Mathematica Slovaca
Mohamed A. El-Shehawey; Bakheet N. Al-Matrafi
On a gambler's ruin problem
Mathematica Slovaca, Vol. 47 (1997), No. 4, 483--488
Persistent URL: http://dml.cz/dmlcz/131714
Terms of use:
© Mathematical Institute of the Slovak Academy of Sciences, 1997
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to
digitized documents strictly for personal use. Each copy of any part of this document must contain
these Terms of use.
This paper has been digitized, optimized for electronic delivery and stamped
with digital signature within the project DML-CZ: The Czech Digital Mathematics
Library http://project.dml.cz
Mathennatica
Slovaca
©1997
. . ., ....
-— f«ri*-,\
ivi
, ,.-..-. _.-..-.
M a t h . SloVaCa, 4 7 ( 1 9 9 7 ) , NO. 4 , 4 8 3 - 4 8 8
ON
Mathematical I n s t i t u t e
Slovák Academy of Sciences
A G A M B L E R ' S RUIN P R O B L E M
MOHAMED A . E L - S H E H A W E Y * — B . N . AL-MATRAFI**
(Communicated by Lubomír Kubáček )
ABSTRACT. The object of this paper is to specify an explicit expression for
the absorption probabilities for a random walk on the integers — 6 , - 6 + 1 ,
..., — 1, 0 , 1 , . . . , a which arises in a gambler's ruin problem proposed by the
authors. There are two barriers, one absorbing at a and the other at —6, such
that the random walk particle is returned to the position j , —6 < 3 < a, whenever
reaches it.
1. Introduction
The classical problem of the gambler's ruin affords a classical illustration of
the simple random walk with absorbing barriers. For example, B a r n e 11 (1964),
Cox
and
Miller
(1965),
Feller
(1968),
Srinivasan
and
M e h a t a (1976), and K a n n a n (1979) have treated this problem. We con­
sider here a more general problem where there are two barriers, one of which is
absorbing and the other is such that the random walk particle is immediately
returned to a certain position upon reaching it. We consider two players A and
B with initial fortunes a and b dollars respectively. The game consists of a
series of independent turns. Let Xu, u > 1, denote the B 's winnings in the uth
turn with
1
with probability p,
xu
—1
with probability q ,
0
with probability r ,
where p + q + r = 1 and 0 < p , r / < l , 0 < r < 1. Thus, we have
s
™
n = z2
u=l
x
v
AMS S u b j e c t C l a s s i f i c a t i o n (1991): Primary 60G40, 60J10, 60J15.
Key w o r d s : classical ruin problem, simple random walk, absorption probabilities.
483
MOHAMED A. EL-SHEHAWEY — B. N. AL-MATRAFI
which represents B 's net gain at the end of n turns. If at any stage Sn = — b,
B is ruined, player A immediately donates j dollars of his fortune to player
B, 0 < j < a. The game ends when B has gained all .A's fortune and A is
ruined, i.e., Sn = a. Thus Sn is a random walk with two barriers, one of which,
at a , is absorbing, and the other, at —6, such that the random walk particle is
returned to the position j , —6 < j < a , whenever reaching it. In this paper, an
explicit expression for the absorption probabilities is derived. C o x and M i l l e r
[3; p. 33] (1965) solved the classical random walk problem with absorbing barriers
at the points —b and a; however, the probability of absorption at time n is found
to be erroneous. The correct formula can be found in G u l a t i and H i l l [6]
(1981). The special case, when the barriers are symmetrically placed on either
side of the origin has been given by M u n f o r d [6] (1981). If we replace the
absorbing barrier at —6 by an impenetrable (reflecting) barrier at j = 0, — l ' s
are never accumulated if Sn reaches zero. The absorption probabilities have
been deduced by E l - S h e h a w y [4] (1992).
2. Generating function for t h e absorption probabilities
Let / ( n , a | iQ) denote the probability that the particle is absorbed at a at
time n given that its initial position was iQ, — b < iQ < a. The absorption
probabilities / ( n , a \ iQ) must satisfy
/ ( n , a\iQ)=
P/(n-l,
a | i 0 + l ) + r / ( n - l , a\iQ)+
<?/(n-l, a | z 0 - l )
( n = 1 , 2 , . . . ; iQ = - 6 + 1 , . . . , - 1 , 0 , 1 , . . . , a - 1 ) ,
where
/ ( 0 , a | iQ) = 0 (i Q = - 6 , . . . , - 1 , 0 , 1 , . . . , a - 1),
/(0,a|a) = l,
f(n,-b\iQ)
= f(nJ\iQ)
(n = 1,2,...).
Introducing the generating function
CO
Gio,a(t) = ^f(n,a\i0)tn.
(2)
n=0
Following C o x and M i l l e r [3] (1965) we deduce that
Gio,a(t)
= Hio(t)/Ha(t),
i0 = - & , . . . , - 1 , 0 , 1 , . . . , a ,
(3)
where
Hi0(t) = (K(t))h+i0 - (M*))6+i0 + k/p}b+i0 [(W)?-*
484
- (h2(t))j-i0],
ON A GAMBLER'S RUIN PROBLEM
and h^t),
h2(t) are the roots of the quadratic equation
pth2 -(l-rt)h
+ qt = 0,
(4)
p, q and r are the probabilities of taking one step to the right, to the left
and remaining in position, respectively. Writing X(t) = 1 — rt and Y(t) =
^Apqt2 — (1 — rt)2, we have
h12(t) = (2pt)-1[X(t)±i
Y(t)},
i = V--l,
and
Hio(t) = (2pi)- ( 6 + i o ) [(x(i) + iY(t))b+i°
+ (2pt)^-J[q/p]^o
- (X(t)
[ ( x ( t ) + iY(t))j-io
-iY(t))b+i0]
- (X(t) - iY(t))j-i0]
.
Accordingly, by expanding both the numerator and denominator of (3) in
powers of Y(t) and noting that only odd powers of Y(t) occur, thus Y(t) is
cancelled, leaving a ratio of two polynomials in t, each of degree at most a+b— 1,
and consequently a partial fraction expansion of it is available. Using complex
variables notation, we have
X(t) = \z\ cos (j>,
Y(t) = \z\ sin <fi,
+ Y2(t) = 2^/pqt,
and
where
\z\ = y/X2(t)
<f> = tan~l (Y(t)/X(t))
.
It is useful to observe that
1 -rt
= 2y/pqtcos<£,
and h12(t) = y/q/pe±1<^.
or
t = (r + 2^/pqcos<f))~ ,
(6)
Formula (3) becomes
Gio,a(t)=Tio(<t>)/Ta(4>),
(7)
where
Ti0(cf>)= [ ^ / p ^ ] a - i o [ ( V p ) 6 + i s i n ( 6 + g 0 + ( V g ) 6 + i s i n ( j - ^ o ) < A " .
The denominator of (7) is found to have a + b—I distinct roots. A study of the
function 7(</>), where
/JL v
sin(a
+ b)(j)
r
/—r-,6+j
7(4>) = ~T~T
^ - V<7/P
8)
J
w
sm(a — j)(p L
shows that if a — j -= Ufpjq] J (6 + a), the roots of the equation *y(<f)) = 0 give
distinct roots of Ta(<fi) 6 [0, IT) . If a - j < [y/pjq ] 3 (b + a), there are a + 6 - 1
485
MOHAMED A. EL-SHEHAWEY — B. N. AL-MATRAFI
distinct real roots </>v (v = 1,2,..., a + b — 1) of (8). The corresponding roots
of Ha(t) are then
v
Г + 2у/р^СОБфи
v=l,2,...,a
'
+ b-l
If a — j > [yjp/q]
(b + a), there are only a + b — 2 distinct roots <f>v (v =
2, 3 , . . . , a + b — 1) that give distinct roots i^ of Ha(t). The remaining root of
H
l
is
a( )
given by
1
u1 = v + 2-v/pgcosh<^1
where ^ is the unique root of the equation,
J
sinh(a — j)(f) = [y/p/q]
sinh(a + b)(f).
3. Explicit expression for t h e absorption probabilities
We have
THEOREM.
a+Ь-l
f(n,a\i0)
=-2yJpq
-1
M°(Фг)+ £
^(ФĄ^WU^)
i/=2
п
(9)
^in^J,
(г + 2у/р~саафи)
where
n+1
[r + 2y/pqcos</>1] ,
a-j<
M«(<f>1) = M(4>1)-\ [1-(VP-V-)T+1,
[r + 2Vpqcosh(ß1]n+1,
a-j
(y/p/q)(a
+ b) ,
= (y/pTq)b+Í(a + b),
a - j > (y/p/q)
+J
(a + b),
and
' ^ 0 (^i)[^ 1 U^]" 1 [ r + 2 V7^ cos ^] -2sin ^i'
a-j<(Vp7q)b+Í(a + b),
ҖФi) = <
-2{a-iQ)(a
+ b)-\a-j)-x(2a
= i
J~
) (a + Ь),
2
[r + 2 V м c o s h ^ ] - s i n h ^ ,
a-j>(y/pTq)Ь+І(a
486
+ 2Vpq)
Ь+J
a-j={Vp/
-Гji^)p^%
+ Ъ-j)-l[r
+ Ъ).
\
ON A GAMBLER'S RUIN PROBLEM
Proof.
in the form
Employing the partial fraction method, formula (3) can be written
oo
G
/ a+b— 1
U*)=E E
\
^r>
where the constants T)v are given by,
V» =
~Hi0(t)
дHgjt)
(10)
дt
t = tu
Differentiating the quadratic equation (4) with respect to t we get
d
^=t-^i2(t)[o-phi2(t)r1-
(ii)
The study of (8) with (3), (10) and (11) finally yield
% =
- 2 ^ ? [ s i n ( 6 + i0)<f>v + (^/g/p)
a
(V Jp)
b+0
sin(j - i0)<f>v]tl s i n ^ ,
a
*° [( + b) cos(a + b)фu + (yґq/pý*3(j
ІV = 2 , 3 ,
- a) cos(j - a)фџ] '
...,a + ò — 1
(12)
and
Vl
where
cos
Фг = {
---гv^м^),
rU
2y/pqtx
cosh - 1
rt
if a-ú<(Vp/q)
ì-
0+J
if a-j>(^ф) (a
i x is the smallest root in absolute value of Ha(t).
ъ+jt
> + &),
\b+j/
+ b),
Formula (9) follows.
D
We see t h a t with the appropriate change of notation, expression (9) agrees
with t h a t of M u n f o r d [8] (1981) in the case j = 0, p > q\ and with t h a t of
W e e s a k u T s [10] (1961) and B i a s i [2] (1976) for the particular case j = 0,
b = 1, r = 0, replacing iQ, a, <\>v, r)v with b — u, b, a^, p^ and interchanging
p and g respectively.
REFERENCES
[1] BARNETT, V. D.: A three-player extension of the gambler's ruin problem, J. Appl.
Probab. 1 (1964), 321-334.
[2] BLASI, A.: On a random walk between a reflecting and an absorbing barrierђ Ann. Probab.
4 (1976), 695-696.
487
MOHAMED A. EL-SHEHAWEY — B. N. AL-MATRAFI
[3] C O X , D. R — M I L L E R , H. D . : The Theory of Stochastic Processes, Methuen, London,
1965.
[4] E L - S H E H A W Y , M. A . : On absorption probabilities for a random walk between two different barriers, Ann. Fac. Sci. Toulouse M a t h . (5) I (1992), 95-103.
[5] F E L L E R , W . : An Introduction
to Probability Theory and Its Applications.
Vol. 1 (3rd
ed.), Wiley, New York, 1968.
[6] G U L A T I , C. M.—HILL, J. M . : A note on an alternative derivation of random walk probabilities, Sankhya Ser. A 4 3 (1981), 379-383.
[7] K A N N A N , D . : An Introduction
to Stochastic Processes, New York, 1979.
[8] M U N F O R D , A. G . : A first passage problem in a random walk with a quality
control
application, J. Roy. Statist. Soc. Ser. B 4 3 (1981), 142-146.
[9] SRINIVASAN, S. K . — M E H A T A , K. M . : Stochastic Processes, M c G r a w Hill, New Delhi,
1976.
[10] W E E S A K U L , B . : Tlie random walk between a reflecting and an absorbing barrier, Ann.
M a t h . Statist. 3 2 (1961), 765-769.
Received August 15, 1994
Revised April 23, 1995
' Department
Faculty of
Damietta
EGYPT
of
Mathematics
Science
** Department
of
Mathematics
Faculty of Applied
Sciences
Umm Al-Qura
University
Makkah
SAUDI
ARABIA
488