On the independence of the value function for stochastic differential

games of the probability space

We show that the value function in a stochastic differential game does not

change if we keep the same space (Ω,F) but introduce probability measures

by means of Girsanov’s transformation depending on the policies of the players.

We also show that the value function does not change if we allow the driving

Wiener processes to depend on the policies of the players. Finally, we show that

the value function does not change if we perform a random time change with the

rate depending on the policies of the players.

Motivation

First I explain why changing probability space and Wiener process can be

advantageous.

Let D = x ∈ RRd : |x| < 1, let g a Borel bounded function on ∂D, and for

x ∈ D let τ = τ (x) be the first exit time of

xt := x + wt

from D, where wt is a d-dimensional Wiener process. Define

v(x) = Exg(xτ ).

As is well known v is infinitely differentiable in D and satisfies

∆v = 0 in D. (1)

From the explicit representation of v by means of the Poisson formula it follows

that there exists a constant N, depending only on d, such that

|vx(0)| ≤ N sup∂D|g|. (2)

We want to show that this result can be easily obtained probabilistically without

referring to equation (1). However, we only do this for smooth g and v.

For |x| ≤ ε < 1 and

τε = inft ≥ 0 : |wt| ≥ 1− ε

we have (by the strong Markov property)

v(x) = Ev(x + wτε).

Differentiating this at 0 and letting ε ↓ 0 yields (next to) nothing.

However, fix a unit ξ ∈ RRd and take an adapted RRd-measurable say bounded

process πt.

By Girsanov’s theorem for small δ we have

v(δξ) = Ev(δξ + wτε − δ∫ τε

0

πt dt)exp(δ

∫ τε

0

πt dwt − [δ2/2]

∫ τε

0

|πt|2 dt).

Differentiate at δ = 0:

v(ξ)(0) = E

[v(ξτε)

(wτε) + v(wτε)

∫ τε

0

πt dwt

],

where

ξt = ξ −∫ t

0

πs ds. (3)

We choose πt = r(wt)ξt, where ξt is defined by

ξt = ξ −∫ t

0

r(ws)ξs ds, ξt = ξexp(−∫ t

0

r(ws)ds),

∫ τ0

0

r(ws)ds =∞.

Introduction

Let RRd = x = (x1, ...,xd) be a d-dimensional Euclidean space and let d1 ≥ d

be an integer. Assume that we are given separable metric spaces A and B, and

let, for each α ∈ A, β ∈ B, the following functions on RRd be given:

(i) d× d1 matrix-valued σαβ(x) = σ(α,β,x) = (σαβij (x)),

(ii) RRd-valued bαβ(x) = b(α,β,x) = (bαβi (x)), and

(iii) real-valued functions cαβ(x) = c(α,β,x) ≥ 0, fαβ(x) = f(α,β, x), and

g(x).

Under natural assumptions which will be specified later, on a probability space

(Ω,F ,P) carrying a d1-dimensional Wiener process wt one associates with

these objects and a bounded domain G ⊂ RRd a stochastic differential game

with the diffusion term σαβ(x), drift term bαβ(x), discount rate cαβ(x), running

cost fαβ(x), and the final cost g(x) payed when the underlying process first exits

from G.

After the order of players is specified in a certain way it turns out (see our

Remark 0.2) that the value function v(x) of this differential game is a unique

continuous in G viscosity solution of the Isaacs equation

H[v] = 0 (4)

in G with boundary condition v = g on ∂G, where for a sufficiently smooth

function u = u(x)

H[u](x) = sup infα∈A β∈B

[Lαβu(x) + fαβ(x)], (5)

Lαβu(x) := aαβij (x)Diju(x) + bαβi (x)Diu(x)− cαβ(x)u(x),

aαβ(x) := (1/2)σαβ(x)(σαβ(x))∗, Di = ∂/∂xi, Dij = DiDj.

We will assume that σ and b are uniformly Lipschitz with respect to x, σσ∗ is

uniformly nondegenerate, and c and f are uniformly bounded. In such a situation

uniqueness of continuous viscosity solutions or even continuous Lp viscosity

solutions of (5) is shown in [5]

R. Jensen and A. Swiech, Uniqueness and existence of maximal and minimal

solutions of fully nonlinear elliptic PDE , Comm. on Pure Appl. Analysis, Vol. 4

(2005), No. 1, 199–207.

and therefore the fact of the independence of v of the probability space seems

to be obvious.

Roughly speaking, the goal of this talk is to show that the value function does

not change even if we keep the same space (Ω,F) but introduce probability

measures by means of Girsanov’s transformation depending on the policies of

the players. We also show that the value function does not change if we allow

the driving Wiener processes to depend on the policies of the players. Finally,

we show that the value function does not change if we perform a random time

change with the rate depending on the policies of the players.

These facts are well known for controlled diffusion processes and play there

a very important role, in particular, while estimating the derivatives of the value

function. A rather awkward substitute of them for stochastic differential games

was used for the same purposes in [12].

N.V. Krylov, On regularity properties and approximations of value functions for

stochastic differential games in domains, Annals of Probability, Vol. 42 (2014),

No. 5, 2161–2196.

Applying the results presented here one can make many constructions in [12]

more natural and avoid introducing auxiliary “shadow” processes.

However, not all proofs in [12] can be simplified using our present methods.

We deliberately avoided discussing the way to use the external parameters in

contrast with [12] just to make the presentation more transparent.

Our proofs do not use anything from the theory of viscosity solutions and are

based on a version of Swiech’s ([14])

A. Swiech, Another approach to the existence of value functions of stochastic

differential games, J. Math. Anal. Appl., Vol. 204 (1996), No. 3, 884–897.

idea as presented in [11]

N.V. Krylov, On the dynamic programming principle for uniformly nondegen-

erate stochastic differential games in domains, Stochastic Processes and their

Applications, Vol. 123 (2013), No. 8, 3273–3298.

and a general solvability theorem in class C1,1 of Isaacs equations from [9].

N.V. Krylov, On the existence of smooth solutions for fully nonlinear elliptic

equations with measurable “coefficients” without convexity assumptions, Meth-

ods and Applications of Analyis, Vol. 19 (2012), No. 2, 119–146.

Main result

Assumption 1. (i) The functions σαβ(x), bαβ(x), cαβ(x), and fαβ(x) are contin-

uous with respect to β ∈ B for each (α,x) and continuous with respect to α ∈ A

uniformly with respect to β ∈ B for each x. The function g(x) is bounded and

continuous.

(ii) The functions cαβ(x) and fαβ(x) are uniformly continuous with respect to x

uniformly with respect to (α,β) ∈ A×B and for any x ∈ RRd and (α,β) ∈ A×B

‖σαβ(x)‖, |bαβ(x)|, |cαβ(x)|, |fαβ(x)| ≤ K0,

where K0 is a fixed constants and for a matrix σ we denote ‖σ‖2 = trσσ∗,

(iii) For any (α,β) ∈ A×B and x,y ∈ RRd we have

‖σαβ(x)− σαβ(y)‖+ |bαβ(x)− bαβ(y)| ≤ K0|x− y|.

Let (Ω,F ,P) be a complete probability space, let Ft, t ≥ 0 be an increasing

filtration of σ-fields Ft ⊂ F such that each Ft is complete with respect to F ,P.

The set of progressively measurable A-valued processes αt = αt(ω) is de-

noted by A. Similarly we define B as the set of B-valued progressively measur-

able functions. By B we denote the set of B-valued functions β(α·) on A such

that, for any T ∈ (0,∞) and any α1· ,α

2· ∈ A satisfying

P(α1t = α2

t for almost all t ≤ T) = 1, (6)

we have

P(βt(α1· ) = βt(α

2· ) for almost all t ≤ T) = 1.

Definition 0.1. A function pα·β·t = pα·β·

t (ω) given on A × B × Ω × [0,∞) with

values in some measurable space is called a control adapted process if, for any

(α·,β·) ∈ A×B, it is progressively measurable in (ω, t) and, for any T ∈ (0,∞),

we have

P(pα1

· β1·

t = pα2

· β2·

t for almost all t ≤ T) = 1

as long as

P(α1t = α2

t ,β1t = β2

t for almost all t ≤ T) = 1.

Assumption 2. For each α· ∈ A and β· ∈ B we are given control adapted

processes

(i) wα·β·t , t ≥ 0, which are standard d1-dimensional Wiener process relative to

to the filtration Ft, t ≥ 0,

(ii) rα·β·t , t ≥ 0, and πα·β·

t , t ≥ 0, which are real-valued and RRd1-valued, respec-

tively,

(iii) for all values of the arguments

δ−11 ≥ rα·β·

t ≥ δ1, |πα·β·t | ≤ K1,

where δ1 > 0 and K1 ∈ (0,∞) are fixed constants.

Finally we introduce

aαβ(x) := (1/2)σαβ(x)(σαβ(x))∗,

fix a domain G ⊂ RRd, and impose the following.

Assumption 3. G is a bounded domain of class C2 and there exists a constant

δ ∈ (0,1) such that for any α ∈ A, β ∈ B, and x,λ ∈ RRd

δ|λ|2 ≤ aαβij (x)λiλj ≤ δ−1|λ|2.

Remark 0.1. As is well known, if Assumption 3 is satisfied, then there exists a

bounded from above Ψ ∈ C2loc(RR

d) such that Ψ > 0 in G, Ψ = 0 on ∂G, and for

all α ∈ A, β ∈ B, and x ∈ G

LαβΨ(x) + cαβΨ(x) ≤ −1. (7)

For α· ∈ A, β· ∈ B, and x ∈ RRd consider the following Itô equation

xt = x +

∫ t

0

rα·β·s σαsβs(xs)dwα·β·

s

+

∫ t

0

[rα·β·s ]2

[bαsβs(xs) + σ

αsβs(xs)πα·β·s

]ds. (8)

Observe that equation (8) satisfies the usual hypothesis, that is for any α· ∈ A,

β· ∈ B, x ∈ RRd, and T ∈ (0,∞) it has a unique solution on [0,T] denoted by

xα·β·xt and xα·β·x

t is a control adapted process for each x.

Set

φα·β·xt =

∫ t

0

[rα·β·s ]2cαsβs(xα·β·x

s )ds,

ψα·β·xt =

∫ t

0

rα·β·s πα·β·

s dwα·β·s + (1/2)

∫ t

0

[rα·β·s πα·β·

s |2 ds,

define τα·β·x as the first exit time of xα·β·xt from G, and introduce

v(x) = inf supβ∈B α·∈A

Eα·β(α·)x

[ ∫ τ

0

r2t f(xt)e

−φt−ψt dt + g(xτ )e−φτ−ψτ

], (9)

where the indices α·, β, and x at the expectation sign are written to mean that

they should be placed inside the expectation sign wherever and as appropriate,

that is

Eα·β·x

[ ∫ τ

0

r2t f(xt)e

−φt−ψt dt + g(xτ )e−φτ−ψτ

]:= E

[g(xα·β·x

τα·β·x)e−φα·β·x

τα·β·x−ψα·β·xτα·β·x

+

∫ τα·β·x

0

[rα·β·xt ]2fαtβt(xα·β·x

t )e−φα·β·xt −ψα·β·x

t dt].

Observe that, formally, the value xτ may not be defined if τ = ∞. In that case

we set the corresponding terms to equal zero.

Here is our main result.

Theorem 1. Under the above assumptions the function v(x) is independent

of the choice of the probability space, filtration and control adapted process

(r,π,w)α·β·t , it is bounded and continuous in G.

Remark 0.2. Once we know that v(x) is independent of the choice of the proba-

bility space, filtration and control adapted process (r,π,w)α·β·t , we can take any

probability space carrying a d1-dimensional Wiener process wt and construct

v(x) by setting wα·β·t = wt, r ≡ 1, π ≡ 0. In that case we are in the position to

apply the results of [1], [6], and [10]:

Fleming and P. E. Souganidis, On the existence of value functions of two-

player, zero-sum stochastic differential games, Indiana Univ. Math. J., Vol. 38

(1989), No. 2, 293–314.

J. Kovats, Value functions and the Dirichlet problem for Isaacs equation in a

smooth domain, Trans. Amer. Math. Soc., Vol. 361 (2009), No. 8, 4045–4076.

N.V. Krylov, On the dynamic programming principle for uniformly nondegener-

ate stochastic differential games in domains and the Isaacs equations, Probab.

Theory Relat. Fields, Vol. 158 (2014), No. 3, 751–783.

according to which v is continuous in G and satisfies the dynamic programming

principle. Then it is a standard fact that v is a viscosity solution of (4) (see, for

instance, [1], [6], [14]).

[14] A. Swiech, Another approach to the existence of value functions of sto-

chastic differential games, J. Math. Anal. Appl., Vol. 204 (1996), No. 3, 884–

897.

Indeed, if a smooth function ψ(x) is such that ψ(x) ≥ v(x) in a neighborhood

of x0 ∈ G and ψ(x0) = v(x0), then by defining γα·β·ε , ε > 0 as the first exit time

of xα·β·x0

t from the ε-neighborhood of x0 for all small ε we have

ψ(x0) = v(x0) = inf supβ∈B α·∈A

Eα·β(α·)x0

[ ∫ γε

0

f(xt)e−φt dt + v(xγε)e

−φγε]

≤ inf supβ∈B α·∈A

Eα·β(α·)x0

[ ∫ γε

0

f(xt)e−φt dt +ψ(xγε)e

−φγε].

On the other hand set H[ψ] = −h and observe that by Theorem 4.1 of [11]

N.V. Krylov, On the dynamic programming principle for uniformly nondegen-

erate stochastic differential games in domains, Stochastic Processes and their

Applications, Vol. 123 (2013), No. 8, 3273–3298.

ψ(x0) = inf supβ∈B α·∈A

Eα·β(α·)x0

[ ∫ γε

0

(f + h)(xt)e−φt dt +ψ(xγε)e

−φγε].

It follows that

inf infα·∈A β·∈B

Eα·β·x0

∫ γε

0

h(xt)e−φt dt ≤ 0, (10)

and if we assume that H[ψ](x0) < 0, then h > 0 in an ε-neighborhood of x0 and

(10) is impossible, since c is bounded and σ and b are bounded so that Eα·β·x0γε

is bounded away from zero. Hence H[ψ](x0) ≥ 0 and v is a viscosity subsolution

by definition. Similarly one shows that it is a viscosity supersolution.

Provided that we know that continuous viscosity solutions are unique the above

argument proves the fact that the value function is independent of the probability

space (if we drop out r and π and take w independent of the policies). Jensen

[2]

R.R. Jensen, The maximum principle for viscosity solutions of fully nonlinear

second order partial differential equations, Arch. Rational Mech. Anal., Vol. 101

(1988), No. 1, 1–27.

proved uniqueness for Lipschitz continuous viscosity solutions to the fully non-

linear second order elliptic PDE not explicitly depending on x in bounded do-

mains. Related results in the same year with H depending on x were published

in Jensen-Lions-Souganidis [4].

R. Jensen, P.-L. Lions, and P.E. Souganidis, A uniqueness result for viscos-

ity solutions of second order fully nonlinear partial differential equations, Proc.

Amer. Math. Soc., Vol. 102 (1988), No. 4, 975–978.

In what concerns uniformly nondegenerate Isaacs equations, Trudinger in [15]

N.S. Trudinger, On regularity and existence of viscosity solutions of nonlinear

second order, elliptic equations, Partial differential equations and the calculus of

variations, Vol. II, 939–957, Progr. Nonlinear Differential Equations Appl., Vol. 2,

Birkhäuser Boston, Boston, MA, 1989.

proves the existence and uniqueness of continuous viscosity solutions for Isaacs

equations if the coefficients are continuous and a is 1/2 Hölder continuous uni-

formly with respect to α,β (see Corollary 3.4 there). Uniqueness is also stated

for Isaacs equations with Lipschitz continuous a as Corollary 5.11 in [3] (1998):

R.R. Jensen, Viscosity solutions of elliptic partial differential equations, Pro-

ceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998).

Doc. Math. 1998, Extra Vol. III, 31–38.

Jensen and Swiech in [5]

R. Jensen and A. Swiech, Uniqueness and existence of maximal and minimal

solutions of fully nonlinear elliptic PDE , Comm. on Pure Appl. Analysis, Vol. 4

(2005), No. 1, 199–207.

further relaxed the requirement on a to Hölder 1/2− and proved uniqueness of

continuous even Lp-viscosity solutions.

An application

Take G ∈ C3, bounded regular cαβ ≥ 0, σαβ, bαβ, fαβ, and Ψ from the

previous section and assume that on a probability space (Ω,F ,P) we are given

four d1-dimensional independent Wiener processes w(i)t , i = 1, ...,4. Recall that

Ψ > 0 in G, Ψ = 0 on ∂G and

cαβ := −LαβΨ = −[aαβij DijΨ + bαβi DiΨ− cαβΨ] ≥ 1.

We will work in the space RRd × RR4 = z = (x,y) : x ∈ RRd,y ∈ RR4. Set

Ψ(x,y) = Ψ(x)− |y|2 and in RRd × RR4 consider the surface

Γ = z : Ψ(z) = 0.

Denote by DΨ the gradient of Ψ which we view as a column-vector and set

cαβ(x) = −[aαβij DijΨ + bαβi DiΨ].

Next, for α ∈ A,β ∈ B, z = (x,y) ∈ RRd × RR4, and i = 1, ...,4 we define the

functions

σαβ(z), bαβ(i)(z), bαβ(z)

in such a way that on Γ they coincide with

(1/2)[DΨ(x)]∗σαβ(x), −(1/2)yicαβ(x),

|y|2bαβ(x) + 2aαβ(x)DΨ(x),

respectively, and are Lipschitz continuous functions of z with compact support

with Lipschitz constant and support independent of α and β.

Next, we take α· ∈ A, β· ∈ B, z = (x,y) ∈ RRd × RR4 and define

zα·β·zt = (x,y)α·β·z

t

by means of the system

xt = x +

∫ t

0

σαsβs(xs)yis dw(i)

s +

∫ t

0

bαsβs(zs)ds, (11)

yit = yi +

∫ t

0

σαsβs(zs)dw(i)s +

∫ t

0

bαsβs(i)(zs)ds, (12)

i = 1, ...,4.

It turns out that, if z0 ∈ Γ, then zα·β·zt ∈ Γ for all t ≥ 0.

Now we introduce a value function

v(z) = inf supβ∈B α·∈A

Eα·β(α·)z

∫ ∞0

f(xt)e−φt dt,

where

φα·β·zt =

∫ t

0

cαtβt(zα·β·zs )ds.

On G we also have the value function

v(x) = inf supβ∈B α·∈A

Eα·β(α·)x

∫ τ

0

f(xt)e−φt dt,

where xt = xα·β·xt and φt = φ

α·β·xt are such that

dxt = σαtβt(xt)dwt + bαtβt(xt)dt, x0 = x, φt =

∫ t

0

cαsβs(xs)ds,

wt is a d1-dimensional Wiener process on Ω,F ,P), and τ = τα·β·x is the first

exit time of xt from G.

Here is a fundamental fact relating the original differential game in domain G,

which is a domain with boundary, with the one on Γ, which is a closed manifold

without boundary.

Theorem 2. Suppose that g ≡ 0. Then for x ∈ G and y ∈ RRd such that |y|2 =

Ψ(x) we have v(x,y) = v(x)/Ψ(x).

Scetch of the proof. For z0 ∈ Γ equation (11) is rewritten as

dxt =√

Ψ(xt)σαtβt(xt)dwα·β·

t + Ψ(xt)bαtβt(xt)dt + 2aαtβt(xt)DΨ(xt)dt

= rα·β·t σαtβt(xt) dwα·β·

t dt + |rα·β·t |2

[bαtβt(xt) + σ

αtβt(xt)πα·β·t

]dt,

where

dwα·β·t = [Ψ(xt)]

−1/2yit dw

(i)t , rα·β·

t =√

Ψ(xt),

πα·β·t = [Ψ(xt)]

−1[σαtβt(xt)

]∗DΨ(xt).

Itô’s formula shows that

exp(− φα·β·z

t

)= Ψ−1(x)|rα·β·

t |2exp(− φα·β·z

t

)exp

(− ψα·β·z

t

),

where

φα·β·zt =

∫ t

0

|rα·β·t |2cαsβs(xs)ds,

ψα·β·zt = −

∫ t

0

rα·β·s πα·β·

s dwα·β·s − (1/2)

∫ t

0

|rα·β·s πα·β·

s |2 ds.

Hence for |y|2 = Ψ(x)

v(x,y) = Ψ−1(x) inf supβ∈B α·∈A

Eα·β(α·)z

∫ ∞0

r2t f(xt)e

−φt−ψt dt.

Now we can set π = 0, r = 1 and then the right-hand side becomes Ψ−1v.

Further applications

We use Theorem 1 to prove a result to state which we need a few new objects.

In the end of Section 1 of [9] (2012) a function P(uij,ui,u) is constructed defined

for all symmetric d × d matrices (uij), RRd-vectors (ui), and u ∈ RR such that it is

positive-homogeneous of degree one, is Lipschitz continuous, and at all points

of differentiability of P for all values of arguments we have Pu ≤ 0 and

δ|λ|2 ≤ Puijλiλj ≤ δ−1|λ|2,

where δ is a constant in (0,1) depending only on d,K0, and δ. For smooth

enough functions u(x) introduce

P[u](x) = P(Diju(x),Diu(x),u(x))

Part of Theorem 1.1 of [9] says the following. Here G ∈ C1,1.

Theorem 3. Let g ∈ C1,1(RRd). Then for any K ≥ 0 the equation

max(H[u],P[u]−K) = 0 (13)

in G (a.e.) with boundary condition u = g on ∂G has a unique solution u ∈

C0,1(G) ∩C1,1loc(G).

The following result is proved in 2014.

Theorem 4. Denote by uK the function from Theorem 3 and assume that G and

g are of class C3. Then there exists a constant N such that |v − uK| ≤ N/K on

G for K ≥ 1.

A very week version of this theorem was already used in [13] (2014) for estab-

lishing a rate of convergence of finite-difference approximations for solutions of

Isaacs equations.

Probabilistic representation of uK

In this section we suppose that all assumptions in the previous section are

satisfied. Set

A1 = A

and let A2 be a separable metric space having no common points with A1 and

such that

P(uij,ui,u) = supα∈A2

[aαijuij + bαi ui − cαu

].

Extend aαβ...from A1 ×B to

A = A1 ∪A2.

by setting aαβ = aα ...if α ∈ A2. However, fαβ := 0 if α ∈ A2.

Then we introduce A as the set of progressively measurable A-valued pro-

cesses and B as the set of B-valued functions β(α·) on A such that, for any

T ∈ [0,∞) and any α1· ,α

2· ∈ A satisfying

P(α1t = α2

t for almost all t ≤ T) = 1,

we have

P(βt(α1· ) = βt(α

2· ) for almost all t ≤ T) = 1.

Next, take a constant K ≥ 0 and set

vK(x) = inf supβ∈B α·∈A

vα·β(α·)K (x),

where

vα·β·K (x) = Eα·β·

x

[ ∫ τ

0

fK(xt)e−φt dt + g(xτ )e

−φτ]

=: vα·β·(x)−KEα·β·x

∫ τ

0

Iαt∈A2e−φt dt,

fαβK (x) = fαβ(x)−KIα∈A2.

It turns out that

uK = vK.

REFERENCES

[1] W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic

differential games, Indiana Univ. Math. J., Vol. 38 (1989), No. 2, 293–314.

[2] R.R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential

equations, Arch. Rational Mech. Anal., Vol. 101 (1988), No. 1, 1–27.

[3] R.R. Jensen, Viscosity solutions of elliptic partial differential equations, Proceedings of the International Con-

gress of Mathematicians, Vol. III (Berlin, 1998). Doc. Math. 1998, Extra Vol. III, 31–38.

[4] R. Jensen, P.-L. Lions, and P.E. Souganidis, A uniqueness result for viscosity solutions of second order fully

nonlinear partial differential equations, Proc. Amer. Math. Soc., Vol. 102 (1988), No. 4, 975–978.

[5] R. Jensen and A. Swiech, Uniqueness and existence of maximal and minimal solutions of fully nonlinear

elliptic PDE , Comm. on Pure Appl. Analysis, Vol. 4 (2005), No. 1, 199–207.

[6] J. Kovats, Value functions and the Dirichlet problem for Isaacs equation in a smooth domain, Trans. Amer.

Math. Soc., Vol. 361 (2009), No. 8, 4045–4076.

[7] N.V. Krylov, “Controlled diffusion processes”, Nauka, Moscow, 1977 in Russian; English translation Springer,

1980.

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