Nonzero-sum risk-sensitive stochastic differential games

Systems & Control Letters 172 (2023) 105443

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Systems & Control Letters

journal homepage:www.elsevier.com/locate/sysconle

Nonzero-sum risk-sensitive stochastic differential games: A multi-parameter eigenvalue problem approach

^✩

Mrinal K. Ghosh

^a

, K. Suresh Kumar

^b

, Chandan Pal

^c

, Somnath Pradhan

^d,∗

aDepartment of Mathematics, Indian Institute of Science Bangalore, Bengaluru, India

bDepartment of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

cDepartment of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India

dDepartment of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada

a r t i c l e i n f o

Article history:

Received 22 June 2022

Received in revised form 22 November 2022 Accepted 20 December 2022

Available online xxxx Keywords:

Risk-sensitive cost criterion

Parametric family of Markov generators Principal eigenvalue

Nash equilibrium

Hamilton–Jacobi–Bellman equations

a b s t r a c t

We study nonzero-sum stochastic differential games with risk-sensitive ergodic cost criterion. Under certain conditions, using multi parameter eigenvalue approach, we establish the existence of a Nash equilibrium in the space of stationary Markov strategies. We achieve our results by studying the relevant systems of coupled Hamilton–Jacobi–Bellman (HJB) equations. Exploiting the stochastic representation of the principal eigenfunctions we completely characterize Nash equilibrium points in the space of stationary Markov strategies. The complete characterization of Nash equilibrium points is established under an additive structural assumption on the running cost and the drift term.

©2022 Elsevier B.V. All rights reserved.

1. Introduction

We study non zero-sum risk-sensitive stochastic differential games in a multi parameter eigenvalue problem framework. In the literature of stochastic differential games, one usually con- siders the expectation of the integral of costs ([1–3] etc.). This is the so called risk-neutral situation where the players (i.e., the decision makers or controllers) ignore the risk. If the players are risk-sensitive (i.e., risk-averse or risk-seeking), then one of the most appropriate cost criteria is the expectation of the exponen- tial of the integral of costs as it leads to certainty equivalence [4].

Since the cost criterion is the expectation of the exponential of the integral costs, it is multiplicative as opposed to the additive nature of the cost criterion in the expectation of the integral costs case. Due to this, the analysis of the risk-sensitive case is significantly different from its risk-neutral counterpart. To our knowledge, the risk-sensitive criterion was first introduced by Bellman [5]; see [6] and the references therein. Though this criterion has been studied extensively for stochastic optimal con- trol problems [7–21], the corresponding literature in the context of stochastic differential games is rather limited. Some excep- tions are [22–25]. Basar [22] proves the existence of a Nash equilibrium for finite horizon nonzero-sum risk sensitive games.

✩ This paper is dedicated to the memory of Ari Arapostathis.

∗ Corresponding author.

E-mail addresses: mkg@iisc.ac.in(M.K. Ghosh),suresh@math.iitb.ac.in (K.S. Kumar),cpal@iitg.ac.in(C. Pal),sp165@queensu.ca(S. Pradhan).

El-Karoui and Hamadene [25] study risk-sensitive control, zero- sum and nonzero-sum game problems. They prove the existence of an optimal control, a saddle-point and a Nash equilibrium point for relevant cases. In [25], authors use Pontryagin’s mini- mum principle to characterize the optimality condition and the adjoint problem leads to some special backward stochastic dif- ferential equations. Basu and Ghosh [23] study infinite horizon risk-sensitive zero-sum stochastic differential games and estab- lish the existence of saddle points which are mini–max selectors of the associated Hamilton–Jacobi–Isaacs (HJI) equation. In a re- cent work Biswas and Saha [24] consider risk-sensitive zero-sum stochastic differential games for controlled diffusion process in R^d. Under fairly general conditions on the drift and the diffusion coefficients (e.g., the coefficients are locally Lipschitz continuous and have some global growth condition), they study the ergodic cost criterion. They completely characterize saddle point equi- libria in the space of stationary Markov strategies, under the assumption that running cost function satisfies either small cost condition or dominated by some inf-compact function.

In the framework of reflecting diffusions Ghosh and Prad- han [26] (in bounded domain), [27] (in orthant) have studied similar nonzero-sum game problem for risk-sensitive ergodic cost criterion. They studied the game problems by studying the asso- ciated system of coupled HJB equations. In the reflecting diffusion setup, the associated coupled systems are semi-linear elliptic pdes with some oblique boundary conditions. The authors used the principal eigenvalue approach to completely characterize all possible Nash equilibria in the space of stationary Markov strate- gies. Due to the presence of these nontrivial boundary conditions,

https://doi.org/10.1016/j.sysconle.2022.105443 0167-6911/©2022 Elsevier B.V. All rights reserved.

in order to establish the existence of principal eigenpair to as- sociated coupled HJB equations, the authors in [27] crucially used the fact that the drift term, diffusion matrix are uniformly bounded and the running cost function satisfies certain small cost condition. Together with the ergodic cost criterion, in [27] the authors studied the game problem for discounted cost criterion as well.

For controlled diffusion models, similar game problem under discounted cost criterion is studied in [28]. By studying the as- sociated system of coupled HJB equations, which is in this case is a coupled system of semi-linear parabolic pdes, they have established the existence of Nash equilibrium points in the class of eventually stationary Markov strategies. The uniform bound- edness assumptions on the diffusion coefficients and the running cost functions play important role in the analysis of this game problem.

In this paper, we address the existence of Nash equilibria for stochastic differential games where the state of the system is governed by a controlled diffusion processes in the whole space R^d. We consider the risk-sensitive ergodic cost evaluation crite- rion. We analyze this game problem by analyzing the associated system of coupled HJB equation, which is a system of coupled semi-linear elliptic pdes in R^d. Compared to [26–28], under a relatively weaker set of assumptions on diffusion coefficients (e.g., the drift term and diffusion matrix are locally Lipschitz continuous and have some global growth condition) (seeAssump- tion 1), using principal eigenvalue approach we establish the existence of a Nash equilibrium in the space of stationary Markov strategies. Also, in this present study, we are allowing our running cost function to be unbounded as well (seeAssumption 2(ii)).

In order to establish the existence of principal eigenpair of the associated coupled system of Hamilton–Jacobi–Bellman (HJB) equation, we first study the corresponding Dirichlet eigenvalue problem on smooth bounded domains inR^d. Applying a version of non-linear Krein–Rutman theorem we show that principal eigenpair exists for Dirichlet eigenvalue problem. Then increas- ing these domains toR^dand employing Fan’s fixed point theo- rem [29], we establish the existence of principal eigenpair to the associated coupled system of HJB equation in the whole spaceR^d, which lead to the existence of a Nash equilibrium. Furthermore, exploiting the stochastic representation of the principal eigen- functions we completely characterize all possible Nash equilibria in the space of stationary Markov strategies. Thus, the main results of this article can be roughly described as follows.

•

Existence and uniqueness of solution to the coupled HJB equa- tion:Using Principal eigenvalue approach, we establish the existence and uniqueness of solution to the associated cou- pled HJB equation in an appropriate function space.

•

Characterization of Nash equilibrium:Using Fan’s fixed point theorem we first establish the existence of Nash equilibrium in the space of stationary Markov strategies. Then utilizing the stochastic representation of the principal eigenfunctions we completely characterize all possible Nash equilibria in the space of stationary Markov strategies.

The rest of this paper is organized as follows. Section2deals with the problem description. In Section 3 we discuss the principal eigenvalue problem for controlled diffusion operators on smooth bounded domains. Section4is devoted to study the eigenvalue problem for controlled diffusion operator in whole spaceR^d. The complete characterization of Nash equilibrium in the space of stationary Markov strategies is presented in Section5.

2. Problem description

For the sake of notational simplicity we treat two player case.

Let U_i

,

ⁱ

=

1

,

2 be compact metric spaces and V_i

=

P(U_i), the space of probability measures on the compact metric space U_i with the topology of weak convergence. Let_b

¯ =

(_b

¯

₁

, . . . ,

_b

¯

_d)

:

R^d

×

U₁

×

U₂

→

_R^d,

¯

r_i

:

_R^d

×

U₁

×

U₂

→ [

0

, ∞

)

,

ⁱ

=

1

,

^2,

σ :

_R^d

→

_R^d×dbe given functions satisfyingAssumption 1(to be described below).

Defineb

=

(b₁

, . . . ,

^bd)

:

_R^d

×

V₁

×

V₂

→

_R^d

,

^ri

:

_R^d

×

V₁

×

V₂

→ [

0

, ∞

) by

b_k(x

, v

1

, v

2)

=

∫

U2

∫

U1

b

¯

_k(x

,

^u1

,

^u2)

v

1(du₁)

v

2(du₂)

,

r_i(x

, v

1

, v

2)

=

∫

U2

∫

U1

r

¯

_i(x

,

^u1

,

^u2)

v

1(du₁)

v

2(du₂)

,

^x

∈

_R^d

, v

1

∈

V₁

, v

2

∈

V₂

,

^k

=

1

, . . . ,

^d

,

ⁱ

=

1

,

²

.

We consider a nonzero-sum stochastic differential game whose state is evolving according to a controlled diffusion process given by the solution of the following stochastic differential equation (s.d.e.)

dX(t)

=

b(X(t)

, v

1(t)

, v

2(t))dt

+ σ

(X(t))dW(t)

,

^(2.1) where W(

·

) is anR^d-valued standard Wiener process,

v

i(

·

) is a V_i-valued process which is a non-anticipative functional of the state processX(

·

), i.e.,

v

i(t)

=

f_i(t

,

^X(

[

0

,

^t

]

)) whereX(

[

0

,

^t

]

)(s)

=

X(s

∧

t) for alls

∈ [

0

, ∞

) andf_i

: [

0

, ∞

)

×

C(

[

0

, ∞

)

;

_R^d)

→

V_i. Such a strategy is called an admissible strategy. Fori

=

1

,

^2,Ai

denotes the space of all admissible strategies of Playeri. In order to ensure the existence of a solution to Eq.(2.1)and the existence of Nash equilibrium (to be describe in(2.6)), we impose following conditions on the drift termb, and the dispersion matrix

¯ σ

^. Assumption 1.

(i) Local Lipschitz continuity: The function

σ =

[

σ

^ij]

:

_R^d

→

R^d×d and _b

¯ :

_R^d

×

U₁

×

U₂

→

_R^d are locally Lipschitz continuous inx(uniformly with respect to the rest), i.e., for each R

≥

0, there exists a constantC_R

>

0 depending on R

>

0, such that

|¯

b(x

,

^u1

,

^u2)

− ¯

b(y

,

^u1

,

^u2)

|

²

+ ∥ σ

^(x)

− σ

^(y)

∥

²

≤

C_R

|

x

−

y

|

² for all x

,

^y

∈

B_R (

:= {

x

∈

_R^d

: |

x

| <

^R

}

), i

=

1

,

² and (u₁

,

^u2)

∈

U₁

×

U₂, where

∥ σ ∥ :=

√

tr(

σ σ

^{T) and} b

¯ =

(_b

¯

1

, . . . ,

_b

¯

d)^T. Also, we assume that b

,

^ri are jointly continuous in (x

,

^u1

,

^u2) fori

=

1

,

^2.

(ii)Affine growth condition: _b

¯

_and

σ

satisfy a global growth condition of the form

sup

u1∈_U1,u2∈_U2

⟨¯

b(x

,

^u1

,

^u2)

,

^x

⟩

⁺

+∥ σ

^(x)

∥

²

≤

C₀( 1

+|

x

|

²)

∀

x

∈

_R^d

,

for some constantC₀

>

^0.

(iii) Nondegeneracy:For eachR

>

0, it holds that

d

∑

i,j=₁

a^ij(x)z_iz_j

≥

C_R⁻¹

|

z

|

²

∀

x

∈

B_R

,

and for allz

=

(z₁

, . . . ,

^zd)^T

∈

_R^d, wherea

= [

a^ij

] :=

¹

2

σ σ

^T. Also, we assume that the running cost functions_r

¯

_i

:

_R^d

×

U₁

×

U₂

→

_R+i

=

1

,

2 are jointly continuous in (x

,

^u1

,

^u2) and locally Lipschitz continuous in x (uniformly with respect to the rest), i.e., for allR

≥

0 andx

,

^y

∈

B_R there exists a constant C_R

>

⁰ depending onR

>

0, such that

|¯ri(x,^u1,^u2)− ¯ri(y,^u1,^u2)|² ≤ CR|x−y|² for all (u1,^u2)∈U1×U2.

2

M.K. Ghosh, K.S. Kumar, C. Pal et al. Systems & Control Letters 172 (2023) 105443

It is well known that, underAssumption 1, for any (

v

1

, v

2)

∈

A₁

×

A₂ and initial conditionX(0)

=

x, the s.d.e.(2.1)admits a unique weak solution which is a strong Markov process (see [30, Theorem 2.2.11, p.42]). For the stochastic differential game, the controlled diffusion given by (2.1) has the following inter- pretation: The ith player controls the state dynamics, i.e., the controlled diffusion given above, through the choice of her/his strategy

v

i. The functionr

¯

_irepresents the running cost function of Player i. If the strategy

v

ihas the form

v

i(t)

= ¯ v

i(t

,

^X(t))

,

^t

≥

0 for some

v ¯

i

: [

0

, ∞

)

×

_R^d

→

V_i, then

v

ior by an abuse of notation

v ¯

i

is called a Markov strategy for Playeri. LetM_i

= { v

i

: [

0

, ∞

)

×

R^d

→

V_i

| v

iis measurable

}

be the set of all Markov strategies for Playeri. Under a pair of Markov strategies the s.d.e.(2.1)admits a unique strong solution which is a strong Markov process (see [30, Theorem 2.2.12, p.45]). If

v

i does not have explicit depen- dence ont, i.e.,

v ¯

i(t

,

^x)

= ¯ v

i(x)

,

^x

∈

_R^d

,

^t

≥

0, it is said to be a stationary Markov strategy for Playeri. The set of all stationary Markov strategies for Player iis denoted by S_i

,

ⁱ

=

1

,

^{2. We} topologize S_i

,

ⁱ

=

1

,

2, using a metrizable weak* topology on L^∞(R^d

;

M_s(U_i)), where M_s(U_i) denotes the space of all signed measures on U_i with weak* topology. Since S_i is a subset of the unit ball of L^∞(R^d

;

M_s(Ui)), it is compact under the above weak* topology. One also has the following characterization of the topology given by the following convergence criterion: For i

=

1

,

^2,

v

_iⁿ

→ v

i inS_iasn

→ ∞

if and only if

lim

n→∞

∫

R^d

f(x)

∫

U_i

g(x,^ui)viⁿ(x)(du_i)dx =

∫

R^d

f(x)

∫

U_i

g(x,^ui)vi(x)(du_i)dx, (2.2) for allf

∈

L¹(R^d)

∩

L²(R^d)

,

^g

∈

C_b(R^d

×

U_i); see [30, p. 57], for details.

For

v

i

∈

V_i

,

ⁱ

=

1

,

^{2, letLv1,v2}

:

C²(R^d)

→

C(R^d), be given by L^v1,v2f(x)

=

a^ij(x)

∂

^2f(x)

∂

^xi

∂

^xj

+

bi(x

, v

1

, v

2)

∂

^f(x)

∂

^xi

,

^f

∈

C²(R^d)

,

^(2.3) where Einstein summation convention is used. Further, let G₁^v2f

=

inf

v1∈V1

[

L^v₁^1,v2f

+

r₁(x

, v

1

, v

2(x))f

] , v

2

∈

S₂

,

^(2.4)

G₂^v1f

=

inf

v2∈_V2

[

L^v₂^1,v2f

+

r₂(x

, v

1(x)

, v

2)f

] , v

1

∈

S₁

,

^f

∈

C²(R^d)

,

where forf

∈

C²(R^d),

L^v₁^1,v2f(x)

=

L^v1,v2(x)f(x)

∀ v

1

∈

V₁

, v

2

∈

S₂ and

L^v₂^1,v2f(x)

=

L^v1(x),v2f(x)

∀ v

1

∈

S₁

, v

2

∈

V₂

.

For (

v

1

, v

2)

∈

S₁

×

S₂, it is easy to see that

L^v₁^1(x),v2f(x)

=

L^v₂^1,v2(x)f(x)

=

L^v1(x),v2(x)f(x)

=

a^ij(x)

∂

^2f(x)

∂

^xi

∂

^xj

+

b_i(x

, v

1(x)

, v

2(x))

∂

^f(x)

∂

^xi

.

The analysis of our game problem will be based on the analysis of the eigenvalue problems of the above defined operators.

2.1. Ergodic cost criterion

Given the running cost functionsr_i

:

_R^d

×

V₁

×

V₂

→

_R+

,

ⁱ

=

1

,

2, for any (

v

1

, v

2)

∈

A₁

×

A₂, the associated risk-sensitive ergodic cost of Playeriis defined by

ρ

i(x

, v

1

, v

2)

=

lim sup

T→∞

1

T logE^vx^1,v2

[ e

∫T

0 ri(X(t),v1(t),v2(t))dt]

,

ⁱ

=

1

,

²

.

(2.5)

The definition of a Nash equilibrium is standard, i.e., (

v

₁^∗

, v

₂^∗)

∈

A₁

×

A₂ is a Nash equilibrium among the class of admissible strategies if

ρ

1(x

, v

^∗1

, v

2^∗)

≤ ρ

1(x

, v

1

, v

^∗2)

,

^{for all}

v

1

∈

A₁

,

^(2.6)

ρ

2(x

, v

^∗₁

, v

₂^∗)

≤ ρ

2(x

, v

₁^∗

, v

2)

,

^{for all}

v

2

∈

A₂

,

^{for allx}

∈

_R^d

.

We assume that our running cost functions r_i,i

=

1

,

^{2 satisfy} Assumption 1(i). Now for each (

v

1

, v

2)

∈

A₁

×

A₂, define

λ

1(x

, v

2)

=

inf

v^′₁∈A₁

ρ

1(x

, v

₁^′

, v

2)

, λ

1(

v

2)

=

inf

x∈R^d

λ

1(x

, v

2)

,

^(2.7) Λ1(x

, v

2)

=

inf

v^′1∈S₁

ρ

1(x

, v

₁^′

, v

2)

,

Λ1(

v

2)

=

inf

x∈R^d

Λ1(x

, v

2)

, λ

2(x

, v

1)

=

inf

v^′2∈A₂

ρ

2(x

, v

1

, v

^′2)

, λ

2(

v

1)

=

inf

x∈R^d

λ

2(x

, v

1)

,

Λ2(x

, v

1)

=

inf

v^′₂∈S₂

ρ

2(x

, v

1

, v

^′2)

,

Λ2(

v

1)

=

inf

x∈R^d

Λ2(x

, v

1)

.

Now we outline our program for establishing the existence of a Nash equilibrium. We analyze our game problem by analyzing the corresponding system of coupled Hamilton–Jacobi–Bellman (HJB) equations. Suppose that one of the players, say Player 2 announces his strategy

v

2

∈

S₂in advance, then Player 1 tries to minimize associated cost

ρ

1(x

, v

1

, v

2) (see, Eq.(2.5)) over all

v

1

∈

A₁, which is a (stochastic) optimal control problem for Player 1.

Such an optimal control problem has been studied in [7,13,14]

and it is shown that one can characterize the optimal value and optimal controls by analyzing the corresponding HJB equation given by

λ

1

ψ

1(x)

=

G^v₁²

ψ

1(x) with

ψ

1(0)

=

1

.

^(2.8) It is well known that (see [7]) the principal eigenvalue of the HJB equation is the optimal value

λ

1(

v

2) and any minimizing selector of (2.8) (which is same as the minimizing selector of (2.4)), i.e., any

v

₁^∗

∈

S₁which satisfies

λ

1(

v

2)

ψ

1(x)

=

G₁^v2

ψ

1(x)

=

L^v

∗ 1,v2

1

ψ

1

+

r₁(x

, v

^∗1(x)

, v

2(x))

ψ

1

,

is an optimal control for Player 1. In particular,

v

₁^∗

∈

S₁ is an optimal response for Player 1 corresponding to the announced strategy

v

2of Player 2. Note that

v

1^∗depends on

v

2and the map

v

2(

∈

S₂)

→

the optimal responses of Player 1

may be multi-valued. Analogous result holds for Player 2 if Player 1 announces his strategy

v

1

∈

S₁ in advance. From the above discussion, it is easy to see that for any given pair of strategies (

v

1

, v

2)

∈

S₁

×

S₂, one can construct a set of pairs of optimal responses

{

(

v

^∗₁

, v

₂^∗)

∈

S₁

×

S₂

}

from their corresponding HJB equations. Clearly any fixed point of this multi-valued map is a Nash equilibrium. The above discussion leads to the following program for finding a pair of Nash equilibrium strategies for ergodic cost criterion. Suppose that there exist a pair of stationary strategies (

v

₁^∗

, v

₂^∗)

∈

S₁

×

S₂, a pair of scalars (

λ

1

, λ

2) and a pair of functions (

ψ

1

, ψ

2) in an appropriate function space satisfying the following coupled HJB equations

λ

1

ψ

1

=

G^v

∗ 2 1

ψ

1

=

L^v

∗ 1,v^∗₂

1

ψ

1

+

r₁(x

, v

₁^∗(x)

, v

^∗₂^(x))

ψ

1

λ

2

ψ

2

=

G^v

∗ 1 2

ψ

2

=

L^v

∗ 1,v^∗₂

2

ψ

2

+

r₂(x

, v

₁^∗(x)

, v

^∗₂^(x))

ψ

2

,

then (

v

₁^∗

, v

₂^∗) will be a pair of Nash equilibrium. The above discus- sion leads us to study the principal eigenvalues associated with the above coupled equations in the subsequent sections.

3. Dirichlet eigenvalue problem for controlled diffusion oper- ators

In this section, we discuss the principal eigenvalue problem associated with the nonlinear operatorsG_i^vj on smooth bounded

3

domains D

⊂

_R^d. The generalized principal eigenvalue of the semi-linear operatorG^v_i^j with Dirichlet boundary condition onD is defined by

λ

⁺_i ⁽

v

j

,

^D)

=

inf

{ λ ∈

_R

|

for some

ϕ ∈

W^2,p(D)

∩

C(_D)

¯ ,

^p

>

^d

, ϕ >

⁰

,

^G_i^vj

ϕ ≤ λϕ

^inD

} ,

^(3.1) fori

̸=

j,i

,

^j

=

1

,

2. Now we prove the existence of the principal eigenvalues of a certain parametric family of semi-linear elliptic pdes.

Theorem 3.1. Suppose thatAssumption1holds. Let

v

j

∈

S_jand D be a bounded smooth domain inR^d. Then there exists (unique up to a scalar multiplication)

ψ

D

∈

W^2,p(D)

∩

C(_D)

¯ ,

^p

>

^d

, ψ

D

>

^0such that

G_i^vj

ψ

D

= λ

⁺_i ⁽

v

j

,

^D)

ψ

D

,

^(3.2)

ψ

D

=

0 on

∂

^D

,

ⁱ

,

^j

=

1

,

^{2 with i}

̸=

j

.

Proof. We takei

=

1

,

^j

=

2. Supposer₁

≤

0 (this will be dropped shortly). For

φ ∈

C₀¹(D)(

:=

C₀(_D)

¯ ∩

C¹(D)),f

∈

L^p(D), let

Γ1(φ,^f)(x)= − inf v1∈V1

{b_i(x, v1, v2(x))∂φ^(x)

∂^xi

+r₁(x, v1, v2(x))φ^(x)}+f(x),

and consider a_ij(x)

∂

²

φ ˆ

^(x)

∂

^xi

∂

^xj

=

_Γ1(

φ,

^f)(x)

,

^with

φ ˆ =

0 on

∂

^D

.

^(3.3) Then by [31, Theorem 9.15, p.241], [31, Theorem 9.14, p.240], there exists a unique solution

φ ˆ ∈

W^2,p(D)

∩

C(_D),

¯

_p

>

^d, satisfying

∥ ˆ φ ∥

_W2,^p(D)

≤ κ

1(

∥ ˆ φ ∥

∞

+ ∥

_Γ1(

φ,

^f)

∥

_Lp(D))

,

^(3.4) for some positive constant

κ

1

= κ

1(p

,

D) which is independent of

φ ˆ

^,

φ

^,f. From [31, Theorem 9.1, p.220], we deduce that

∥ ˆ φ ∥

∞

≤ κ

2

∥

_Γ1(

φ,

^f)

∥

_Ld(D)

,

for some constant

κ

2

>

0. Hence, from(3.4), we obtain

∥ ˆ φ ∥

_W2,^p(D)

≤ κ

3

∥

_Γ1(

φ,

^f)

∥

_Lp(D) (3.5) for some positive constant

κ

3. Now consider an operatorTmap- ping

φ ∈

C₀¹(D) to the corresponding solution

φ ˆ

^of (3.3), i.e., T(

φ

⁾

= ˆ φ

. Since the embeddingW^2,p(D)

↪ →

C^1,α(D) forp

>

^d and

α ∈

(0

,

¹

−

^d_p) is compact, the operator Tis compact and continuous. Now we want to show that the following space of functions

{ φ ∈

C₀¹(D)

: φ = ν

^T(

φ

^{) for some}

ν ∈ [

0

,

¹

]} ,

is bounded inC₀¹(D). Suppose that there exists a sequence (

φ

n

, ν

n) with

∥ φ

n

∥

_C1

0(D)

→ ∞

and

ν

n

→ ν ∈ [

0

,

¹

]

asn

→ ∞

. Scaling

φ

n

appropriately we assume that

∥ φ

n

∥

_C1

0(D)

=

1. Hence, in view of the estimate(3.5), extracting a suitable subsequence, there exists a nontrivial

φ ˜

^satisfying

a_ij(x)∂²φ˜^(x)

∂^xi∂^xj

= −ν ^inf

v1∈V1

{b_i(x, v1, v2(x))∂φ˜^(x)

∂^xi

+r₁(x, v1, v2(x))φ˜^(x)}, with

φ ˜ =

0 on

∂

D. This is a contradiction to the strong maximum principle [31, Theorem 9.6, p.225]. This implies that the above space is bounded. Hence, by the Leray–Schauder fixed point theorem [31, Theorem 11.3, p.280], it follows that Tadmits a fixed point

ϕ ∈

W^2,p(D)

∩

C(D) , i.e., we have

¯

G₁^v2

ϕ

^(x)

=

f(x)

,

^with

ϕ =

0 on

∂

^D

.

Also, by the strong maximum principle [31, Theorem 9.6] it is clear that

ϕ

satisfying the equation is unique.

LetX

=

C₀(D) andCthe cone of non-negative functions inX.

Now define an operator_T

ˆ

_{which maps f}

∈

Xto corresponding solution

ϕ ∈

W^2,p(D)

∩

C(D) satisfying

¯

G₁^v2

ϕ

^(x)

= −

f(x)

,

^with

ϕ =

0 on

∂

^D

.

From the above discussion it is easy to see that the opera- tor _T

ˆ

is well defined. Thus, combining [31, Theorem 9.1] and [31, Theorem 9.14], we deduce that

∥ ϕ ∥

_W2,^p(D)

≤ κ

1sup

D

| ϕ | ,

^(3.6)

for some positive constant

κ

1. From (3.6), it is clear that _T

ˆ

_is compact and continuous. Also, from the definition one can see that _T

ˆ

is 1-homogeneous (i.e.,_T(

ˆ λ ˜

^f)

= ˜ λ

_T(f

ˆ

_{) for all}

λ ˜ ≥

0).

Suppose _T(f

ˆ

_k)

= ϕ

k, k

=

1

,

^{2, with f}1

≤

f₂. Thus, we have G₁^v2

ϕ

1(x)

≥

G^v₁²

ϕ

2(x). SinceG₁^v2 is concave, it follows thatG₁^v2(

ϕ

2

− ϕ

1)(x)

≤

0. Hence, applying [32, Theorem 3.1] we obtain

ϕ

2

≥ ϕ

1

and iff₁

<

^f2(i.e.,f₁

≤

f₂ andf₁

̸=

f₂) then we have

ϕ

2

> ϕ

1

(see [32, Lemma 3.1]). This implies that_T

ˆ

is order preserving. Let

φ ˜ ∈

Cbe nontrivial nonnegative function with compact support, hence from the above discussion we deduce that_T(

ˆ φ ˜

⁾

>

^{0. Thus,} one can choose

κ

2

>

0 such that

κ

2_T(

ˆ φ ˜

⁾

− ˜ φ >

^{0 in}D. Therefore, by Krein–Rutman theorem (seeTheorem A.1), we conclude that there exists (

λ, ψ ˆ

D)

∈

_R+

×

W^2,p(D)

∩

C(_{D) with}

¯ ψ

D

>

0 satisfying G₁^v2

ψ

D

= ˆ λψ

D in D

,

^and

ψ

D

=

0 on

∂

^D

.

^(3.7) Where

ψ

D is unique up to scalar multiplication. Now, r₁

≥

0 (which is the case by our assumption), since r₁ is bounded in D

¯

replacingr₁by (r₁

− ∥

r₁

∥

∞,^D), following the above arguments there exists (

λ

D

, ψ

D)

∈

_R

×

W^2,p(D)

∩

C(_{D) with}

¯ ψ

D

>

0 satisfying (3.7).

Next, we show that

λ

D

= λ

⁺1(

v

2

,

^D)

.

Clearly,

λ

D

≥ λ

⁺1(

v

2

,

^D)

.

^(3.8)

Suppose

λ

⁺1(

v

2

,

^D)

< λ

D. Then for each

ε >

0, there exists

ε

^′

≤ ε

and

ϕ

^′

∈

W^2,p(D)

∩

C(_D)

¯ , ϕ

^′

>

0 such that

G₁^v2

ϕ

^′

≤

(

λ

⁺₁⁽

v

2

,

^D)

+ ε

^′)

ϕ

^′

.

^(3.9) Choose

ε

^′

>

0 small enough such that

λ

⁺₁⁽

v

2

,

^D)

+ ε

^′

< λ

D. Also, we have

G₁^v2

ψ

D

−

(

λ

⁺₁⁽

v

2

,

^D)

+ ε

^′)

ψ

D

>

^G₁^v2

ψ

D

− λ

D

ψ

D

=

0

.

^(3.10) Hence byTheorem A.3, it follows that

ψ

D

=

t

ϕ

^{′for somet}

>

^0.

This gives a contradiction. Therefore we get

λ

D

= λ

⁺₁⁽

v

2

,

^{D). This} completes the proof. □

4. Eigenvalue problem for controlled diffusion operators inR^d In this section we explore the existence of principal eigenvalue of the controlled diffusion operatorG_i^vj

, v

j

∈

A_jin the whole space R^d and establish their relations with the risk-sensitive ergodic optimal control problem. The generalized principal eigenvalue of G_i^vj in the whole space is defined by

λ

⁺_i ⁽

v

j)

=

inf

{ λ ∈

_R

|

for some

ϕ ∈

W_loc^2,p(R^d)

∩

C(R^d)

,

^p

>

^d

, ϕ >

⁰

,

^Gi^vj

ϕ ≤ λϕ

^a

.

^e

. } .

^(4.1)

In order to study our game problem we enforce following Foster–Lyapunov condition on the dynamics.

4

M.K. Ghosh, K.S. Kumar, C. Pal et al. Systems & Control Letters 172 (2023) 105443

Assumption 2.

(i) In bounded cost case:There existV

∈

C²(R^d) with inf_RdV

≥

1, constants

δ, α > ˜

0 and a compact set_K

˜

_{such that}

sup

ui∈_Ui,ⁱ=₁,²

L^u1,u2V

≤ ˜ α

^I_K^˜

− δ

^V

.

^(4.2) and max_i=₁,2

∥

r_i

∥

∞

< δ

^.

Or,

(ii) In unbounded cost case: There exist V

∈

C²(R^d) with inf_RdV

≥

1, an inf-compact positive

ℓ ∈

C(R^d) (i.e., the sublevel sets

{ ℓ ≤ κ }

are compact, or empty, inR^dfor each

κ ∈

_R), a constant

α > ˜

0 and a compact set_K

˜

_{such that}

sup

ui∈_Ui,i=₁,2

L^u1,u2V

≤ ˜ α

^I_K^˜

− ℓ

^V

,

^(4.3) and fori

=

1

,

²

ℓ

^(x)

−

sup

ui∈_Ui,i=₁,2

r

¯

_i(x

,

^u1

,

^u2) is inf-compact

.

^(4.4) As noted in [7,9], if a and b are bounded, it might not be possible to find an unbounded function

ℓ

which satisfies (4.3).

In view of this, we are assuming(4.2).

Fori

̸=

j, it is easy to see that underAssumption 2(i)

vsup1∈A₁ vsup2∈A₂

lim sup

T→∞

1

T logE^vx^1,v2

[ e

∫T

0ri(X(t),v1(t),v2(t))dt]

≤ ∥ri∥∞<∞. Also, underAssumption 2(ii), applying Itô–Krylov formula, it fol- lows that

vsup1∈A₁ vsup2∈A₂

lim sup

T→∞

1

T logE^vx^1,v2

[ e

∫T 0ℓ(X(t))dt]

≤ α ˜

min_K˜V

.

From(4.4), it is clear that sup_{uk∈Uk,k=1,2}

¯

r_i(

· ,

^u1

,

^u2)

≤ κ

1

+ ℓ

⁽

·

), for some positive constant

κ

1. Therefore, we obtain

sup v1∈A₁

sup v2∈A₂

lim sup

T→∞

1

TlogE^vx^1,v2

[

e^{∫0Tri(X(t),v1(t),v2(t))dt}]

≤κ1+ α˜ min_K˜V.

(4.5) Now we proceed to prove the existence of the principal eigenpair to certain semi-linear elliptic pdes in the whole spaceR^d. Theorem 4.1. LetAssumptions1and2hold. Suppose

v

j

∈

S_j, then there exists a unique

ψ ∈

W_loc^2,p(R^d)

∩

C(R^d)

,

^p

>

^d

, ψ >

^0such that

G_i^vj

ψ = λ

⁺i (

v

j)

ψ

^with

ψ

⁽⁰⁾

=

1

.

^(4.6) Moreover

λ

⁺_i ⁽

v

j)is simple and satisfies

λ

⁺_i ⁽

v

j)

≤ λ

i(

v

j)

,

^{for i}

̸=

j

,

ⁱ

,

^j

=

1

,

²

.

^(4.7) Proof. Takei

=

1

,

^j

=

2. LetD

=

B_n

,

ⁿ

≥

1, denote the open ball centered at the origin with radiusn. FromTheorem 3.1, there exists a (unique)

ψ

n

∈

W^2,p(Bn)

∩

C(_B

¯

_n)

, ψ

n

>

^{0 inB}n with

ψ

n(0)

=

1 satisfying

G₁^v2

ψ

n

= λ

n

ψ

n

ψ

n

=

0 on

∂

^Bn

,

^(4.8)

where

λ

n

= λ

⁺1(

v

2

,

^Bn). Choose

v

1

∈

A₁, since

ψ

n

=

0 on

∂

^Bn

applying Ito–Dynkin’s formula we obtain

ψ

n(x)

≤

_E^v_x^1,v2

[ e

∫T

0(r1(X(t),v1(t),v2(X(t)))−λn)dt

ψ

n(X(T))I{_T≤τ}

]

≤ ∥ ψ

n

∥

∞,^BnE^vx^1,v2

[

e^{∫0T(r1(X(t),v1(t),v2(X(t)))−λn)dt} ]

for all (T

,

^x)

∈

_R+

×

B_n

,

where

τ

is the first exit time of the process X(t) fromB_n and

∥ ψ

n

∥

∞,Bn

:=

sup_x∈_Bn

ψ

n(x). Thus, taking logarithm on both sides of the inequality, dividing byT and lettingT

→ ∞

, it follows that

λ

n

≤

lim sup

T→∞

1

T logE^vx^1,v2

[

e^{∫0Tr1(X(t),v1(t),v2(X(t)))dt}]

< ∞ .

^(4.9) Since

λ

nis nondecreasing inn(see,(3.1)), it follows that lim_n

λ

n

= λ

^exists.

Now using Harnack inequality (see [31, Corollary 8.21, p.199]) and the interior estimates [31, Theorem 9.11, p.235], we get for each bounded domainD, there existsn₀such that

sup

n≥_n0

∥ ψ

n

∥

_W2,^p(D)

< ∞ .

^(4.10)

Hence, by a standard diagonalization procedure and Banach–

Alaoglu theorem, we can extract a subsequence

{ ψ

nk

}

such that for some

ψ ∈

W_loc^2,p(R^d)

∩

C(R^d)

,

^p

≥

2

{ψnk → ψ ^{in W}loc^2,p(R^{d) (weakly)}

ψnk → ψ ^{in C1,α(K)} (strongly) for all compact setK⊂_R^d, (4.11) where 0

< α <

¹

−

^d

p. Now multiplying both sides of(4.8)by a test function

ϕ ∈

C_c^∞(R^d), integrating, and then lettingn

→ ∞

, we deduce that

ψ ∈

W_loc^2,p(R^d)

∩

C(R^d)

,

^p

≥

2 satisfies

G₁^v2

ψ = λψ

ⁱⁿR^d

.

^(4.12)

From(4.9), it follows that

λ ≤ λ

1(

v

2)

.

Since for eachn

∈

_Nwe have

ψ

n

>

0 it is clear that

ψ ≥

0 inR^d and since

ψ

n(0)

=

1 for alln, we have

ψ

⁽⁰⁾

=

1. Thus, applying Harnack’s inequality we deduce that

ψ >

^{0 in}R^d.

Next from the definition of the generalized principal eigen- value, it is immediate that

λ ≥ λ

⁺₁⁽

v

2)

.

^(4.13)

Also from the definition of the generalized principle eigenvalue (see Eq.(3.1)), it follows that

λ

n

= λ

⁺₁⁽

v

2

,

^Bn)

≤ λ

⁺₁⁽

v

2)

.

^(4.14) Thus, combining(4.13)and(4.14)we get

λ = λ

⁺1(

v

2)

.

Next we show that any eigenvalue of G₁^v2 corresponding to a positive eigenfunction in the class W_loc^2,p(R^d)

∩

C(R^d) is simple.

This, in particular, would impliy the simplicity of the generalized principal eigenvalue

λ

⁺1(

v

2).

Let

ψ

k

∈

W_loc^2,p(R^d)

∩

C(R^d)

,

^k

=

1

,

2 be positive eigenfunctions corresponding to an eigenvalue

λ

(in particular, we are interested in

λ = λ

⁺₁⁽

v

2)) satisfying

ψ

k(0)

=

1. Let t₀

>

0 be such that

ψ

1

−

t₀

ψ

2

≥

0 in_B

¯

R.

Let

v

1be a minimizing selector ofG₁^v2

ψ

1. Thus L^v₁^1,v2

ψ

1

+

r₁(x

, v

1(x)

, v

2(x))

ψ

1

=

G^v₁²

ψ

1

= λψ

1

L^v₁^1,v2

ψ

2

+

r₁(x

, v

1(x)

, v

2(x))

ψ

2

≥

G^v₁²

ψ

2

= λψ

2

.

This gives us the following inequality

L^v₁^1,v2(

ψ

1

−

t₀

ψ

2)

+

r₁(x

, v

1(x)

, v

2(x))(

ψ

1

−

t₀

ψ

2)

≤ λ

⁽

ψ

1

−

t₀

ψ

2)

.

Since

ψ

1

−

t₀

ψ

2

≥

0 in_B

¯

R, it follows that

L^v₁^1,v2(