in solving ergodic control problems: for the risk sensitive optimization see Fleming andMcEneaney[9],and Zawisza[31]for the consumption-investment problem.

Equation(1.1)canbeused not only to solve Isaacs equations, but also to other non-standard control problems. In finance, it canbeapplied to solve recursive utility problems, for example those consideredbyKraft et al.[17].Wefocus on stochastic control problems with state dependent bounds for the control set.Atthe end of the second sectionwepresent some particular optimaldividendproblemlinkedtothisissue.

Apart from stochastic control applications, our paper has some useful applications in other fields. First of all, for the last few decades, many researchers have investigated the theory of parabolic equations with un- bounded coefficients. For recent contributions in this field see Kunze et al.[21],Angiuli and Lunardi[3]and the survey paper of Lorenzi[24].Our Theorem2.3provides some new existence results in this area.

In addition, our work might be helpful in proving the existence results for forward-backward stochastic systems. The detailed analysis is contained in Ma and Yong[25,Chapter 4]. The link between backward equations and quasilinear equations is mutual,

Existence results for Isaacs equations 3 i.e. some results concerning existence theo- rems for partial differential equations can be proved by applying backward stochastic equations. One of the most general results concerning existence of solutions to equations(1.1)and(1.2)can be deduced from theW^2,1theo-

−

rem proved by BSDE methods in Delarue and Guatteri[7].Their results arestrong enough to cover as well our existence results under our Assumption1(Theorem2.2).However, the importance of our proof lies in the fact that we use the fixed point method with respect to a norm, which ensures that the solution can be uniformly approximated by solutions to linear equations and guarantees relatively fast convergence together with the first derivative.

During the peer revision processwehavealso discovered that the sameset of conditions (Assumption 1) is largely coveredbythe recent result of Addona et al.[2,Theorem 3.6]proved byexploiting the fixed point approach.However,those authors use a slightly different technique which operates on a solution defined on the small timeinterval(T δ, T]and theyhavenotprovedglobal uniform convergence to the fixed point. Moreover, they assumeC^1+αregularity in the space variable for the second order coefficienta.

Thereareofcoursesomeotherrelatedworks.KruzhkovandOle˘ınik’s[2 0]andFriedman’s[15]results work for many Isaacs equations but with trivial second order term (a=I).Rubio[26]considered only stochastic control formulation which is not directly applicable to the max-min framework and other semilinear equations mentioned in this paper. In addition, our last re- sult (Theorem3.3)is strong enough to extend Rubio’s[26]results to the case when the functionsfandβsatisfy the exponential growth condition and the functionhhas linear growth. Ma andYong’stheorem[25,Chapter 4]

holds under smoothness conditions which are not precisely indicated. In addition, it is worth mentioning that standard stochastic controlbookssuch as Flem- ing and Rishel[10]and Fleming and Soner[11] provide general results, but they are not sufficient for many applications.Weshould also mention here the work of Addona[1]where some existence results concerning so called mild solutions to equation(1.1)are considered, and Fleming and Sougani- dis[12]where thevaluefunction of a suitable game

isprovedtobea viscosity

solutionto(1.2)underaglobalLipschitzconditionforcoefficients.

Our paper is structured as follows. First,weprovean existence result under conditions which allow us to apply the approach based on the fun- damental solution and fixed point arguments.

The fixed point approach canbeuseful in obtaining numerical solutions to our equation.Further, weex- tend it to allow some local conditionsbymaking some approximations and transforming the equation into a form which enables us to use a stochastic representation. Such type of approximationwasused earlierbyZawisza[32]toprovean existence result for some infinite

horizon control problems.Atthe endwefocusonanexplicitIsaacsequationforastochasticgameformulation.

2. General results.Westartbyproving an existence theorem under the conditions listed in Assumption1.Further,wewill apply it toprovea suitableresultundertheconditionsgiveninAssumption2.

×

×

b

l l

b l·l

2 x

Assumption1.

(A1) The matrixais of the forma=σσ^T, where the coefficientsσi,j(x, t),i,j=1,...,n,areuniformlybounded,Lipschitzcontinuousoncom- pact subsets inR^N[0, T], and Lipschitz continuous inxuniformly with respect tot. In addition there exists a constantµ >0such that foranyξ∈R^N,

N

ai,j(x,t)ξiξj≥µ|ξ|², (x,t)∈R^N× [0,T].

i,j=1

(A2) The functionβis bounded and uniformly Lipschitz continuous. (A3) The functionHis Hölder continuous on compact subsets ofR^2N+1

[0, T). Moreover, there existsK >0such that for all(p, u, x, t)and (p¯,u¯,x,t)inR^2N+1×[0,T],

(2.1) |H(p,u,x,t)|≤K(1+|u|+|p|),

|H(p,u,x,t)−H(p¯,u¯,x,t)|≤K(|u−u¯|+|p−p¯|).

LetC^1,0standforthespaceofallfunctionswhicharecontinuous,bounded

andhavethe first derivative with respect toxwhich is also continuous and bounded.Thespaceisequippedwiththefamilyofnorms (2.2

) uκ:= sup

(x,t)∈R^N×(0,T]

+ sup

e^−κ(T−t)|u(x, t)|

e^−κ(T−t)|Dxu(x, t)|.

(x,t)∈R^N×(0,T)

Note thatC^1,0with eachκis a Banach space. This norm was inspiredbythe work of Becherer andSchweizer[4]. They use this definition of norm, but without the gradient term. In their paper some semilinear equations are solved, but their setting excludes nonlinearity in the gradient part. The norm(2.2) has also been usedbyBerdjane and Pergamenshchikov[5]to solve semilinear equations in the consumption investment problem, but the nonlinearityintheirequationinvolvesonlythezeroordertermu.

We consider first the linear equation (ut+¹ T

r(a(x,t)D²u)+f(x,t)=0, (x,t)∈R^N×[0,T),

u(x, T) =β(x), x∈R^N.

It is well known (seeFriedman[13,Chapter 1, Theorem 12]) that under (A1)and(A2),forfboundedandlocallyHöldercontinuousinxuniformly withrespecttotoncompactsubsetsofR^n×[0,T),thereexistsaunique

x

∈

b,h

b

b,h

l·l

γ

bounded classical solution given by

T

u(x, t) =\β(y)Γ(x, t, y, T)dy+\ \Γ(x, t, y,s)f(y, s)dy ds,

R^N tR^N

whereΓ(x, t, y, s)is the fundamental solution to the problem Γt+¹Tr(a(x, t)D²Γ) = 0.

2 x

Moreover,

(2.3) \

R^N

Γ(x,t,y,s)dy=1, forx∈R^N,

0≤t<s≤T,

the functionsΓ,Γt,DxΓ,D²Γare continuous on the set ofx, y∈R^Nand 0≤t < s≤T, and there existc, C >0such that

(2.4)

C

|Γ(x,t,y,s)|≤

(s−t)^N/2exp

(−c |y−x|

2s−t C

|DxΓ(x, t, y, s)| ≤

(s−t)^(N+1)/2exp (−c |y−x|

2s−t

(seeFriedman[14,Chapter 6, Theorem 4.5]). In fact Theorem 12 inFried-

man[13]requiresthatfbeHöldercontinuousinxuniformlywithrespecttot[ 0 ,T].No netheless,foruniformityrestrictedtocompactsubsetsof[0,T)the result canbeprovedin the sameway,because fort < T0< Twecan write

T

\ \Γ(x, t, y,s)f(y, s)dy ds

tR^N T0

=\ \

T

Γ(x,t,y,s)f(y,s)dyds+\

\Γ(x, t, y,s)f(y, s)dy ds.

tR^N T0R^N

The first integral on the right hand side can be treated as in Friedman’s proof. In the second one, there is no singularity and standard theorems about differentiation under the integral sign can be applied.

We also consider the subspaceC^1,0of all functionsusuch that:

(1)u∈C^1,0,

(2) for any pair of compact setsB⊂R^NandU⊂(0, T)there existL >0 andγ∈(0,1]such that

|Dxu(x,t)−Dxu(x¯,t)|≤L|x−x¯|, (x,t),(x¯,t)∈B×U.

Note thatC^1,0might notbeclosedin κandthereforeitisnotgenerallya Banach space.Wecan define themapping

,

,

b,h b,h

lfl\√

s−tds= 2C

t

lfl T−t, (2.5

) Tu(x, t) :=\

R^N

β(y)Γ(x, t, y, T)dy

T

+\ \ _H(Dxu(y, s), u(y, s), y, s)Γ(x, t, y, s)dy ds.

tR^N

Proposition2.1.Under Assumption1the operatorTmapsC^1,0into C^1,0and

thereexistsκ>0suchthatTisacontractionwithrespecttol ·lκ. Proof.Suppose that the functionfis continuous, bounded and locally Hölder continuous inxuniformly with respect tot∈U, for any compact setU⊂(0, T). Set

T

v1(x, t) :=\β(y)Γ(x, t,y,T)dy, v2(x, t) :=\ \Γ(x, t, y,s)f(y, s)dy ds.

R^N tR^N

Both functions are bounded and continuous. By the Feynman–Kac formula,

T

v1(x, t) =Ex,tβ(XT), v2(x,t)=Ex,t\f(Xs,s)ds,

t

wheredXt=σ(Xt)dWt,σσ^T=aandEx,tstands for the expected value when the system starts at(x, t). Standard estimates for diffusion processes (see Friedman[14,Chapter 5, Lemma 3.3]) ensure thatv1(x, t)is globally Lipschitz continuous inxuniformly with respect tot. Forv2we have

T

Dxv2(x, t) =\ \DxΓ(x, t, y, s)f(y, s)dy ds,

tR^N

(see Friedman[13,Chapter 1, Theorem 3]). From (2.4)and (2.6)

\ c l_N/2

exp(−c |x−y|² dy=1, s>t,x∈R^N,

we get

R^N4π(s−t)

T

s−t

C (

|x−y|²

(2.7

) |Dxv2(x, t)| ≤\ \_(s−t)^{(N+1)/2exp−c} _s−t f(y,s)dy ds

tR^N

4πl_N/2 c

T 1

4πl_N/2 √ c

c c

≤C

b, T

h

2 x

wherelflstands for the sup norm off. Foru∈C^1,0we can setf(x, t) :=H(Dxu(x, t), u(x, t), x, t). We already know thatw:=uis a classical solu-tion to

wt+¹Tr(a(x, t)D²w)+H(Dxu(x, t), u(x, t), x, t) = 0,(x, t)∈R^N×(0, T),

b,h b,

T h

κ(T−s) ≤ l−l

(

3κ

with the terminal conditionw(x, T)=β(x). In particularDxuis Lipschitz continuous on compact subsets ofR^N×(0, T). This fact together with in-

equality (2.7)ensures that the operatorTm a p s C^1,0intoC^1,0.The next twoestimates wills h o w that is a contraction for sufficiently

largeκ> 0.Using inequality(2.1),property(2.3)and

T

e^−κ(T−t)

\

e^κ(T−s)ds=1

e^−κ(T−t)[e^κ(T−t)−1]≤1

κ , κ

t

we get

e^−κ(T−t)|Tu(x, t)− Tv(x, t)|

T

≤e^−κ(T−t)K

\ \(|u(y, s)−v(y, s)|+|Dxu(y, s)−Dxv(y, s)|)

tR^N

T ×Γ(x,t,y,s)dyds

≤e^−κ(T−t)Klu−vlκ\ Γ(x,t,y,s)e dyds ^K u vκ. κ

In addition,

tR^N

e^−κ(T−t)|Dx(Tu(x, t)− Tv(x, t))|

T

≤e^−κ(T−t)K

\ \|H(Dxu(y, s), u(y, s), y, s)−H(Dxv(y, s), v(y, s), y, s)|

tR^N

T × |DxΓ(x, t, y, s)|dy ds

≤e^−κ(T−t)K

\ \⁽|u(y, s)−v(y, s)|+|Dxu(y, s)−Dxv(y, s)|)

tR^N

C

×(s−t)^(N+1)/2exp (−c

|y−x|

2s−t dy ds.

Onceagain,(2.6)impliesthatthereexistsM¯>0suchthat

e^−κ(T−t)|Dx(Tu(x,t)−Tv(x,t))|≤M¯e^κt

T

lu−vlκ\

t

e^−κs

√s−tds

We have

≤M¯e^κtlu−vlκ T

\(s−t)^−3/4ds

t

T\ e^−3κsds

t

1/3.

e^κt

T\ e^−3κsds

t

1/3

=e^κt1 3κ [e^−3κt−e^−3κT]

1/3 1

≤√₃ .

2/3

\

Therefore, there exists a constantL >0, depending only on the time hori-

∩C

∈C

b,h

b

T

γ

k

∈ C

b,h

l·l

b,h

∈ l·l

zonT, such that sup

(x,t)∈R^N×[0,T) e^−κ(T−t) Dx(Tu(x, t)− Tv(x, t)) L

≤√₃

κlu−vlκ. Theorem2.2.Under Assumption1,thereexists as o l u t i o n u

2,1(R^N(0,T)) (R^N

(0, T])to(1.1)which in addition isboundedtogetherwithDxu.

Proof.As in the proof of the Banach Theorem, we can take anyu1∈C^1,0and defineun+1=Tun,n∈N. Because the operatorTis a contraction in norm, there existsδ∈(0,1)such that

n

Henc e,

lun+1−unlκ≤δ

m−1

lu2−u¹lκ, n∈N.

lum−unlκ≤ δ^klu2−u1lκ, m >n,

k=n

which implies thatunis aCauchysequence and consequently it is convergent touC^1,0inκ.The norm convergence implies that the sequenceDxunconverges uniformly to somev(R^N[0, T)). In particular,v=Dxu.Moreover,uis a fixed point of.

Tocomplete the reasoning it is sufficient toprovethatualso belongs toC^1,0.First note thatunis convergent inκ(forκlarge enough). Therefore,unandDxunare bounded uniformly with respect ton.Wecannowcombine (E8), (E9) from Fleming and Rishel[10,Appendix E] toprovea uniform bound on compact subsets for the Hölder norm ofDxun,i.e. for allk∈Nthere existLk>0andγk∈(0,1]such that for alln ∈N,

|Dxun(x,t)−Dxun(x¯,t)|≤Lk|x−x¯|, (x,t),(x¯,t)∈Bk×[δ^k,tk], whereBk={x∈R^N| |x| ≤k}and{δk}_k∈Nand{tk}_k∈Nare sequences converging to0andTrespectively. Lettingn→ ∞proves thatDxu∈C^1,0.

Now we describe the second set of conditions.

Assumption2.

(B1) The matrixai,j(x, t)is Lipschitz continuous on compact subsets inR^N×[0, T]. In addition, there exists a constantµ

>0such that foranyξ∈R^N,

N

ai,j(x,t)ξiξj≥µ|ξ|², (x,t)∈R^N×[0,T].

i,j=1

(B2) The functionHis Hölder continuous on compact subsets ofR^2N+1×[0, T]. Moreover, there existK >0and a

set{Km,n>0 :m, n∈N}suchthatforallx,x¯,p,p¯∈R^N,u,u¯∈R,andt∈[0,T],

∈ −

−

k

m

l





2−|p|/l ifl≤ |p| ≤2l, (2.8)

(2.9) (2.10 ) (2.1 1) (2.12 )

|H(0,0, x, t)| ≤K,

H(0,u,x,t)−H(0,u¯,x,t)≤K(u−u¯)ifu>u¯,

|H(p,u,x,t)−H(p,u¯,x,t)|≤Km,n|u−u¯|if|u|,|u¯|≤m,|x|≤n,

|H(0, u, x, t)| ≤Km,nif|u| ≤m,|x| ≤n,

|H(p,u,x,t)−H(p¯,u,x,t)|≤Km,n|p−p¯|if|u|≤m,|x|≤n.

(B3) The functionβis bounded and Lipschitz continuous on compact sub- sets ofR^N.

Theorem2.3.Under Assumption2, there exists a bounded solutionu∈C^2,1(R^N×[0, T))∩ C(R^N×[0, T])to(1.1).

Proof.Note that forε >0wecan defineaandHalsofort [ε, T]bytheformula

a(x, t) :=a(x,0),

H(p,u,x,t):=H(p,u,x,0), t∈[−ε,0),(p,u,x)∈R^2N+1. Notice thatH(pN, p_N−1, . . . , p1, u, x, t)can be written as (2.13)

H(pN,pN−1,...,p1,u,x,t)= ^N

i=1

Hⁱ(pi,u,x,t)−Hⁱ⁻¹(p_i−1,u,x,t)pi

H

(0 , u, x, t ) (0H , 0 , x, t )

+ u+H(0,0,x,t),

u

whereHⁱ(pi, u, x, t) :=H(0, . . . ,0, pi, . . . , p2, p1, u, x, t). Consider now a newHamiltonian of the form

H_k,m,l(p, u, x, t) :=ξ¹(x)ξ²(u)ξ³(p)H(p,u,x, t), k,m,l∈N,

where ^{k m l}

ξ¹(x) :=

ξ²(u) :=

ξ³(p) :=

1 if|x| ≤k,

2−|x|/k ifk≤ |x|≤2k,0 if|x| ≥2k,

1 if|u| ≤m,

2− |u|/mifm≤ |u| ≤2m, 0 if|u| ≥2m,

1 if|p|≤l,

0 if|u|≥2l.











p

i

Notice that for a fixed compact setB⊂R^2N+1×[−ε, T]there exists a

(

k k

(

2 x

i−1

u(x,t)

0, u(x,t) = 0.

k

collection of sufficiently large indices such that

Hk,m,l(p, u, x, t) =H(p,u,x,t), (p,u,x,t)∈B.

Moreover, for fixedk, m, l∈Nthere existsL(k, m, l)>0such that for all (p,u,x,t),(p¯,u¯,x,t)∈R^2N+1×[0,T]wehave

|Hk,m,l(p, u, x, t)| ≤L(k, l, m)(1 +|p|+|u|),

|Hk,m,l(p,u,x,t)−Hk,m,l(p¯,u¯,x,t)|≤L(k,l,m)(|u−u¯|+|p−p¯|).

Therefore, Theorem2.2can be applied for the HamiltonianHk,m,l. Suppose thatσis the unique positive definite square root ofa. By Friedman[14,Chapter 6, Lemma 1.1],σis Lipschitz continuous on compact subsets ofR^N×[0, T]. Define

σ(x,t):= σ(x,t) if|x|≤k,

σ(kx/|x|,t) if|x|>k, ak:=σk σ^T, β_k

(x) :=ξ¹(x)β(x).

This implies that there exists a bounded solutionu_k,m,l∈C^2,1(R^N×(−ε, T))

∩ C(R^N×(−ε, T])to ut+¹ Tr(

a_k(x,t)D²u)+H_k,m,l(Dxu,u,x,t)=0, (x,t)∈R^N×(−ε,T),

u(x, T) =β_k(x), x∈R^N.

Our reasoning here is based on Arzelà–Ascoli’s Lemma, so we need to prove some bounds for derivatives ofuk,m,l. Taking advantage of(2.13)we can find Borel measurable functionsb_k,m,l,h_k,m,landf_k,m,lsuch thatu_k,m,lis a solution to

ut+¹Tr(a_k(x, t)D²u) +b_k,m,l(x, t)Dxu+h_k,m,l(x, t)u+f_k,m,l(x, t) = 0

2 x

with the terminal conditionu(x, T)=βk(x). Namely, let fk,m,l(x, t) :=Hk,m,l(0,0, x, t),

[Hⁱ (ux(x,t),u(x,t),x,t)−H (u

x (x, t), u(x, t), x, t)]

ik,m,l (x, t) := ^k,m,l

i

k,m,l

ux_i(x, t)

i−1

(ifux_i(x, t)/= 0and0otherwise), and (_{[ H}

k ,m,l (0 ,u ( x,t ) ,x,t ) − H k ,m,l , 0 ,x,t(0 )]

, u(x,t)/=0, Conditions(2.8)and(2.9)imply

h_k,m,l(x,t)≤K, |fk,m,l(x,t)|≤K, b

hk,m,l(x, t) :=

∈ ∈ ×−

\ e

\

thk,m,l(Xl,l)dlf_k,m,l(Xs, s)ds+e

\

thk,m,l(Xl,l)dlβ_k(XT)

t

for allk,m , l Nand(x,t) R^N (ε, T].WecannowusethestandardFeynman–

Kactypetheoremtoobtainastochasticrepresentationoftheform

T s T

uk,m,l(x,t)=Ex,t ,

k

2 x

{∈ | | | ≤}

× −

l l

l l

≤ | |l|− |

∈

wheredXt=b_k,m,l(Xt, t)dt+σ_k(Xt, t)dWt,σkσ^T=a_k. The existence of a strong solution to this stochastic differential equationwas provedbyVereten- nikov[29].Since the functionsβandf_k,m,lare bounded, andhk,m,lis bounded above, there existsm^∗>0independent ofkandmsuch that

|uk,m,l(x, t)| ≤m^∗.

This indicates thatuk,l(x, t) :=uk,m^∗,l(x, t)is a solution to (ut+¹ T

r(ak(x,t)D²u)+Hk,l(Dxu,u,x,t)=0, (x,t)∈R^N×(−ε,T),

u(x, T) =β(x), x∈R^N,

whereHk,l(p, u, x, t) :=ξ¹(x)ξ³(p)H(p, u, x, t). Repeating the procedure

de- k l

scribed above, we can find Borel measurable functionsbk,l,hk,landfk,lsuch thatuk,lis a solution to

ut+¹Tr(ak(x, t)D²u) +bk,l(x, t)Dxu+hk,l(x, t)u+fk,l(x, t) = 0

2 x

with the terminal conditionu(x, T)=β_k(x). We have h_k,l(x,t)≤K, |fk,l(x,t)|≤K an

d

|h_k,l(x, t)|≤Km^∗,n, (x,t)∈Bn×[0,T].

Westillneedaboundwhichisindependentofk,l.ToapplyArzelà–Ascoli’s Lemma it is sufficient toprovesuch a bound for each setBn[δn,tn],w h e r e Bn=xR^Nx nandδnandtnare sequences converging toεandTrespectively.Toget the estimates,wefirst consideranyfunctionϕwhichsatisfiestheuniformLipschitzconditionwi thconstantL>0:

|ϕ(z)−ϕ(z¯)|≤L|z−z¯|, z,z¯∈R^N. The Lipschitz condition implies linear growth:

|ϕ(z)|≤L|z|+|ϕ(0)|, z∈R^N.

Next,weneedtoestimate|ξ³(z)ϕ(z)−ξ³(z¯)ϕ(z¯)|f o r z,z¯∈ R^N.Wecan assumethat|z|≤ 2 lor|z¯|≤ 2 l.Otherwise|ξ³(z)ϕ(z)−ξ³(z¯)ϕ(z¯)|= 0.

Withoutlossofgeneralitywecanassumethat|z¯|≤2l.Wehave

3 3 3 3 3

|ξ_l(z)ϕ(z)−ξ^l(z¯)ϕ(z¯)|≤|ξ^l(z)||ϕ(z)−ϕ(z¯)|+|ϕ(z¯)||ξ^l(z)−ξ₃ ^l(z¯)|₃

≤L|z−z¯|+(2lL+|ϕ(0)|)|ξ_l(z)−ξ^l(z¯)|

L+1 (2

lL+ϕ(0))z z¯, z,z¯

R^N. l

k,l

i k,

l

l ^{m ,n}

l ^{m ,n m ,n m ,n x}_i ⁿ

for(x, t)∈Bn×[−δn, tn].

x

x

∈

× −

×−

× −

× −

× −

2 x

x

x

Therefore, using additionally(2.11)and(2.12),we get Hⁱ

(ux(x,t),u(x,t),x,t)−

Hⁱ⁻¹(

ux (x, t), u(x, t), x, t)

≤1

(2lK^∗ +|H(0, u, x, t)|) +K^∗ l|u(x,t)|

≤1(2

lK^∗ +K^∗)+K^∗l|u(x,t)|, (x,t)∈B×[0,T],k>n.

This implies that the coefficientbk,lis uniformly boundedonBn

[δn,tn] for sufficiently largel.

So far we have obtained uniform bounds forb_k,l,h_k,l,f_k,lonBn[δn, tn].

To find bounds foruk,l,(uk,l)t,Dxuk,l,D²uk,land their Hölder norms uni- formly on every setBn×[0, tn], we make the following reasoning:

(1) Weuse Lieberman [13, Ths. 7.20, 7.22] to get uniform bounds forL^p(Bn[δn,tn])norms ofu_k,l,(u_k,l)t,Dxu_k,l,D²u_k,l.Fora more generalandmorereadableresult,seeCrandalletal.[6,Theorem9.1].

(2) Weuse Fleming and Rishel[10,Appendix E, E9] to get uniform bounds foru_k,l,Dxu_k,land their Hölder normsonBn [δn,tn].

(3) Weu s e b o u n d s f o r u_k,landD^xu_k,lt o e n s u r e t h a t f o r fi x e d n for sufficiently largek, lwe have Nand

Hk,l

(Dxuk,l(x, t), uk,l(x, t), x, t)

=H(

Dxuk,l(x, t), uk,l(x, t), x, t)

(4) Wecan use this fact to obtain a uniform bound on the Hölder normon

Bn [δn, tn]for the familyHk,l(Dxuk,l(x,t),uk,l(x,t), x,t).

(5) Wealready know thatuk,lis a classical solution to thep r o b l e m ut+¹Tr(ak(x, t)D²u)+Hk,l(Dxuk,l, uk,l, x, t) = 0,(x, t)∈Bn×[−δn, tn].

(6) Now, it is sufficient to apply Fleming and Rishel[10,Appendix E, E10] (which is in fact due to Ladyzhenskaya et al.

[22,Chapter IV, Theorem 10.1]) to get uniform bounds for the remaining derivatives and theirHöldernorms.

The bounds for the derivatives ensure thatuk,l, (u_k,l)t,Dxu_k,l,D²u_k,lareuniformly bounded, while the bounds for the Hölder norms ensure equicon-tinuity ofuk,l, (u_k,l)tDxu_k,l,D²u_k,lonBn×[0,tn].Thus,wecan use the Arzelà–

AscoliLemmaoneachsetBn×[0,tn]todeducethatforeachgivense-

m ,n x

i i−1

x n µ

quence(kn,ln,n∈N)thereexistsasubsequence(knµ,lnµ,µ∈N)suchthat the sequences(u_knµ,lnµ, µ∈N),((u_knµ,lnµ)t,µ∈N),(Dxu_knµ,lnµ, µ∈N ),

(D²uk ,

l_nµ , µ∈N)are uniformly convergent onBn×[0, tn]. By the stan- dard diagonal argument, there exists a sequence(knµ, lnµ, µ∈N)such that(uk_nµ,l_nµ, µ∈N)converges locally uniformly together with suitable deriva- tives to a functionu∈C^2,1(R^N×[0, T)).

×{}

x,t

≥ ⊂

≥

∈

≤≤ | −

|

k,l

x,t,T

k,l

≤IEx,t[Z

x,t,T

I

Now, we need only prove thatuis continuous on the boundaryR^NT. Let us apply the Itô rule to the functionuk,land the stochastic systemdXt(k)=σ_k(Xt(k), t)dWt, and write

E^k,lu_k,l(X_T∧τ(x,t)(k), T∧τ_k(x, t)) =u_k,l(x, t)

x,t ^k

k,l x,t

T∧τk(x,t)

\

t

[−hk,l(Xs(k), s)u_k,l(Xs(k), s)−fk,l(Xs(k), s)]ds, whereτk(x, t)=inf{s≥t|Xs(k)(x, t)/∈B}for a sufficiently large closed ballB. The symbolE^k,lis used to denote the expected value under the

measure given by the Girsanov transform

k,l _{T∧τ(x,t) T∧τ(x,t)}

dQ :=Z^k,l :=e\^t k

σ⁻¹b_k,l(Xs(k),s)dWs−^1\_t

k

|σ⁻¹b_k,l(Xs(k),s)|²ds.

k 2 k

dP ^x,t,T

Notethatthedefinitionofτdoesnotdependonkbecausethereexistsk0N such that for allk k0we haveBBk,and consequently ifk,l k0thenby Friedman[14,Theorem 2.1, Section 5]wegetP(τk(x,t) =τl(x,t)) = 1andP(sup_{t s τk(x,t)}Xs(k)Xs(l)= 0) = 1. Therefore,wewill further omit the variablekin the notation for the processXand the stopping time

τ(x,t). Until random timeτ(x, t)the processXtakes itsvaluesinB,and the coefficientsbk,l,hk,l,fk,lare uniformly bounded onB×[0, T].Takeany(x,t)∈B×[0,T],andsupposethatx¯∈IntBand(x,t)∈IntB×[0,T].

The nThe

n|uk,l(x,t)−β(x¯)|≤|Ex,tZ_x,t,Tu^k,l(X_T∧τ(x,t),T∧τ(x,t))−β(x¯)|

T∧τ(x,t)

Furtherm ore,

k,l

+E_x,tZ^k,l

\ |hk,l(Xs(k),s)uk,l(Xs,s)+fk,l(Xs,s)|ds.

t

|Ex,tZ_x,t,Tu^k,l(X_T∧τ(x,t),T∧τ(x,t))−β(x¯)|

≤Ex,t Z_x,t,T|u^{k ,l} (X_T∧τ(x,t),T∧τ(x,t))−β(x¯)|

k,lx, t,T

The random variable[Z^k,l

]² Ex,t|uk,l(X_T∧τ(x,t),T∧τ(x,t))−β(x¯)|². ]²can be rewritten as a product of the Girsanov

exponent and a uniformly bounded random variable. In addition, Ex,t|uk,l(X_T∧τ(x,t),T∧τ(x,t))−β(x¯)|²

+E

t≤s≤T∧τ(x,t) s B

=Ex,t|β(X_T∧τ(x,t))−β(x¯)|²χ_{{sup |X|<R}}

+Ex,t|uk,l(X_T∧τ(x,t),T∧τ(x,t))−β(x¯)|²χ_{sup

=:I1+I2,

t≤s≤T∧τ(x,t) |Xs|≥RB}

→ | |≤

≥

\σ(Xr)dWr≥RB− |x|

t

≤R

Ex,t

−|x|

\

t

σ(Xr, r)dWr.

Ex, t

Tr(σ(Xr, r)σ(Xr, r))dr

t

∈/

≤

whereRBdenotes the radius ofB.The expressionI1is independent ofk,lfork,l k0andbythe standard diffusion estimates it converges to0as(x,t)

( x¯,T).ThesameholdsforI2becauseuk,l(x,t)m^{∗a n d}bymartingaleinequalities wehave

s

1 ^T∧τ(x,t)

B

Additionally, by the Itô isometry and the Cauchy–Schwarz inequality,

Ex, t

T∧τ(x,t)

\ σ(Xr, r)dWr t

T∧τ(x,t)

≤

\

t

1/2

≤

Asaconsequence,|uk,l(x,t)−β(x¯)|

admitsanestimatewhichisindependentofk,landconvergesto0as(x,t)→(x¯,T ).Therefore,thesameholdsfor

|u(x,t)−β(x¯)|.Thisimpliesthecontinuityofu.

As mentioned in the Introduction, our result can be applied to models with state dependent bounds for the control set. Let us consider the follow- ing example describing a variant of the optimal dividend payment problem, which is one of the most important actuarial control problems. Let us definethe insurer surplus process:

dXt= [µ−dt]dt+σ dWt,

whereµ, σR,σ= 0andWis a one-dimensional Brownian motion. The

progressively measurable processdtis the

dividendpaymentintensity.Weassumethatdtcannotexceedsomefractionofthes urplusprocess,andthere is nopaymentat all when the surplus is P_x,

t

sup

t≤s≤T∧τ(x,t)|Xs| ≥RB ≤P_x,t sup

t≤s≤T∧τ(x,t)

P_x,

t

sup

t≤s≤T∧τ(x,t)|Xs| ≥RB ≤P_x,t sup

t≤s≤T∧τ(x,t)

B

→0 ast→T.

negative. So,weshouldalwayshave0dtκX⁺.The problem of the

insurer is to maximize the

overalldiscountedutilityofdividendpayments,i.e.

T

Ex,t\e^−w(k−t)f(d_k)dk,

t

where the functionfcanbeconsidered as a utility function andwas a discountrate.Intheformulationoftheproblemwecanalsousesomepenalty functiontopenalizetheobjectiveforallowingthesurplustobenegative,but

−

×

∈

∈

×

0≤

− )

¯

d≤m(x, t

−₀

≤ )

0≤d≤m(x,t) 0≤d≤m(x¯,¯

t)

0≤ ) −₀

≤ )

≤₀

≤ )

that is not crucial to our analysis. The HJB equation for this problem is

ut+¹ σ 2D²u

+ max [(µ−d)Dxu+f(d)]−wu=0, u(x,T) = 0.

2

x

In this case

0≤d≤κx⁺

(2.14) H(p,u, x, t)= max[ ( µ d)p+f(d) wu].

0≤d≤κx⁺

More generally, we assume that

(2.15) H(p,u,x,t)= max

0≤d≤m(x,t)h(p, u, x, t, d).

Proposition2.4.Letthe functionhbeLipschitz continuous oncom-

pactsubsets ofR³[0, T]R, satisfy

conditions(2.8)and(2.9)uniformlywithrespecttodRand conditions(2.10)–

(2.12)uniformly withrespecttod UforallcompactsUR, and let the functionmbeLipschitz contin-uous oncompactsubsets ofR[0, T]. Then the functionHgiven by(2.15)satisfies(B2).

Proof.Almost all conditions in(B2)concerning the variablespandu are trivial or very easy to prove by just using the inequality

d≤m(x,max

t

h(p,u,x,t,d) max

) 0≤d≤m(x,t

≤0≤max

h(p¯,u¯,x,t,d)

)|h(p,u,x,t,d)−h(p¯,u¯,x,t,d)|.

d≤m(x,t

Local Lipschitz continuity in(x, t)is much harder to prove. For fixed

(p¯,u¯)∈R^N+1wehave

max h(p¯,u¯,x,t,d)− max h(p¯,u¯,x¯,t¯,d)

0≤d≤m(x,t)

≤0≤max

)

+ max

0≤d≤m(x¯,t)

h(p¯,u¯,x,t,d) max

d≤m(x,t

h(p¯,u¯,x¯,t¯,d)− max

h(p¯,u¯,x¯,t¯,d) h(p¯,u¯,x¯,t¯,d).

The first term on the right hand side canbeestimatedbyusing the assumed local Lipschitz continuity: for a given compact setB⊂R³×[0, T]thereexistsLB>0suchthatforall(p¯,u¯,x,t),(p¯,u¯,x¯,t¯)∈B, (2.16) max

d≤m(x,t h(p¯,u¯,x,t,d) max

d≤m(x,t h(p¯,u¯,x¯,t¯,d)

d≤m(x,tmax |h(p¯,u¯,x,t,d)−h(p¯,u¯,x¯,t¯,d)|≤LB(|x−x¯|+|t−t¯|).

To estimate the second term we will consider three cases.

CaseI:Themaximumofh(p¯,u¯,x¯,t¯,d)over[0,m(x¯,t¯)]isattainedatsomep ointd^∗/=m(x¯,t¯).The n byl ocalL ip s chitzco ntinuityo fm andh we

0≤d≤m(x,t) 0≤d≤m(x¯,¯ t)

−

⊂ ×

∈

¯

∈

∈

∈

canfindasufficie ntlysmallneighbourhoodof(x¯,t¯)suchthatthemaxi mumofh(p¯,u¯,x¯,t¯,d)over0≤d≤m(x,t)isstillattainedatd^∗.Inthatcase

max h(p¯,u¯,x¯,t¯,d)− max h(p¯,u¯,x¯,t¯,d)=0.

CaseII:Themaximumofh(p¯,u¯,x¯,t¯,d)isattainedatd^∗=m(x¯,t¯)andm(

x,t)<m(x¯,t¯).Thentherestillexistsaneighbourhoodsuchthatmax_{0≤d≤m(x,t)} h(p¯,u¯,x¯,t¯,d)=h(p¯,u¯,x¯,t¯,d^∗).

CaseIII:Themaximumofh(p¯,u¯,x¯,t¯,d)isattainedatd^∗=m(x¯,t¯)andm(

x,t)>m(x¯,t¯).Thenthemaximumover[0,m(x,t)]isattainedatdˆ∈[m(x¯,t¯),m(

x,t)].Inthatcase max

h(p¯,u¯,x¯,t¯,d) max

¯ h(p¯,u¯,x¯,t¯,d)

0≤d≤m(x¯,

t) 0≤d≤m(x,t)

=|h(p¯,u¯,x¯,t¯,d^∗)−h(p¯,u¯,x¯,t¯,dˆ)|.

ThefunctionhisLipschitzcontinuousoncompactsubsetsofR³×[0,T]×R, so for every compact setB⊂R³×[0, T]there existsL >0such that for all(p¯,u¯,x¯,t¯)∈B,

|h(p¯,u¯,x¯,t¯,d^∗)−h(p¯,u¯,x¯,t¯,dˆ)|≤L|d^∗−

dˆ|≤L|m(x¯,t¯)−m(x,t)|.

It remains to apply the assumed local Lipschitz continuity ofm.

Collecting all inequalities together,wefind that for any compact setBR³[0,T]thereexistsaconstantL>0suchthatforany(p¯,u¯,x¯,t¯)Bthereis asmallneighbourhoodU(p¯,u¯,x¯,t¯)suchthat

(2.17) max

h(p,u,x,t,d)− max h(p¯,u¯,x¯,t¯,d)

0≤d≤m(x,t) 0≤d≤m(x¯,t)

≤L|(p,u,x,t)−(p¯,u¯,x¯,t¯)|

forall(p,u,x,t,d)U(p¯,u¯,x¯,t¯).ThefactthattheconstantL>0dependsonlyonth ecompactsetBandnotontheparticularchoiceof(p¯,u¯,x¯,t¯)implies local LipschitzcontinuityofH.Namely,letB⊂R³×[0, T]be

a compact and convex set of the form{x∈R³| |x| ≤R} ×[0, T]. Fix z=(p,u,x,t),z¯=(p¯,u¯,x¯,t¯)∈B andconsiderthecompactset(theline connectingzandz¯)

O_[z,z¯]={z+α(z¯−z)|α∈[0,1]}⊂B.

Foreach pointz Bthere existsUz(wemay assume it is an open ball)onwhich(2.17)holds.CompactnessofO[z,z¯]impliesthatthereexistfinitel ymanypointsz=z1,z2,...,zn=z¯O_[z,z¯](wecanorderthemaccordingtothe

increasing euclidean distance fromz)such that for every ordered pair

( ) zi,zi+1wehavezⁱ,zi+1∈U¯^ziorzⁱ,zi+1∈U¯^zi+1.Inthatcase

n n

|H(z)−H(z¯)|≤|

H(zi)−H(zi−1)|≤L|

zi−zi−1|=L|z−z¯|.

3. Isaacs equation.Now, our primary concern is to solve the semilinear equation

(3.1) ut+¹Tr(a(x,t)D²u)

2 x

+maxmini(x,t,δ,η)Dxu+h(x,t,δ,η)u+f(x,t,δ,η)= 0,

δ∈D η∈Γ

(x, t)∈R^N×[0, T), with the terminal conditionu(x, T)=β(x).

Assumption3.

(C1)T h e matrix[ai,j(x,t)],i,j=1,...,N,issymmetric,anditscoefficients areLipschitzcontinuousoncompactsubsetsinR^N×[0,T].Inaddition thereexistsaconstantµ>0suchthatforanyξ∈R^N,

N

ai,j(x,t)ξiξj≥µ|ξ|², (x,t)∈R^N× [0,T].

i,j=1

(C2) The functionsf,h,iare continuous, and there exists a strictly positive sequence{Ln}n∈Nsuch that for allζ=f,h, iand allδ∈D,η∈Γ,(x, t)∈Bn×[0, T],

|ζ(x,t,δ,η)−ζ(x¯,t¯,δ,η)|≤Ln(|x−x¯|+|t−t¯|).

(C3) The functionβis uniformly bounded and Lipschitz continuous on compact subsets ofR^N.

(C4) The functionfis uniformly bounded andhis bounded above.WecannowpresentanimmediateconsequenceofTheorem 2.3.

Corollary3.1.Under Assumption3, there exists a bounded classicalsolutionu∈ C^2,1(R^N×[0, T))∩ C(R^N×[0, T])to(3.1).

Proof.For the proof it is sufficient to define

H(p, u, x, t) := max minΠ(p, u, x, t, δ, η),

δ∈D η∈Γ

where

Π(p,u, x, t, δ,η)=i(x, t, δ,η)p+h(x, t, δ,η)u+f(x, t, δ,η), and use the inequality

i=2 i=

i= 2

2 i=

2

|H(p,u,x,t)−H(p¯,u¯,x¯,t¯)|≤maxmax|Π(p,u,x,t,δ,η)−Π(p¯,u¯,x¯,t¯,δ,η)|.

δ∈D η∈Γ

In some cases it is possible to extend the above result to the case when

fandgmay be unbounded. We first need the following lemma:

Lemma3.2.Assume thatX(n)is a strong solution to dXt=bn(Xt, t, ω)dt+σn(Xt, t, ω)dWt, wherebnandσnare sequences of continuous functions such that

bn:R^N×[0, T]×Ω→R^N,

σn:R^N×[0, T]×Ω→L(R^N,R^N) and there existK, M>0such that for allx∈R^N,n∈Nandω∈Ω,

|bn(x,t,ω)|≤K(1+|x|), |σn(x,t,ω)|≤M.

ThenforallA>0thereexistsacontinuousfunctionRˆ

n∈Nand(x, t)∈R^N×[0, T], such that for all

Ex,tsup

t≤s≤Te^A|Xs(n)|≤Rˆ(x).

Proof.Westartbyproving some pathwise inequalities which hold almost surely inΩ.Ifbnhas linear growth, then there existsK >0such that for allk∈[t, T],

k k

|Xk(n)|≤|x|+KT+ K\|Xs(n)|ds+sup \σn(Xs(n),s,ω)dWs.

Therefor e,

t 0≤k≤Tt

k

wher e

|Xk(n)| ≤AT+K\|Xs(n)|ds, t≤k≤T,

t