Proceedings of the International Congress of Mathematicians
Vancouver, 1974
Stochastic Differential Games with Stopping Times
and Variational Inequalities
Avner Friedman
Consider a system of 77 stochastic differential equations
(1)
dz{t) = f{x{t)91, y9 z) dt + o{x{t)91) dw{t)
where a{x91) is an n x n matrix, f{x919 y9 z) is an w-vector, both uniformly Lipschitz
continuous m{x9 t)eRn x [0,00), and w{t) is «-dimensional Brownian motion. The
variables y, z are viewed as control functions. They are taken to be measurable
functions y = y{x91)9 z = {x9 t) with values in compact sets Y and Z respectively.
We say that Y and Z are the control sets for the players y and z respectively. We
are also given a pay-off
PzT{y9 S; z9 T) = E^ | j exp y J k{x9 s9 y9 z) ds h{x919 y9 z) dt
(2)
+
ex
P |_ J k(x> U y>z)
dt
gi(x(S)> S) XS^T
+ exp [ J k{x9 u y, z) dt g2{x{T)9 T) } XT<S
where k9 h9 gh g2 are, say, smooth functions with bounded second derivatives. Here
S and T are stopping times with range in [z9 T0] for the process (1) with y = y{x91)9
z = z{x91) and TQ is a fixed positive number.
The aim of the player y is to choose {y{x91)9 S) so as to maximize the pay-off, and
the aim of the player z is to choose {z{x91)9 T) so as to minimize the pay-off. We
shall refer to (1), (2) as a stochastic differential game with stopping time.
A pair {{y*{x91)9 S*)9 {z*{x, t)9 T*)} is called a saddle point if
p^{y9 s\ z*9 T*) ^ PUy*> s*i z*> T1*) ^ PUy*> S*'> z> T)
© 1975, Canadian Mathematical Congress
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AVNER FRIEDMAN
for any pairs {y9 S) and (z, T). If such a saddle point exists, then the number
v{ç9z) = pay*,s*;z*,T*)
is called the value of the game.
Let (7* be the transpose of a9 and let a = aa*. Set
Lu = £ 2 aiß^ujdxßXj
where a = (a,7). It is easy to see that if a saddle point exists then
(3)
g2{x91) ^ gl{x91)9
g2{x9 TQ) = gl{x9 F 0 ).
We shall now assume that (3) holds, that
2 au{x , * ) « , ^ a | f | 2 for all (*, 0, f e i ? » ( a > 0),
(4)
and that the minimax condition holds :
max min [h{x919 y9z) + p •/(*, f, >>, z) + uk{x91, y9 z)]
(5)
= min max [h{x91, y9z) + p •/(*,19 y9 z) + uk{x919 y, z)]
z<=Z
=
y*BY
H{x9t9u9p).
Consider the nonlinear parabolic variational inequality
u e LP{09 r 0 ; W*>P*) [\ L°°(0, T0; W^)9
dußt G LP{0, TQ ; V*'**),
gi^u^
g2,
[du/dt + Lu + H{x919 u9 ux)] (v - u) ^ 0 a.e. for every v, gi S v g g2>
W (*,T 0 ) = ^ i ( x , r 0 )
with any/? > w. Here JF'»^ is 0^(1?*) with any density function e~fi]xl, p > 0.
(6)
1. (i) There exists a unique solution u of {6).
(ii) Let y*{x91)9 z*{x91) be any control functions which realize the maxy and minz
in (5) when p = ux{x91). Let S* be the exit time from the set {u{x91) > gi{x91)} fl
{z < t g TQ} and let T* be the exit time from the set {u{x91) < g2{x91)} f| {z < t
^ TQ}. Then {{y*9 5*), (z*, T*)} is a saddle point.
(iii)K(f,r) = u{£,z).
THEOREM
Theorem 1 is due to Bensoussan and Friedman [1]. The special case where there
are no stopping times was proved earlier by Friedman [5]. The special case where
there are no controls y{x91)9 z{x91) was proved by Bensoussan and Lions [2] and
Friedman [6], [7] (in [2] there is only one player); Krylov [11] has considered the
corresponding stationary case.
Consider next the case where a{x91) = 0, i.e., (1) is replaced by a deterministic
dynamical system
(7)
dx{t)=f{x9t9y9z)dt
and Pç,T{y9 S; z, T) is defined by (2), but with E^tT removed. The stopping times
S, T are now any numbers in the interval [z9 TQ]. Since the assertion (i) of Theorem 1
STOCHASTIC DIFFERENTIAL GAMES WITH STOPPING TIMES
341
is false in the present case, we proceed in a different manner :
In the theory of differential games [4] one takes control functions y{t)9 z{t) and
defines the concepts of upper value, value, 5-strategy, strategy, saddle point, etc.
Then one proves that the upper value V+ exists and, under some assumptions, the
value exists. The differential game setting is that corresponding to taking S s= TQ,
T = Jo. Bensoussan and Friedman [1] have generalized the basic concepts and
existence theorems in the theory of differential games to differential games with
stopping times. In particular, they proved :
2. (i) The upper value V+{x91) exists and satisfies a.e. thefirstorder
nonlinear variational inequality
THEOREM
(dV+
T
dV+
lì
m x
+
[—gf- + min a \Kx919 y9 z) + -^- • f{x919 y9 z) + V k{x919 y9 z)j |(v - V+)
^ 0 a.e.for any v9 gi ^ v ^ g2.
(ii) Iff = fx{x919 y) + f2{x919 z), h = h\{x919 y) + h2{x919 z) then the value exists.
Consider now the case where o = el91 the identity matrix. Denote the value
occurring in Theorem 1 by VB{x91).
In the special case where there are no stopping times, i.e., S = TQ, T = TQ9 it
is known [3], [8] that
(8)
V&9T)-+V{C,T)
ife^O
where V{t~, z) is the value of the deterministic differential game (with dynamics (7)).
Bensoussan and Friedman [1] have proved (8) in case S is any stopping time but
T = T09 provided
* + $ - + •£•/+»!.
sa
We return to the situation of Theorem 1. It is of great interest to study the
domains of continuation C\ = {u > gi}9 C2 = {u < g2}.
In the case where there is only one player, say y9 and, furthermore, there are no
control functions y = y{x91) in (1), (2), there are some recent results on the shape
and smoothness of the boundary of C\ [12], [13], [10], [9]. It would be of interest to
obtain such results in the case where there is a control function y in the system (1)
and in the pay-off (2).
References
1. A. Bensoussan and A. Friedman, Nonlinear variational inequalities and differential games
with stopping times, J. Functional Analysis 16 (1974), 305-352.
2. A. Bensoussan and J.-L. Lions, Problèmes de temps d'arrêt optimal et inéquations variationellesparabolique, Applicable Anal. 3 (1973), 267-295.
3. J. Elliott and N. J. Kaiton, Upper values of differential games, J. Differential Equations 14
(1973), 89-100.
4. A. Friedman, Differential games, Wiley, New York, 1971.
5.
, Stochastic differential games, J. Differential Equations 11 (1972), 79-108. MR 45
#1613.
342
AVNER FRIEDMAN
6. A. Friedman, Stochastic games and variational inequalities, Arch. Rational Mech. Anal. 51
(1973), 321-346.
7.
, Regularity theorems for variational inequalities in unbounded domains and applications
to stopping time problems, Arch. Rational Mech. Anal. 52 (1973), 134-160.
8.
, Differential games, CBMS Regional Conference Series in Math., no. 18, Amer.
Math. Soc, Providence, R. I., 1974.
9.
, Parabolic variational inequalities in one space dimension and the smoothness of the
free boundary, J. Functional Anal. 18, 151-176.
10. A. Friedman and D. Kinderleherer, A one-phase Stefan problem, Indiana Univ. J. Math.
(to appear).
11. N. V. Krylov, Control of Markov processes and W-spaces, Izv. Akad. Nauk SSSR Ser. Mat.
35 (1971), 224-255 = Math. USSR Izv. 5 (1971), 233-266. MR 45 #4493.
12. P. van Moerbeke, Optimal stopping and free boundary problems, Arch. Rational Mech.
Anal, (to appear).
13.
, An optimal stopping problem for linear reward, Acta Math. 132 (1974), 1-41.
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