Continuous-Time Evolutionary Finance
with Fix-Mix Strategies∗
Zhaojun Yang1,2 Christian-Oliver Ewald1 Klaus Reiner Schenk-Hoppe´1,3
1School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
(e-mail: ewald@)
2School of Mathematics and Econometrics,Hunan University,
410082, Changsha, China(e-mail: zjyang@)
3Business School, University of Leeds, Leeds LS2 9JT, UK,
(Email: -Hoppe@)
November 26, 2005
Abstract
This paper develops a continuous-time evolutionary model of financial markets
with endogenous prices. Investment strategies are fix-mix. Our main goal is to
understand the wealth dynamics. In particular we seek to identify evolutionary
stable investment strategies, . those strategies that prevent entrants to the
financial market from gaining wealth in the long run.
Key words: evolutionary finance, continuous-time portfolio theory, endogenous asset
prices
JEL-Classification: G11, D52, D81.
1 The Discrete Model
Let us consider the following dynamic system model with short time interval ∆t:
wit+∆t =
K∑
k=1
(
p0t+∆tD
k
t+∆t + λ
′
k(∆t)wt+∆t
)
θit,k, (1)
∗Correspondence to: Klaus Reiner Schenk-Hoppe´. Research Supported by Natural Science Fund (No. 04JJ3009) of
Hunan Province, China.
1
where we assume
λ0(∆t) = c∆t, (2)
λik(∆t) = λ
i
k(1− c∆t), (3)
K∑
k=1
λik = 1, 0 < λ
i
k < 1, i = 1, · · · , I, (4)
λk(∆t) = (λ
1
k(∆t), · · · , λIk(∆t))′, wt+∆t = (w1t+∆t, · · · , wIt+∆t)′,
θit,k =
λik(∆t)w
i
t
λ′k(∆t)wt
=
λikw
i
t
λ′kwt
, (5)
and Dkt+∆t is the amount of consumption goods which is taken as dividend in the time
interval of [t, t+∆t] and the price of consumption goods is p0t+∆. In addition, ”
′” means
the matrix transit operator and c is the consumption rate in a unit time.
We derive from the market clearing condition on consumption goods that
I∑
i=1
K∑
k=1
p0t+∆tD
k
t+∆tθ
i
t,k =
I∑
i=1
c∆twit+∆t. (6)
Noting that
I∑
i=1
θit,k = 1, (7)
It follows from (6) that
p0t+∆tDt+∆t = c∆tWt+∆t, (8)
where
Dt+∆t ≡
K∑
k=1
Dkt+∆t, Wt+∆t ≡
I∑
i=1
wit+∆t. (9)
Undoubtedly, what we care about more are the traders’ wealth shares than their wealth.
Thus, denoting rit =
wit
Wt
, we conclude from (1) and (8) that
rit+∆t =
K∑
k=1
(
c∆tdkt+∆t + λ
′
k(∆t)rt+∆t
)
θit,k, (10)
where
dkt+∆t ≡
Dkt+∆t
Dt+∆t
2
is the relative dividend payment of asset k and rt+∆t ≡ (r1t+∆t, · · · , rIt+∆t)′.
let
Θt =
(
θit,k
)
I×K , Λ =
(
λik
)
K×I ,
then we get from (10) that
rt+∆t = c∆t [Id− (1− c∆t)ΘtΛ]−1 (Θtdt+∆t) , (11)
where dt+∆t ≡ (d1t+∆t, · · · , dKt+∆t)′ and Id represents the I-by-I identity matrix.
Remark 1. Eqn. (11) is well-defined since the matrix Id− (1− c∆t)ΘtΛ is invertable
by Evstigneev, Hens and Schenk-Hoppe´ (2006) for every ∆t > 0.
2 The Continuous Model
Denoting
A = (ai,j)I×I ≡ ΘtΛ,
then we have the following lemma:
Lemma 1.
rt+∆t =
(Id− (1− c∆t)A)∗Θtdt+∆t∣∣∣∣∣∣∣∣∣
1 1 · · · 1
(c∆t− 1)a2,1 1 + (c∆t− 1)a2,2 · · · (c∆t− 1)a2,I
...
... · · · ...
(c∆t− 1)aI,1 (c∆t− 1)aI,2 · · · 1 + (c∆t− 1)aI,I
∣∣∣∣∣∣∣∣∣
, (12)
where ”*” means the matrix adjoint operator.
Proof Obviously, it is concluded from Eqn. (11) that
rt+∆t = c∆t
[Id− (1− c∆t)ΘtΛ]∗
|[Id− (1− c∆t)ΘtΛ]| (Θtdt+∆t) . (13)
Noting from (5) and (4) that
I∑
i=1
θit,k = 1,
I∑
i=1
ai,j = 1, j = 1, · · · , I (14)
3
we conclude that the sum of each column in matrix [Id− (1− c∆t)ΘtΛ] equal to c∆t.
In the end, the lemma can be directly derived from determinant qualities.¤
Thus, we have
rit+∆t =
∣∣∣∣∣∣∣
1 + (c∆t− 1)a1,1 · · · z1∆t · · · (c∆t− 1)a1,I
...
...
... · · · ...
(c∆t− 1)aI,1 · · · zI∆t · · · 1 + (c∆t− 1)aI,I
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
1 · · · 1 · · · 1
(c∆t− 1)a2,1 · · · (c∆t− 1)a2,i · · · (c∆t− 1)a2,I
...
...
... · · · ...
(c∆t− 1)aI,1 · · · (c∆t− 1)aI,i · · · 1 + (c∆t− 1)aI,I
∣∣∣∣∣∣∣∣∣
, (15)
where z∆t = (z
i
∆t)I×1 ≡ Θtdt+∆t.
On account of that
I∑
i=1
zi∆t = 1, (16)
it follows from (14) that
lim
∆t→0+
rit+∆t = lim
∆t→0+
∣∣∣∣∣∣∣
1 + (c∆t− 1)a1,1 · · · z1∆t · · · (c∆t− 1)a1,I
...
...
... · · · ...
(c∆t− 1)aI,1 · · · zI∆t · · · 1 + (c∆t− 1)aI,I
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
1 · · · 1 · · · 1
(c∆t− 1)a2,1 · · · (c∆t− 1)a2,i · · · (c∆t− 1)a2,I
...
...
... · · · ...
(c∆t− 1)aI,1 · · · (c∆t− 1)aI,i · · · 1 + (c∆t− 1)aI,I
∣∣∣∣∣∣∣∣∣
=
B1,i
|B| ,
(17)
where
B ≡ (bi,j)I×I =
1 1 · · · 1
−a2,1 1− a2,2 · · · −a2,I
...
... · · · ...
−aI,1 −aI,2 · · · 1− aI,I
,
and B1,i is the algebraic cofactor of element b1,i in matrix B.
Theorem 1.
B1,i
|B| = r
i
t, i = 1, · · · , I. (18)
4
(Just a guess, we can throw it away also) ¤
As one can see above, it is too complicated to acquire any new insight from this
general situation. In the following, therefore, we turn on the case of I = 2 that is simple
but not loss of the essence of the problems.
More formally, let (Ω,FT , (Ft)0≤t≤T , P) be a stochastic basis, where (Ft)0≤t≤T is a
filtration representing the information exposed with time. Assume that all random
variables and stochastic processes are defined on Ω, and filtration (Ft)0≤t≤T satisfies
the usual hypotheses, where F0 is trivial and all stochastic processes are adapted to
this filtration.
Theorem 2. If I =2, then we have{
r1t+∆t =
c∆tz1∆t−(c∆t−1)a1,2
(a1,2+a2,1)+c∆t(a2,2−a2,1)
r2t+∆t = 1− r1t+∆t
. (19)
and
lim
∆t→0+
r1t+∆t = r
1
t , lim
∆t→0+
r2t+∆t = 1− r1t = r2t . (.) (20)
Proof It follows directly from (15) that
r1t+∆t =
∣∣∣∣ z1∆t (c∆t− 1)a1,2z2∆t 1 + (c∆t− 1)a2,2
∣∣∣∣∣∣∣∣ 1 1(c∆t− 1)a2,1 1 + (c∆t− 1)a2,2
∣∣∣∣ .
Thus,(19) holds evidently since z1∆t + z
2
∆t = 1. From (19), we get that
lim
∆t→0+
r1t+∆t =
a1,2
a1,2 + a2,1
. ()
Noting that
a1,2 =
(
K∑
k=1
λ1kλ
2
k
λ′krt
)
r1t , (21)
and
a2,1 =
(
K∑
k=1
λ1kλ
2
k
λ′krt
)
(1− r1t ), (22)
5
we get
a1,2 + a2,1 =
K∑
k=1
λ1kλ
2
k
λ′krt
=
a1,2
r1t
. (23)
Thus (20) is proved. ¤
Theorem 3. Assume the relative dividend processes (dkt )t>0, k = 1, · · · , K are exoge-
nous process with continuous paths, then the stochastic process of wealth share r1t and
r2t are solutions to the following differential equations with stochastic coefficients respec-
tively:
dr1t
dt
=
cz10 − cr1t
a1,2 + a2,1
= c
(∑K
k=1
dkt λ
1
k
λ1kr
1
t+λ
2
k(1−r1t )
)
− 1∑K
k=1
λ1kλ
2
k
λ1kr
1
t+λ
2
k(1−r1t )
r1t (.) (24)
and
dr2t
dt
= −dr
1
t
dt
=
cz20 − cr2t
a1,2 + a2,1
= c
(∑K
k=1
dkt λ
2
k
λ1k(1−r2t )+λ2kr2t
)
− 1∑K
k=1
λ1kλ
2
k
λ1k(1−r2t )+λ2kr2t
r2t . (.) (25)
Proof According to theorem 2, and L’ Hopital Theorem, it isn’t difficult to prove
that
lim
∆t→0
r1t+∆t − r1t
∆t
=
cz10 − cr1t
a1,2 + a2,1
. (.)
Thus (24) follows from (23) and the definition of z∆t just after (15). In addition, (25)
can be proved similarly .¤
Theorem 4. There exists a unique mean square solution in both stochastic differential
equations (24) and (25) for any initial condition r10, r
2
0 ∈ [0, 1] respectively. The unique
solution can be written respectively as follows:
r1t = r
1
0 + c
∫ t
0
(∑K
k=1
dkuλ
1
k
λ1kr
1
u+λ
2
k(1−r1u)
)
− 1∑K
k=1
λ1kλ
2
k
λ1kr
1
u+λ
2
k(1−r1u)
r1udu, (26)
and
r2t = r
2
0 + c
∫ t
0
(∑K
k=1
dkuλ
2
k
λ1k(1−r1u)+λ2kr2u
)
− 1∑K
k=1
λ1kλ
2
k
λ1k(1−r2u)+λ2kr2u
r2udu, (27)
where the integrals are understood to be the mean square integrals.
6
Proof On account of the similarity between (24) and (25) and that both (26) and
(27) hold obviously, it is enough to prove that there exists a unique mean square solution
to one of (24) and (25), say (25) as chosen by us in the following.
Define function
f(X) ≡ c
(∑K
k=1
dkt λ
2
k
λ1k(1−X)+λ2kX
)
− 1∑K
k=1
λ1kλ
2
k
λ1k(1−X)+λ2kX
X. (28)
Thus stochastic differential equation (25) can be written as
dX
dt
= f(X), X0 = r
2
0,∈ [0, 1]. (29)
Noting that function f(·) is continuously differential in [0, 1] and f(0) = f(1) = 0, one
can know that the phase space of (29) is a subset of interval [0, 1] for every ω ∈ Ω. For
this reason, according to Theorem in Soong(1973), one only need to show that
function f : Ξ→ L2 satisfies the following mean square Lipschitz condition:
‖f(X)− f(Y )‖ ≤ α ‖X − Y ‖ , α > 0, (30)
where Ξ and L2 represent respectively the set of all random variables with values in
[0,1] and a L2-space with norm ‖X‖ ≡ (E[X2])1/2 , see Soong(1973).
In fact, on account of that
0 < min(λ1k, λ
2
k) ≤ λ1k(1−X) + λ2kX ≤ max(λ1k, λ2k), k = 1, · · · , K, (31)
and
0 < min(λ1k, λ
2
k) ≤
λ1kλ
2
k
λ1k(1−X) + λ2kX
≤ max(λ1k, λ2k), k = 1, · · · , K, (32)
for every X,Y ∈ Ξ, we have
|f(X)− f(Y )| ≤
∣∣∣∣∣∣cX
(∑Kk=1 dkt λ2kλ1k(1−X)+λ2kX)−1∑K
k=1
λ1
k
λ2
k
λ1
k
(1−X)+λ2
k
X
−
(∑K
k=1
dkt λ
2
k
λ1
k
(1−Y )+λ2
k
Y
)
−1∑K
k=1
λ1
k
λ2
k
λ1
k
(1−Y )+λ2
k
Y
∣∣∣∣∣∣
+
∣∣∣∣∣∣c
(∑K
k=1
dkt λ
2
k
λ1
k
(1−Y )+λ2
k
Y
)
−1∑K
k=1
λ1
k
λ2
k
λ1
k
(1−Y )+λ2
k
Y
(X − Y )
∣∣∣∣∣∣ .
(33)
7
Define
λm ≡ min{λ1k, λ2k|k = 1, · · · , K}, λM ≡ max{λ1k, λ2k|k = 1, · · · , K}, (34)
where 0 < λm ≤ λM < 1 in light of (4). Therefore, by lengthy and elementary calcula-
tions, it follows from (33) that
|f(X)− f(Y )| ≤ 2cλmλM + cλ
2
M
Kλ4m
|X − Y |. (35)
Thus if we take
α =
2cλmλM + cλ
2
M
Kλ4m
, (36)
then (30) is proved by properties of norm.¤
Remark 2. If the relative dividend processes are almost surely continuous, by almost
the same way as the proof of this theorem, one can show that there exists a unique
solution in both (24) and (25) for almost every ω ∈ Ω.
Corollary 1. The hyperplane is invariant. That is ,given ri0 = 0, i = 1, 2, then r
i
t = 0
for all t ≥ 0; and given ri0 > 0, i = 1, 2, then rit > 0 (.) for all t ≥ 0 (no sudden
death or bankruptcy). It is also true for the case of ri0 = 1, i = 1, 2.
Proof It is enough to discuss the case of ri0 = 0, i = 1, 2. Obviously, both SDE
(24) and (25) have two fixed points rit = 0 or r
i
t = 1, i = 1, 2 respectively. Thus if
ri0 = 0, i = 1, 2, then r
i
t = 0 for all t ≥ 0. Moreover, if ri0 > 0, i = 1, 2, but it isn’t true
that rit > 0 for all t ≥ 0, say rit0 = 0 for some t0 > 0. However, this shows that there
are two different solutions to (24) or (25) passing point 0, one is rit and the other is the
fixed point 0. Thus one gets a contradiction with the uniqueness from Remark 2.¤
3 Globally Asymptotic Evolutionary Stability
This section assumes the relative dividend processes (dkt )t>0, k = 1, · · · , K are stationary
in mean. That is, E(dKt ) ≡ d¯k are constant for every k = 1, · · · , K. It is clear that
8
∑K
k=1 d¯k = 1 and 0 < d¯k < 1. We are going to show the optimal portfolio is λ
∗ in all
fix-mix strategies . Here λ∗ is defined by λ∗k ≡ d¯k, k = 1, · · · , importantly, this
section will show that, if the first player take the optimal portfolio λ∗, he must take up
the whole market at last no matter what initial wealth 0 < r10 < 1 is and no matter
what portfolio is chosen by the second player only if it is simple and different from λ∗
also. On account of that, in discrete time, Evstigneev, Hens and Schenk-Hoppe´ (2006)
shows this is true only if r10 is less than 1 and extremely close to 1 as well, the results
in this section are more insightful.
In order to derive the main results, we first present some lemmas in the following.
Define function
G(λ1, λ2, x) ≡ c
(∑K
k=1
d¯kλ
2
k
λ1k(1−x)+λ2kx
)
− 1∑K
k=1
λ1kλ
2
k
λ1k(1−x)+λ2kx
x, (37)
then we have:
Lemma 2. G(λ∗, λ2, x) < 0 for all x ∈ (0, 1) and λ2 6= λ∗, here λ2 is arbitrarily simple
portfolio.
Proof For a arbitrarily given x ∈ (0, 1) , let us find λ2k, k = 1, · · · , K to solve
max G(λ∗, λ2, x) = c
(∑K
k=1
d¯kλ
2
k
λ∗
k
(1−x)+λ2
k
x
)
−1∑K
k=1
λ∗
k
λ2
k
λ1
k
(1−x)+λ2
k
x
x
.
{ ∑K
k=1 λ
2
k = 1,
0 < λ2k < 1, k = 1, · · · , K.
(38)
It is clear that (38) is a nonlinear convex programming and, by a typical method, one
can easily find the unique globally optimal solution as follows:
λ2k = d¯k, k = 1, · · · , K,
and the optimal target value G(λ∗, λ∗, x) = 0. Therefore, this lemma is proved on
account of that the target function is strictly concave.¤
9
Corresponding to SDE (25), we consider the deterministic ordinary differential equa-
tion on variable r¯2t defined as follows:
dr¯2t
dt
= G(λ1, λ2, r¯2t ) = c
(∑K
k=1
d¯kt λ
2
k
λ1k(1−r¯2t )+λ2k r¯2t
)
− 1∑K
k=1
λ1kλ
2
k
λ1k(1−r¯2t )+λ2k r¯2t
r¯2t . , 0 < r¯
2
0 < 1. (39)
Lemma 3. There exists a unique solution to ODE (39). In addition, if λ1 = λ∗ and
λ2 6= λ1 then the solution (r¯2t ) is monotonically decreasing and satisfies
lim
t→∞
r¯2t = 0, 0 < r¯
2
t < 1 (t > 0) (40)
for every initial value 0 < r¯20 < 1.
Proof Firstly, just by the same way with the proof of (35), one can show G(λ1, λ2, ·)
satisfies the Lipschitz condition in interval (0, 1). That is
|G(λ1, λ2, x)−G(λ1, λ2, y)| ≤ α|x− y|, 0 < x, y < 1,
where α is defined in (36). Thus, the existence and uniqueness holds.
Secondly, noting that 0 and 1 are two fixed solutions of (39), one gets 0 < r¯2t < 1
for all t > 0 for the similar reasons stated in the proof of Corollary 1.
Finally, the solution r¯2t satisfies
dr¯2t
dt
= G(λ∗, λ2, r¯2t ) < 0.
It means that the function r¯2t is decreasing as time goes and, therefore, limt→∞ r¯
2
t exists
since the function is also bounded. Moreover, It is clear that the limit is nothing but
zero because G(λ∗, λ2, r¯2t ) < 0 once 0 < r¯
2
t < 1.¤
Theorem 5. Regard the relative dividend dkt , k = 1, · · · , K as new information that
are independent on the present wealth (rit), i = 1, 2. Then, if λ
1 = λ∗, λ2 6= λ1 and
0 < r20 < 1, we have
lim
t→∞
r2t = lim
t→∞
r¯2t = 0, (.) (41)
10
and
lim
t→∞
r1t = lim
t→∞
1− r¯2t = 1, (.) (42)
where . represents convergence in mean square.
Proof It is clear that we only need to prove (41). At first, (25) and (37) imply
that
dr2t
dt
= G(λ∗, λ2, r2t ) + c
∑K
k=1
(dkt−d¯k)λ2k
λ1k(1−r2t )+λ2kr2t∑K
k=1
λ1kλ
2
k
λ1k(1−r2t )+λ2kr2t
r2t . (.) (43)
According to the assumption on independence, the expectation of the last part, new
information, in (43) disappears. Therefore, it follows from Lemma 2 that
d
dt
E(r2t ) = E
(
dr2t
dt
)
= E(G(λ∗, λ2, r2t )) < 0. (44)
That means process E(r2t ) decreases strictly and monotonously. On the other hand,
E(r2t ) ≥ 0 since r2t > 0. Thus the expectation process E(r2t ) is convergent and the
limitation is nothing but zero. Finally, (41) holds obviously on account of that r2t > 0
at the same time. ¤
References
[1] Evstigineev, ., T. Hens, and . Schenk-Hoppe´ (2006): “Evolutionary Stable
Stock Markets,” Economic Theory, 27(), 449-468.
[2] Soong, .(1973): Random Differential Equations in Science and Engineering,
Academic Press, Inc., New York.
11
