Mathematical model of a stock market
In this paper we construct a mathematical model of securities market. The results obtained are a good basis for an analysis of any stock market.
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nasplib_isofts_kiev_ua-123456789-1210082025-02-10T00:14:28Z Mathematical model of a stock market Математична модель фондового ринку Gonchar, N.S. In this paper we construct a mathematical model of securities market. The results obtained are a good basis for an analysis of any stock market. В роботі побудовано математичну модель ринку цінних паперів. Отримані результати є доброю основою для аналізу подій на фондовому ринку. 2000 Article Mathematical model of a stock market / N.S. Gonchar // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 461-496. — Бібліогр.: 2 назв. — англ. 1607-324X DOI:10.5488/CMP.3.3.461 PACS: 02.50.+s, 05.40.+j https://nasplib.isofts.kiev.ua/handle/123456789/121008 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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In this paper we construct a mathematical model of securities market. The
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Mathematical model of a stock market |
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Mathematical model of a stock market |
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Mathematical model of a stock market |
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Mathematical model of a stock market |
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Mathematical model of a stock market |
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mathematical model of a stock market |
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Mathematical model of a stock market / N.S. Gonchar // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 461-496. — Бібліогр.: 2 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT goncharns mathematicalmodelofastockmarket AT goncharns matematičnamodelʹfondovogorinku |
| first_indexed |
2025-12-02T02:17:06Z |
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2025-12-02T02:17:06Z |
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1850361077395619840 |
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Condensed Matter Physics, 2000, Vol. 3, No. 3(23), pp. 461–496
Mathematical model of a stock market
N.S.Gonchar
Bogolubov Institute for Theoretical Physics
of the National Academy of Sciences of Ukraine,
14b Metrolohichna Str., 252143 Kyiv, Ukraine
Received May 30, 2000
In this paper we construct a mathematical model of securities market. The
results obtained are a good basis for an analysis of any stock market.
Key words: random process, effective stock market, option pricing
PACS: 02.50.+s, 05.40.+j
Dedicated to prominent scientist Igor Yukhnovsky, initiator of my perspective
research on economy.
1. Introduction
The aim of this paper is to propose a wide class of random processes to describe
the evolution of a risk active price and to construct a mathematical theory of option
pricing. For this purpose, a general mathematical model of evolution of a risk active
price is proposed on a probability space constructed. On the probability space, an
evolution of a risk active price is described by a random process with jumps that
can have both finite and infinite number of jumps. We introduce a new notion of
non-singular martingale and prove an integral representation for a wide class of
local martingale by a path integral. This theorem is the basic result of the paper
that permits us to introduce the important notion of an effective stock market.
For an effective stock market the mathematical theory of European type options is
constructed. As a result, the new formulas for option pricing, the capital investor
and self-financing strategy corresponding to the minimal hedge are obtained.
2. Some auxiliary results
Hereafter we will use two elementary lemmas the proof of which is omitted.
Lemma 1. For any on the right continuous functions ϕ(x) and ψ(x), that have the
c© N.S.Gonchar 461
N.S.Gonchar
bounded variation on [a, b), the following formula
ϕ(d)ψ(d)− ϕ(c)ψ(c) =
∫
(c,d]
ϕ−(y)dψ(y) +
∫
(c,d]
ψ(y)dϕ(y), (c, d] ⊂ [a, b) (1)
is valid, F−(u) = lim
v↑u
F (v).
By dϕ(y) and dψ(y) we denoted the charges, generated by functions ϕ(y) and ψ(y)
correspondingly, ϕ−(x) = lim
y↑x
ϕ(y).
Lemma 2. The Radon-Nicodym derivative of the measure dg(y), generated by the
function g(y) = (1 − F (y))−1, with respect to the measure dF (y), where F (y) is on
the right continuous and monotonouosly non-decreasing on [a, b) function and such
that F (a) = 0, F (x) < 1, x ∈ [a, b), lim
x→b
F (x) = 1 is given by the formula
dg(y)
dF (y)
=
1
(1− F (y))(1− F−(y))
.
Lemma 3. For on the right continuous and monotoneously non-decreasing function
α(x) such that α(x) <∞, x ∈ [a, b), α(a) = 0, lim
x→b
α(x) = ∞, the representation
α(x) =
∫
[a,x]
dF (y)
1− F−(y)
(2)
is valid for a certain F (x), that is on the right continuous and monotonously non-
decreasing function, satisfying conditions: F (x) < 1, x ∈ [a, b), lim
x→b
F (x) = 1, F (a) =
0, if and only if there exists a positive, on the right continuous and monotoneously
non-decreasing solution of equation
φ(x) =
∫
[a,x]
φ(y)dα(y) + 1 (3)
such that φ(a) = 0, φ(x) <∞, x ∈ [a, b). The function F (x) is given by the formula
F (x) =
φ(x)− 1
φ(x)
. (4)
Proof. The necessity. By definition we put F−(y) = lim
x↑y
F (x). If the representation
(2) holds, then the following equality
∫
[a,x]
dα(y)
1− F (y)
=
∫
[a,x]
dF (y)
(1− F (y))(1− F−(y))
=
1
1− F (x)
− 1
462
Mathematical model of a stock market
is valid. Therefore, the function
φ(x) =
1
1− F (x)
is a positive, on the right continuous and monotonously non-decreasing solution of
equation (3).
The sufficiency. If there exists a solution to (3), satisfying conditions of lemma 3,
then the function (4) satisfies equation
∫
[a,x]
dα(y)
1− F (y)
+ 1 =
1
1− F (x)
.
But
∫
[a,x]
dF (y)
(1− F (y))(1− F−(y))
+ 1 =
1
1− F (x)
.
The latter means that
dα(y)
1− F (y)
=
dF (y)
(1− F (y))(1− F−(y))
,
or
dα(y) =
dF (y)
(1− F−(y))
.
From the latter equality it follows that
α(x) =
∫
[a,x]
dF (y)
(1− F−(y))
.
Lemma 3 is proved.
Let us give the necessary and sufficient conditions for the existence of a solution
to equation (3)
Lemma 4. Nonnegative solution to the equation (3) exists if and only if the series
φ(x) = 1 +
∞
∑
n=1
∫
[a,x]
dα(t1)
∫
[a,t1]
dα(t2) . . .
∫
[a,tn−1]
dα(tn) (5)
converges for all x ∈ [a, b).
Proof. The necessity. If there exists a non-negative solution to (3), then this
solution is the solution to the equation
φ(x) = 1 +
k
∑
n=1
∫
[a,x]
dα(t1)
∫
[a,t1]
dα(t2) . . .
∫
[a,tn−1]
dα(tn)
463
N.S.Gonchar
+
∫
[a,x]
dα(t1)
∫
[a,t1]
dα(t2) . . .
∫
[a,tk−1]
dα(tk)
∫
[a,tk ]
φ(tk+1)dα(tk+1).
From the latter equality there follows the inequality
1 +
k
∑
n=1
∫
[a,x]
dα(t1)
∫
[a,t1]
dα(t2) . . .
∫
[a,tn−1]
dα(tn) 6 φ(x).
Arbitrariness of k, positiveness of every term of the series means the convergence
of (5). The proof of sufficiency follows from the fact that if the series (5) converges
then this series is evidently a solution to the equation (3). The lemma 4 is proved.
Corollary 1. If α(x) is a continuous and monotonously non-decreasing function,
α(x) < ∞, x ∈ [a, b), lim
x→b
α(x) = ∞, α(a) = 0, then the equation (3) has the
solution φ(x) = eα(x).
Corollary 2. If γ(x) is some measurable function on [a, b), which satisfies the in-
equality
∫
[a,x]
γ(y)dα(y) + 1 6 γ(x), x ∈ [a, b),
then there exists a solution to equation (3).
Lemma 5. The solution to the equation (3) exists if jumps of monotonously non-
decreasing and on the right continuous function α(x) is such that ∆α(s) 6= 1. It has
the following form
φ(x) = eα(x)
∏
{s6x}
e−∆α(s)
(1−∆α(s))
.
If 0 6 ∆α(s) < 1, s ∈ [a, b), then this solution is non-negative, on the right contin-
uous and monotonously non-decreasing function, ∆α(s) = α(s)− α−(s).
Proof. First of all the product
∏
{s6x}
e−∆α(s)
(1−∆α(s))
converges, because the estimate
∑
{s6x}
∆α(s) 6 α(x) < ∞, x < b is valid. Let
us verify that φ(x) is a solution to (3) in the case when all jump points of α(x)
are isolated points. It is sufficient to prove that if φ(x) is the solution to (3) on a
certain interval [a, x0] and we prove that φ(x) is the solution to (3) on the interval
(x0, x], x > x0 then it will mean that φ(x) is the solution to (3) on the interval
[a, x]. We assume that the points xi, i = 1, 2, . . . are the jump points of the function
α(x). To verify that φ(x) is the solution to the equation (3) let us assume that we
464
Mathematical model of a stock market
have already proved that on the interval [a, xi), where xi is the jump point of the
function α(x), φ(x) is the solution to equation (3), that is,
∫
[a,xi)
φ(y)dα(y) = φ−(xi)− 1 = eα−(x)
∏
{s<xi}
e−∆α(s)
(1−∆α(s))
− 1.
Let x be any point that satisfies the condition xi < x < xi+1. Since
1 +
∫
[a,x]
φ(y)dα(y) = 1 +
∫
[a,xi)
φ(y)dα(y) +
∫
[xi]
φ(y)dα(y) +
∫
(xi,x]
φ(y)dα(y)
= φ−(xi)− 1 + eα(xi)∆α(xi)
∏
{s6xi}
e−∆α(s)
(1−∆α(s))
+[eα(x) − eα(xi)]
∏
{s6xi}
e−∆α(s)
(1−∆α(s))
+ 1 = eα(x)
∏
{s6x}
e−∆α(s)
(1−∆α(s))
= φ(x).
To complete the proof of the lemma it is necessary to note that on the interval [a, x1)
the solution to (3) is the function eα(x). Let us prove lemma 5 in a general case. If
α(x) satisfies the conditions to lemma 5, then
α(x) = αc(x) +
∑
{s6x}
∆α(s),
where αc(x) is a continuous function on [a, b). Let us introduce the notation
φm(x) = eαm(x)
∏
{s6x}
e−∆αm(s)
(1−∆αm(s))
,
where
αm(x) = αc(x) +
∑
{s6x, ∆α(s)>m−1}
∆α(s).
In the latter sum the summation comes over all jumps of α(x), where the jumps of
α(x) are greater than m−1. It is evident that on any interval [a, x] the set of such
points is finite. Therefore φm(x) satisfies the equation
φm(x) =
∫
[a,x]
φm(y)dαm(y) + 1. (6)
Let d < b, then
sup
x∈[a,d]
|φ(x)− φm(x)| 6 sup
x∈[a,d]
eαc(x)
×
1
∏
{s6d, ∆α(s)>m−1}
(1−∆α(s))
1−
∏
{s6d, ∆α(s)>m−1}
(1−∆α(s))
∏
{s6d}
(1−∆α(s))
6
465
N.S.Gonchar
6 sup
x∈[a,d]
eαc(x)
∑
{s6d, ∆α(s)<m−1}
∆α(s)
∏
{s6d}
(1−∆α(s))
→ 0, m→ ∞.
Moreover,
var
x∈[a,d]
[α(x)− αm(x)] 6
∑
{s6d, ∆α(s)<m−1}
∆α(s) → 0, m→ ∞,
where varx∈[a,d] g(x) means a full variation of the function g(x). From these inequal-
ities we have
∫
[a,x]
[φm(y)− φ(y)]dαm(y) 6 sup
x∈[a,d]
|φm(x)− φ(x)| var
x∈[a,d]
α(x) → 0, m→ ∞,
∫
[a,x]
φ(y)d[αm(y)− α(y)] 6 sup
x∈[a,d]
|φ(x)|| var
x∈[a,d]
[αm(y)− α(y)]| → 0, m→ ∞.
From the equality
φm(x) =
∫
[a,x]
[φm(y)− φ(y)]dαm(y) +
∫
[a,x]
φ(y)d[αm(y)− α(y)] +
∫
[a,x]
φ(y)dα(y) + 1
and from the preceding inequalities there follows the proof of the lemma 5.
Theorem 1. Let ψ(y) be an on the right continuous function of bounded variation
on any interval [a, x], x ∈ [a, b), f(y) be a measurable mapping with respect to the
Borel σ-algebra on [a, b) and bounded function on [a, x], x ∈ [a, b). If, moreover,
α(x) =
∫
[a,x]
dψ(y)
ψ(y)− f(y)
<∞, x ∈ [a, b) (7)
is monotonously non-decreasing and on the right continuous function on [a, b) and
such that
1) 0 6 ∆α(x) < 1, ∆α(x) = α(x)− α−(x), x ∈ [a, b),
2) lim
x→b
α(x) = ∞, α(a) = 0,
3) lim
x→b
ψ(x)e−α(x) = 0,
4)
b
∫
a
|f(x)|e−α−(x)dα(x) <∞,
then for the function ψ(x) the following representation
ψ(x) =
1
(1− F (x))
∫
(x,b)
f(x)dF (x)
is valid for a certain monotonously non-decreasing and on the right continuous func-
tion F (x), such that F (a) = 0, F (x) < 1, x ∈ [a, b), lim
x→b
F (x) = 1.
466
Mathematical model of a stock market
Proof. Let F (x) be the function, which is constructed in the lemma 3. Let us
consider the product [1− F (x)]ψ(x). Then for x < d < b
−[1 − F (x)]ψ(x) + [1− F (d)]ψ(d) =
∫
(x,d]
[1− F−(y)]dψ(y)−
∫
(x,d]
ψ(y)dF (y).
From the lemma 3
dψ(y) = [ψ(y)− f(y)]
dF (y)
(1− F−(y))
.
Therefore,
−[1− F (x)]ψ(x) + [1− F (d)]ψ(d) = −
∫
(x,d]
f(y)dF (y). (8)
Since
[1− F (d)]ψ(d) 6 e−α(d)ψ(d) → 0, d→ b,
∣
∣
∣
∣
∣
∣
∣
∫
(x,d]
f(y)dF (y)
∣
∣
∣
∣
∣
∣
∣
6
∫
(x,d]
|f(y)|dF (y) =
=
∫
(x,d]
|f(y)|[1− F−(y)]dα(y) 6
∫
(x,b)
|f(y)|e−α−(y)dα(y) <∞,
then, taking the limit in the equality (8), we obtain
[1− F (x)]ψ(x) =
∫
(x,b)
f(y)dF (y).
The theorem is proved.
Theorem 2. Let g(u) be a measurable function with respect to B([a, b)) and such
that
∫
[a,b)
|g(y)|dF (y) <∞,
then the following formula
1
(1− F (d))
∫
(d,b)
g(y)dF (y)−
1
(1− F (c))
∫
(c,b)
g(y)dF (y) =
=
∫
(c,d]
1
(1− F (u))
∫
(u,b)
g(y)dF (y)− g(u)
dF (u)
1− F−(u)
, (c, d] ⊂ [a, b) (9)
is valid.
467
N.S.Gonchar
Proof. If we choose
ϕ(x) = (1− F (x))−1, ψ(x) =
∫
(x,b)
g(u)dF (u)
and use lemmas 1 and 2 we obtain the proof of the theorem 2.
3. Probability space
Hereafter we construct a probability space, in which the securities market evo-
lution will be considered. Let α = {aαi }
k(α)+1
i=1 be a sequence from [a, b) ⊆ R1
+, a < b,
that satisfies conditions:
aαi < aαi+1, i = 1, k(α),
k(α)
⋃
i=1
[aαi , a
α
i+1) = [a, b), aα1 = a, aαk(α)+1 = b.
Therefore, the set of intervals {[aαi , a
α
i+1), i = 1, k(α)} forms a decomposition of
interval [a, b) ⊆ R1
+. The number k(α) may be both finite and infinite. Further
on, we consider the family of probability spaces Ωi = [a, b), i = 1, k(α). On every
probability space Ωi a σ-algebra of events F
0
i is given. By definition the σ-algebra F 0
i
is the set of subsets of Ωi = [a, b), that is generated by intervals (c, d) ⊂ [aαi , a
α
i+1).
Let us determine the flow of the σ-algebras F 0,t
i , t ∈ [a, b), F0,t
i ⊆ F0
i , by the
formula
F0,t
i =
{∅, [a, b)}, a 6 t 6 aαi ,
B([aαi , t]), aαi < t < aαi+1,
∨
t∈[aα
i
,aα
i+1)
B([aαi , t]) = F0
i , aαi+1 6 t 6 b,
where we denoted by B([aαi , t]) the σ-algebra of subsets of [a, b) generated by the
subsets of (c, d) ⊂ [aαi , t] and
∨
t∈[aα
i
,aα
i+1)
B([aαi , t]) denotes the σ-algebra, that is the
union of the σ-algebras B([aαi , t]). Let {Ωα,F
0
α} be the direct product of measurable
spaces {Ωi,F
0
i }, i = 1, k(α), and F0,α
t =
k(α)
∏
i=1
F0,t
i be the flow of the σ-algebras on
the measurable space {Ωα,F
0
α}, that is the direct product of the σ-algebras F 0,t
i ,
where Ωα =
k(α)
∏
i=1
Ωi,F
0
α =
k(α)
∏
i=1
F0
i . Let us determine a certain measurable space
{Ω,F0}. Denote by X a set of sequences α = {aαi }
k(α)+1
i=1 from [a, b) that generate
decomposition of [a, b). Let Ω =
∑
α∈X
Ω̄α be the direct sum of the probability spaces
Ω̄α = {α,Ωα}. Elements of Ω̄α are the pairs {α, ωα}, where ωα ∈ Ωα Let us denote
by F̄0
α the σ-algebra of events of the kind Āα = {α,Aα}, where Aα ∈ F0
α, {α,Aα} =
= {{α, ωα}, ωα ∈ Aα}. Analogously, F̄
0,α
t is the flow of the σ-algebras from Ω̄α of
the sets of the kind {α,Aα}, where Aα ∈ F0,α
t . It is evident that Ω̄α∩ Ω̄β = ∅, α 6= β.
468
Mathematical model of a stock market
Let Σ be the σ-algebra of all subsets of X. Introduce a σ-algebra F 0 and the
flow of the σ-algebras F 0
t in Ω. We assume that the σ-algebra F 0 in Ω is the set of
the subsets of the kind
CY =
⋃
α∈Y
Bα, Y ∈ Σ, Bα ∈ F̄0
α.
This follows from the following inclusions
∞
⋃
i=1
CYi
=
⋃
α∈
∞⋃
i=1
Yi
Bα ∈ F0,
∞
⋂
i=1
CYi
=
⋃
α∈
∞⋂
i=1
Yi
Bα ∈ F0, CY1\CY2 =
⋃
α∈Y1\Y2
Bα ∈ F0.
By analogy with the construction of the σ-algebra F 0, the flow of the σ-algebra
F0
t ⊆ F0 is the set of the subsets of the type
CY =
⋃
α∈Y
Bα, Y ∈ Σ, Bα ∈ F̄0,α
t .
Further on we deal with the measurable space {Ω,F 0} and the flow of the σ-
algebras F 0
t ⊆ F0 on it. Hereafter we construct the probability space {Ω,F 0, P}.
Define a probability measure Pα on the measurable space {Ωα,F
0
α}. For this
purpose on every measurable space {Ωi,F
0
i } we determine the family of distribution
functions F α
i (ω
α
i |{ωα}i−1), that at every fixed {ωα}i−1 ∈ Ωi−1 =
i−1
∏
s=1
Ωs is on the
right continuous and non-decreasing function of the variable ωα
i ∈ [a, b),
F α
i (ω
α
i |{ωα}i−1) =
0, a 6 ωα
i 6 aαi , {ωα}i−1 ∈ Ωi−1,
φα
i (ω
α
i |{ωα}i−1), aαi < ωα
i < aαi+1 , {ωα}i−1 ∈ Ωi−1,
1, aαi+1 6 ωα
i < b, {ωα}i−1 ∈ Ωi−1,
where {ωα}i−1 = {ωα
1 , . . . , ω
α
i−1}, ωα = {ωα
1 , . . . , ω
α
k(α)}.
The function φα
i (ω
α
i |{ωα}i−1) satisfies the conditions: 0 6 φα
i (ω
α
i |{ωα}i−1) < 1,
it is on the right continuous and non-decreasing function of the variable ωα
i on
[aαi , a
α
i+1) at every fixed {ωα}i−1 ∈ Ωi−1, moreover, it is a measurable function
from the measurable space {Ωi−1, F̄0
i−1} to the measurable space {[0, 1],B([0, 1])}
at every fixed ωα
i , where B([0, 1]) is the Borel σ-algebra on [0, 1], F̄0
i−1 =
i−1
∏
s=1
F0
s .
Denote by F α
i (dω
α
i |{ωα}i−1) the measure constructed by the distribution function
F α
i (ω
α
i |{ωα}i−1) on the σ-algebra F 0
i at every fixed {ωα}i−1 ∈ Ωi−1. It is evident that
F α
i (dω
α
i |{ωα}i−1) is concentrated on the subset [aαi , a
α
i+1) ⊂ Ωi . Let us determine a
measure on the probability space {Ωα,F
0
α}, having determined it on the set of the
type A1 × . . .× Ak(α), Ai ∈ F0
i by the formula
Pα(A1 × . . .× Ak(α)) =
=
∫
A1
. . .
∫
Ak(α)
F α
1 (dω
α
1 )F
α
2 (dω
α
2 |{ωα}1)× . . .× F α
k(α)(dω
α
k(α)|{ωα}k(α)−1).
469
N.S.Gonchar
The function of the sets so defined can be extended to a certain measure Pα on F0
α
due to Ionescu and Tulcha theorem [1]. We put by definition that on the σ-algebra
F̄0
α the probability measure P̄α is given by the formula P̄α(Āα) = Pα(Aα). Further
on we consider both the probability spaces {Ωα,F
0
α, Pα} and the probability spaces
{Ω̄α, F̄
0
α, P̄α}, that are isomorphic, and the flows of the σ-algebras F 0,α
t ⊆ F0
α and
F̄0,α
t ⊆ F̄0
α on the spaces Ωα and Ω̄α correspondingly. If µ(Y ) is a probability measure
on Σ, we put that on the σ-algebra F 0 the probability measure P is given by the
formula
P (CY ) =
∫
Y
P̄α(Bα)dµ(α), CY =
⋃
α∈Y
Bα, Y ∈ Σ, Bα ∈ F̄0
α.
The latter integral exists, because P̄α(Bα) is a measurable mapping from the mea-
surable space {X,Σ} to the measurable space {R1,B(R1)}, where B(R1) is the Borel
σ-algebra on R1.
Further on we consider the probability space {Ω,F 0, P} and the flow of the
σ-algebras F 0
t ⊆ F0 on it, the probability space {Ω,F , P} and the flow of the σ-
algebras Ft ⊆ F , where F and Ft are the completion of F 0 and F0
t correspondingly
with respect to the measure P. Then we use the same notation P for the extension
of a measure P from the σ-algebra F 0 onto the σ-algebra F , where the σ-algebra
F is the completion of the σ-algebra F 0 by the sets of zero measure with respect to
the measure P given on the σ-algebra F 0.
4. Random processes on the probability space
Definition 1. A consistent with the flow of the σ-algebras F 0
t measurable mapping
ζt({α, ωα}) from the measurable space {Ω,F0} to the measurable space {R1,B(R1)}
belongs to a certain class K if for ζt({α, ωα}) the representation
ζt({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)ζ i,αt ({ωα}i),
ζ
i,α
t ({ωα}i) = fα
i ({ωα}i)χ[aα
i
, t](ω
α
i ) + ψα
i ({ωα}i−1, t)χ(t, aα
i+1)
(ωα
i ),
t ∈ [aαi , a
α
i+1) (10)
is valid, where fα
i ({ωα}i) is a measurable mapping from the measurable space
{Ωi, F̄0
i } to the measurable space {R1,B(R1)} at every fixed α ∈ X0, i = 1, k(α),
ψα
i ({ωα}i−1, t) is a measurable mapping from the measurable space {Ωi−1, F̄0
i−1} to
the measurable space {R1,B(R1)} at every fixed t ∈ [aαi , a
α
i+1), α ∈ X0, i = 2, k(α).
Further we deal with the space X0 that consists of sequences α = {aαi }
k(α)+1
i=1 not
having limiting points on the interval [a, x], ∀x < b, Σ is the σ-algebra of all subsets
of X0. Hereinafter χD(t) denotes the indicator function of the set D from [a, b).
470
Mathematical model of a stock market
Definition 2. By K0 we denote the subclass of the class K of measurable mappings,
satisfying conditions:
1) ψα
i ({ωα}i−1, t) is an on the right continuous function of bounded variation of the
variable t on any interval [aαi , τ ], τ ∈ [aαi , a
α
i+1) at every fixed {ωα}i−1 ∈ Ωi−1, i =
= 1, k(α).
2) fα
i ({ωα}i) is a measurable and bounded mapping from the measurable space
{Ωi, F̄0
i } to the measurable space {R1,B(R1)}.
3) The function
γi,α({ωα}i−1, t) =
∫
[aα
i
, t]
ψα
i ({ωα}i−1, dτ)
[ψα
i ({ωα}i−1, τ)− fα
i ({ωα}i−1, τ)]
, i = 1, k(α), (11)
where fα
i ({ωα}i−1, τ) = fα
i ({ωα}i)|ω
α
i = τ, is monotonously non-decreasing and on
the right continuous function of the variable t on the interval [aαi , a
α
i+1) at every fixed
{ωα}i−1 ∈ Ωi−1, α ∈ X0, satisfying conditions:
a) ∆γi,α({ωα}i−1, t) < 1, {ωα}i−1 ∈ Ωi−1, t ∈ [aαi , a
α
i+1),
∆γi,α({ωα}i−1, t) = γi,α({ωα}i−1, t)− γi,α({ωα}i−1, t−),
γi,α({ωα}i−1, t−) = lim
s↑t
γi,α({ωα}i−1, s).
b) lim
t→aα
i+1
γi,α({ωα}i−1, t) = ∞, γi,α({ωα}i−1, a
α
i ) = 0, {ωα}i−1 ∈ Ωi−1, α ∈ X0.
c) lim
t→aα
i+1
ψα
i ({ωα}i−1, t) exp {−γ
i,α({ωα}i−1, t)} = 0,
d)
∫
[aα
i
, aα
i+1)
|fα
i ({ωα}i−1, t)| exp {−γ
i,α({ωα}i−1, t−)}γ
i,α({ωα}i−1, dt) <∞,
{ωα}i−1 ∈ Ωi−1, α ∈ X0.
We denoted by γi,α({ωα}i−1, dt) the measure on B([aαi , a
α
i+1)), generated
by the monotonously non-decreasing and on the right continuous function
γi,α({ωα}i−1, t) of the variable t at every fixed {ωα}i−1 ∈ Ωi−1, B([aαi , a
α
i+1)) is the
Borel σ-algebra on the interval [aαi , a
α
i+1).
Lemma 6. Any on the right continuous and uniformly integrable martingale on the
probability space {Ω,F , P} with respect to the flow Ft is given by the formula
Mt({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)mi,α
t ({ωα}i), (12)
where
m
i,α
t ({ωα}i) = fα
i ({ωα}i)χ[aα
i
, t](ω
α
i ) + ψα
i ({ωα}i−1, t)χ(t, aα
i+1)
(ωα
i ),
fα
i ({ωα}i) =
∫
Ωi+1
. . .
∫
Ωk(α)
gα({ωα}i, {ωα}[i+1,k(α)])
× F α
i+1(dω
α
i+1|{ωα}i)× . . .× F α
k(α)(dω
α
k(α)|{ωα}k(α)−1),
ψα
i ({ωα}i−1, t) =
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1),
i = 1, k(α), α ∈ X0, (13)
471
N.S.Gonchar
gα({ωα}i, {ωα}[i+1,k(α)]) is a measurable and integrable function on the probability
space {Ω,F , P}, that is,
∫
X0
Mα|g({α, ωα})|dµ(α) <∞.
Proof. Further on, for the mapping gα({ωα}i, {ωα}[i+1,k(α)]) we use one more nota-
tion g({α, ωα})=g
α({ωα}i, {ωα}[i+1,k(α)]), where {ωα}i={ωα
1 , . . . , ω
α
i }, {ωα}[i,k(α)] =
{ωα
i+1, . . . , ω
α
k(α)}, ωα = {ωα
1 , . . . , ω
α
k(α)} = {{ωα}i, {ωα}[i,k(α)]}. Taking into account
the σ-algebra Σ from X0 consists of all subsets of X 0 to prove the lemma 6, it is
sufficient to calculate the conditional expectation
M{g({β, ωβ})|Ft}|β=α =Mα{g({α, ωα})|F̄
0,α
t },
where g({α, ωα}) is a measurable and integrable function on the probability space
{Ω,F , P}, Mα{g({α, ωα})|F̄
0,α
t } is the conditional expectation with respect to the
flow of the σ-algebras F̄0,α
t ⊆ F̄0
α on the probability space {Ω̄α, F̄
0
α, P̄α}. Suppose
that t ∈ [aαi , a
α
i+1). From this it follows that
ϕt
i({ωα}) =Mα{g({α, ωα})|F̄
0,α
t }
is the measurable mapping from {Ω̄α, F̄
0,α
t , P̄α} to {R1,B(R1)}, where F̄0,α
t =
i−1
∏
s=1
F0
s × F0,t
i ×
k(α)
∏
s=i+1
Os, Os = {∅, [a, b)}, s = i+ 1, k(α). Due to the structure
of the σ-algebra F̄0,α
t it follows that ϕt
i({ωα}) depends only on variables {ωα}i and
ϕt
i({ωα}) is a measurable mapping from {Ωi,
i−1
∏
s=1
F0
s ×F0,t
i } to {R1,B(R1)}. Granting
this notation we have
ϕt
i({ωα}) = Qt
i({ωα}i)
= ϕα
i ({ωα}i, t)χ[aα
i
, t](ω
α
i ) + ψα
i ({ωα}i−1, t)χ(t, aα
i+1)
(ωα
i ). (14)
Really,
Qt
i({ωα}i) = Qt
i({ωα}i)χ[aα
i
, t](ω
α
i ) +Qt
i({ωα}i)χ(t, aα
i+1)
(ωα
i ).
Because of the fact that Qt
i({ωα}i)χ[aα
i
,t](ω
α
i ) is a measurable mapping from
{Ωi,
i−1
∏
s=1
F0
s × F0,t
i } to {R1,B(R1)} it follows that Qt
i({ωα}i)χ(t,aα
i+1)
(ωα
i ) is also the
measurable mapping. But this is possible, when Q t
i({ωα}i) does not depend on the
variable ωα
i ∈ (t, b), because the only measurable sets
i−1
∏
s=1
Bs × Ai belong to the
σ-algebra
i−1
∏
s=1
F0
s × F0,t
i , when ωα
i ∈ (t, b), where Bs ∈ F0
s , Ai = [a, aαi ) ∪ (t, b).
Putting
Qt
i({ωα}i) = ϕα
i ({ωα}i, t), {ωα}i ∈ Ωi−1 × [aαi , t],
Qt
i({ωα}i) = ψα
i ({ωα}i−1, t), {ωα}i ∈ Ωi−1 × {[a, aαi ) ∪ (t, b)},
472
Mathematical model of a stock market
we prove the representation (14). Taking into account the definition of the condi-
tional expectation we have
∫
B1
. . .
∫
Bi−1
∫
A
Qt
i({ωα}i)F
α
1 (dω
α
1 )× . . .× F α
i (dω
α
i |{ωα}i−1) =
=
∫
B1
. . .
∫
Bi−1
∫
A
∫
Ωi+1
. . .
∫
Ωk(α)
g({α, ωα})F
α
1 (dω
α
1 )× . . .× F α
k(α)(dω
α
k(α)|{ωα}k(α)−1),
Bs ∈ Ωs, s = 1, i− 1, A ∈ F0,t
i . (15)
Let us introduce the measurable mapping
fα
i ({ωα}i) =
∫
Ωi+1
. . .
∫
Ωk(α)
gα({ωα}i, {ωα}[i+1,k(α)])
×F α
i+1(dω
α
i+1|{ωα}i)× . . .× F α
k(α)(dω
α
k(α)|{ωα}k(α)−1)
from the measurable space {Ωi, F̄0
i } to the measurable space {R1,B(R1)}. It is
evident that
∫
Ω1
. . .
∫
Ωi
|fα
i ({ωα}i)|F
α
1 (dω
α
1 )× . . .× F α
i (dω
α
i |{ωα}i−1) 6Mα|g({α, ωα})| <∞.
From this and (15) it follows that
ϕα
i ({ωα}i, t) = fα
i ({ωα}i),
ψα
i ({ωα}i−1, t) =
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1),
i = 1, k(α), α ∈ X0.
It is evident that between fα
i ({ωα}i) there exists the following relations
fα
i ({ωα}i) =
∫
Ωi+1
fα
i+1({ωα}i+1)F
α
i+1(dω
α
i+1|{ωα}i), i = 1, k(α), α ∈ X0.
The proof of the lemma 6 is completed.
Lemma 7. Let a measurable mapping ζt({α, ωα}) on the measurable space {Ω,F0}
belong to the subclass K0, for every fixed α ∈ X0, i = 1, k(α),
ψα
i ({ωα}i−1, a
α
i ) = fα
i−1({ωα}i−1), {ωα}i−1 ∈ Ωi−1
and there exists a constant A <∞ such that
∫
X0
ψα
1 (a)dµ(α) 6 A
473
N.S.Gonchar
for a certain probability measure µ on Σ. If fα
i ({ωα}i) > 0, then on the measur-
able space {Ω,F0} there exist a measure P on the σ-algebra F0 and a modification
ζ̄t({α, ωα}) of the measurable mapping ζt({α, ωα}), such that ζ̄t({α, ωα}) is a local
martingale on the probability space {Ω,F , P} with respect to the flow of the σ-algebra
Ft, where the σ-algebras F and Ft are the completion of the σ-algebras F0 and F0
t
correspondingly with respect to the measure P.
Proof. The proof of the lemma 7 follows from the theorem 1. Really, all conditions
of the theorem 1 are valid. Therefore, there exists a family of distribution functions
F α
i (ω
α
i |{ωα}i−1), i = 1, k(α), α ∈ X0, {ωα}i−1 ∈ Ωi−1, with the properties, described
at the introduction of the probability space {Ω,F , P}, that for ψα
i ({ωα}i−1, t) the
following representation
ψα
i ({ωα}i−1, t) =
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1),
i = 1, k(α), α ∈ X0,
is valid.
Let us consider the measurable mapping
ζ̄t({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)ζ̄ i,αt ({ωα}i),
ζ̄
i,α
t ({ωα}i}) = fα
i ({ωα}i)χ[aα
i
, t](ω
α
i )
+
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)χ(t, aα
i+1)
(ωα
i ),
on the probability space {Ω,F , P} consistent with the flow of the σ-algebras Ft,
where the σ-algebras F and Ft are the completion of the σ-algebras F 0 and F0
t
correspondingly with respect to the measure P, generated by the family of distribu-
tion functions F α
i (ω
α
i |{ωα}i−1), i = 1, k(α), α ∈ X0 and the measure dµ(α). The
measurable mapping ζt({α, ωα}) differ from the measurable mapping ζ̄t({α, ωα})
on the set Ω\Ω0, P (Ω\Ω0) = 0. Let us construct the set Ω0. Consider the set
Ω0
α = {ωα ∈ Ωα, a
α
i < ωα
i < aαi+1, i = 1, k(α)}, where ωα = {ωα
1 , . . . , ω
α
k(α)} and
show that Pα(Ω
0
α) = 1. Really, since the sequence of the sets
Ωn
α = {ωα ∈ Ωα, a
α
i < ωα
i < aαi+1, i = 1, n, a 6 ωα
i < b, i = n+ 1, k(α)}
has the probability 1, that is, Pα(Ω
n
α) = 1, n = 1, 2, . . . , and taking into account
that Ωn
α ⊃ Ωn+1
α , Ω0
α =
∞
⋂
n=1
Ωn
α, the continuity of the probability measure Pα, we
obtain Pα(Ω
0
α) = 1. As far as there are no more than a countable set of α for which
µ(α) > 0, then there exists a countable subset X0
1 ⊆ X0 such that the direct sum
of the sets Ω0
α, α ∈ X0
1 forms the set Ω0 ⊂ Ω, P (Ω0) = 1.
474
Mathematical model of a stock market
From the condition of the lemma 7 we have the recurrent relations
fα
i−1({ωα}i−1) =
∫
[aα
i
, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)
=
∫
Ωi
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1), i = 1, k(α), α ∈ X0.
As far as
ψα
1 (a) =
∫
Ω1
fα
1 ({ωα}1)F
α
1 (dω
α
1 ) <∞,
then we have
ψα
1 (a) =
∫
Ω1
. . .
∫
Ωi
fα
i ({ωα}i)F
α
1 (dω
α
1 )× . . .× F α
i (dω
α
i |{ωα}i−1).
For every t0 ∈ [a, b) let us introduce the measurable mapping from the measurable
space {Ω,F} to the measurable space {R1,B(R1)}.
gt0({α, ωα}) = fα
i(t0,α)
({ωα}i(t0,α)),
where i(t0, α) = max{i, aαi 6 t0}. From the condition of lemma 7
Mgt0({α, ωα}) =
∫
X0
ψα
1 (a)dµ(α) <∞.
Moreover, it is not difficult to see that
ζ̄t∧t0({α, ωα}) =M{gt0({α, ωα})|Ft}.
The latter equality means that ζt({α, ωα}) is a local martingale since this equality
is valid for any t0 ∈ [a, b). Therefore, we can choose the sequence of stop moments
tn0 → b with probability 1 such that ζt∧tn0 ({α, ωα}) → ζt({α, ωα}) with probability 1.
The lemma 7 is proved.
In a more general case, there holds
Lemma 8. Let a measurable mapping ζt({α, ωα}) on the measurable space {Ω,F0}
belong to the subclass K0. Suppose that for any t0 ∈ [a, b),
∫
X0
dαi(t0,α)dµ(α) <∞
for a certain probability measure dµ(α) on Σ, where dαi = sup
{{ωα}i∈Ωi}
|fα
i ({ωα}i)|,
i(t0, α) = max{i, aαi 6 t0}. If for every fixed i = 1, k(α), α ∈ X0,
ψα
i ({ωα}i−1, a
α
i ) = fα
i ({ωα}i−1), {ωα}i−1 ∈ Ωi−1,
475
N.S.Gonchar
then on the measurable space {Ω,F0} there exist a measure P on the σ-algebra
F0 and a modification ζ̄t({α, ωα}) of the measurable mapping ζt({α, ωα}), such that
ζ̄t({α, ωα}) is a local martingale on the probability space {Ω,F , P} with respect to
the flow of the σ-algebra Ft, where the σ-algebras F and Ft are the completion of
the σ-algebras F0 and F0
t correspondingly with respect to the measure P.
The proof of the lemma 8 is the same as the proof of the lemma 7.
As before, let {Ω,F , P} be the probability space with the flow of the σ-algebras
Ft ⊆ F on it. Suppose that ζt({α, ωα}) is a random process consistent with the flow
of σ-algebras Ft, where
ζt({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)ζ i,αt ({ωα}i),
ζ
i,α
t ({ωα}i}) = fα
i ({ωα}i)χ[aα
i
, t](ω
α
i )
+
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)χ(t, aα
i+1)
(ωα
i ), (16)
satisfying the conditions:
fα
i ({ωα}i) =
∫
Ωi+1
fα
i+1({ωα}i+1)F
α
i+1(dω
α
i+1|{ωα}i),
∫
X0
∫
Ω1
. . .
∫
Ωiα
|fα
iα
({α, ωα}iα)|F
α
1 (dω
α
1 ) . . . F
α
iα
(dωα
iα
|{ωα}iα−1)dµ(α) <∞
for every t0 ∈ [a, b), iα = i(t0, α) = max{i, aαi 6 t0}, then ζt({α, ωα}) is a local
martingale. This assertion can be proved the same way as lemma 7 was proved.
Further on we connect with the local martingale ζt({α, ωα}) on {Ω,F , P} a
stochastic process
ζat ({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)ζ i,αt,a ({ωα}i),
which is consistent with the flow of the σ-algebras Ft, where
ζ
i,α
t,a ({ωα}i) =
= fα
i ({ωα}i−1, t)−
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1).
We shall call the process ζ at ({α, ωα}) as the process associated with the ζt({α, ωα})
process.
476
Mathematical model of a stock market
Definition 3. Realization of the assotiated process ζat ({α, ωα}) is regular if the set
{t ∈ [a, b), ζ
i,α
t,a ({ωα}i) = 0, i = 1, k(α)}
is no more than the countable set.
Definition 4. A local martingal ζt({α, ωα}) is non-singular on {Ω,F , P} if the
set of regular realizations of the associated random process ζat ({α, ωα}) has got the
probability 1.
Lemma 9. On the probability space {Ω,F , P} there always exists a non-singular
local martingale.
Proof. To prove the lemma 9 we construct an example of a martingale on {Ω,F , P}
that is non-singular. Let f α
s (ω
α
s ) > 0, s = 1, k(α), α ∈ X0 be the measurable
mapping with respect to the σ-algebra F 0
s , satisfying conditions:
0 <
∫
Ωs
fα
s (ω
α
s )F
α
s (dω
α
s |{ωα}s−1) <∞,
s = 1, k(α), α ∈ X0, {ωα}s−1 ∈ Ωs−1, (17)
fα
s (t)−
1
1− F α
s (t|{ωα}s−1)
∫
(t, aα
i+1)
fα
s (ω
α
s )F
α
s (dω
α
s |{ωα}s−1) 6= 0,
t ∈ [aαs , a
α
s+1), s = 1, k(α), α ∈ X0, {ωα}s−1 ∈ Ωs−1. (18)
Then the local martingale
ξαt ({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)ξi,αt ({ωα}i)
is not singular, where
ξ
i,α
t ({ωα}i}) = gαi ({ωα}i)χ[aα
i
, t](ω
α
i )
+
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)χ(t,aα
i+1)
(ωα
i ),
gαi ({ωα}i) =
i
∏
s=1
g0,αs ({ωα}s), g0,αs ({ωα}s) =
fα
s (ω
α
s )
∫
Ωs
fα
s (ω
α
s )F
α
s (dω
α
s |{ωα}s−1)
.
If, for example, fα
s (ω
α
s ) > 0, s = 1, k(α), α ∈ X0, are strictly monotonous on
[aαs , a
α
s+1), then the conditions (17), (18) are satisfied. The lemma 9 is proved.
477
N.S.Gonchar
Theorem 3. For any local martingale ζt({α, ωα}) given by the formula (16) and
satisfying conditions sup
{ωα}i∈Ωi
|fα
i ({ωα}i)| = βα
i <∞, i = 1, k(α), α ∈ X0,
∫
[aα
i
, aα
i+1)
|ϕα
i (s|{ωα}i−1)|
F α
i (ds|{ωα}i−1)
1− F α
i (s−|{ωα}i−1)
<∞, {ωα}i−1 ∈ Ωi−1, α ∈ X0,
the following representation
ζt({α, ωα}) =
∫
Ω1
fα
1 (ω
α
1 )F
α
1 (dω
α
1 ) +
∫
[a,t]
ψk(α)(s|ωα)dξs({α, ωα}), t ∈ [a, b)
is valid if the local martingale ξt({α, ωα}) is non-singular, sup
{ωα}i∈Ωi
|gαi ({ωα}i)| =
= δαi <∞, i = 1, k(α), α ∈ X0, and
∫
[aα
i
, aα
i+1)
|ϕ0,α
i (s|{ωα}i−1)|
F α
i (ds|{ωα}i−1)
1− F α
i (s−|{ωα}i−1)
<∞, {ωα}i−1 ∈ Ωi−1, α ∈ X0,
where
ξt({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)ξi,αt ({ωα}i),
ξ
i,α
t ({ωα}i}) = gαi ({ωα}i)χ[aα
i
, t](ω
α
i )
+
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)χ(t, aα
i+1)
(ωα
i ),
ψk(α)(s|ωα) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(s)
ϕα
i (s|{ωα}i−1)
ϕ
0,α
i (s|{ωα}i−1)
,
ϕ
0,α
i (s|{ωα}i−1) =
= gαi ({ωα}i−1, s)−
1
1− F α
i (s|{ωα}i−1)
∫
(s, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1),
ϕα
i (s|{ωα}i−1) =
= fα
i ({ωα}i−1, s)−
1
1− F α
i (s|{ωα}i−1)
∫
(s, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1).
478
Mathematical model of a stock market
Proof. Let us consider on the right continuous version of the random processes
ξ1t ({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)ξi,α,1t ({ωα}i),
ξ
i,α,1
t ({ωα}i) =
=
1
1− F α
i (ω
α
i |{ωα}i−1)
∫
(ωα
i
, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)χ[aα
i
, t)(ω
α
i )
+
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)χ(t, aα
i+1)
(ωα
i ),
ξ2t ({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)ξi,α,2t ({ωα}i),
ξ
i,α,2
t ({ωα}i) = χ[aα
i
, t](ω
α
i )
×
gαi ({ωα}i)−
1
1− F α
i (ω
α
i |{ωα}i−1)
∫
(ωα
i
, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)
.
It is obvious that
ξt({α, ωα}) = ξ1t ({α, ωα}) + ξ2t ({α, ωα}).
All realizations of the random processes ξ it({α, ωα}), i = 1, 2, have got a bounded
variation on any interval [a, t], t < b. Denote by dξ it({α, ωα}), i = 1, 2 and
dξt({α, ωα}) the charges generated by these realizations on the σ-algebra B([a, b)).
To prove the theorem 3, consider those realizations that are continuous at the points
{aαi }
k(α)+1
i=1 , α ∈ X0. The set of realizations satisfying this condition have got the
probability 1. The left and right limits at every point aα
i , i = 1, k(α), α ∈ X0 equal
lim
t↓aα
i
ξt({α, ωα}) = ξ
i,α
aα
i
({ωα}i)
=
∫
[aα
i
, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1), aαi < ωα
i < aαi+1, {ωα}i−1 ∈ Ωi−1,
gαi ({ωα}i−1, a
α
i ), ωα
i = aαi , {ωα}i−1 ∈ Ωi−1,
(19)
lim
t↑aα
i
ξt({α, ωα}) = gαi−1({ωα}i−1), {ωα}i−1 ∈ Ωi−1. (20)
Consider the set Ω0
α = {ωα ∈ Ωα, a
α
i < ωα
i < aαi+1, i = 1, k(α)}, where ωα =
{ωα
1 , . . . , ω
α
k(α)} and show that Pα(Ω
0
α) = 1. Really, since the sequence of the sets
Ωn
α = {ωα ∈ Ωα, a
α
i < ωα
i < aαi+1, i = 1, n, a 6 ωα
i < b, i = n+ 1, k(α)}
479
N.S.Gonchar
has got the probability 1, that is, Pα(Ω
n
α) = 1, n = 1, 2, . . . , and taking into account
that Ωn
α ⊃ Ωn+1
α , Ω0
α =
∞
⋂
n=1
Ωn
α, the continuity of the probability measure Pα, we
obtain Pα(Ω
0
α) = 1. As far as it is no more than a countable set of α for which
µ(α) > 0, then there exists a countable subset X0
1 ⊆ X0 such that the direct sum
of the sets Ω0
α, α ∈ X0
1 forms the set Ω0 ⊂ Ω, P (Ω0) = 1. Since
gαi−1({ωα}i−1) =
∫
[aα
i
, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1) =
=
∫
Ωi
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1), i = 1, k(α), α ∈ X0,
then for every {α, ωα} ∈ Ω0 the realization of a random process ξt({α, ωα}) is con-
tinuous at the points {aαi }
k(α)+1
i=1 . The charge generated by realizations of the random
process ξ1t ({α, ωα}) on the interval [aαi , a
α
i+1) is absolutely continuous with respect
to the measure F α
i (dt|{ωα}i−1) and the Radon-Nicodym derivative equals
dξi,α,1t ({α, ωα})
F α
i (dt|{ωα}i−1)
= χ[aα
i
,ωα
i
](t)
1
1− F α
i (t−|{ωα}i−1)
(21)
×
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)− gαi ({ωα}i−1, t)
,
where
1
1− F α
i (t−|{ωα}i−1)
= lim
τ↑t
1
1− F α
i (τ |{ωα}i−1)
.
The charge dξ2t ({α, ωα}) generated by realizations of the process ξ2t ({α, ωα}) on the
interval [aαi , a
α
i+1) is concentrated at the point t = ωα
i
dξ2t ({α, ωα}) = dξi,α,2t ({ωα}i) = δ(t− ωα
i )ϕ
0,α
i (ωα
i |{ωα}i−1). (22)
Let us calculate
∫
[aα
i
, t]
ϕα
i (s|{ωα}i−1)
ϕ
0,α
i (s|{ωα}i−1)
dξi,αs ({ωα}i) =
∫
[aα
i
, t]
ϕα
i (s|{ωα}i−1)
ϕ
0,α
i (s|{ωα}i−1)
dξi,α,1s ({ωα}i)
+
∫
[aα
i
, t]
ϕα
i (s|{ωα}i−1)
ϕ
0,α
i (s|{ωα}i−1)
dξi,α,2s ({ωα}i) = K
i,α,1
t ({ωα}i) +K
i,α,2
t ({ωα}i).
Using (21) and the theorem 2 we have
K
i,α,1
t ({ωα}i) = −
∫
[aα
i
, t]
χ[aα
i
,ωα
i
](s)ϕ
α
i (s|{ωα}i−1)
F α
i (ds|{ωα}i−1)
1− F α
i (s−|{ωα}i−1)
480
Mathematical model of a stock market
=
1
1− F α
i (ω
α
i |{ωα}i−1)
∫
(ωα
i
, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)χ[aα
i
, t)(ω
α
i )
+
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)χ(t, aα
i+1)
(ωα
i )
−
∫
(aα
i
, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1).
Further,
K
i,α,2
t ({ωα}i) = χ[aα
i
, t](ω
α
i )
×
fα
i ({ωα}i)−
1
1− F α
i (ω
α
i |{ωα}i−1)
∫
(ωα
i
, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)
.
Therefore,
∫
[aα
i
, t]
ϕα
i (s|{ωα}i−1)
ϕ
0,α
i (s|{ωα}i−1)
dξi,αs ({ωα}i) =
= fα
i ({ωα}i)χ[aα
i
, t](ω
α
i )
+χ(t, aα
i+1)
(ωα
i )
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)
−
∫
[aα
i
, aα
i+1)
fα
i ({ωα}i)F
α
i (dω
α
i |{ωα}i−1).
Taking the limit t→ aαi+1 we obtain
∫
[aα
i
, aα
i+1)
ϕα
i (s|{ωα}i−1)
ϕ
0,α
i (s|{ωα}i−1)
dξi,αs ({ωα}i) =
= fα
i ({ωα}i)− fα
i−1({ωα}i−1), i = 1, k(α),
where
fα
0 ({ωα}0) =
∫
(aα1 , aα2 )
fα
1 ({ωα}1)F
α
1 (dω
α
1 ) =
∫
Ω1
fα
1 ({ωα}1)F
α
1 (dω
α
1 ).
Granting this and the definition of ψk(α)(s|ωα) we obtain
∫
Ω1
fα
1 (ω
α
1 )F
α
1 (dω
α
1 ) +
∫
[a,t]
ψk(α)(s|ωα)dξs({α, ωα}) = ζ
i,α
t ({ωα}i)
= ζt({α, ωα}), t ∈ [aαi , a
α
i+1).
The theorem 3 is proved.
481
N.S.Gonchar
Theorem 4. Let ξ0t ({α, ωα}) be a local martingale on {Ω,F , P},
ξ0t ({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)ξi,αt ({ωα}i),
ξ
i,α
t ({ωα}i}) = gαi ({ωα}i)χ[aα
i
, t](ω
α
i )
+
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)χ(t, aα
i+1)
(ωα
i ),
satisfying conditions:
sup
{ωα}i∈Ωi
|gαi ({ωα}i)| = βα
i <∞, i = 1, k(α), α ∈ X0,
∫
[aα
i
, aα
i+1)
|ϕ0,α
i (s|{ωα}i−1)|
F α
i (ds|{ωα}i−1)
1− F α
i (s−|{ωα}i−1)
<∞,
{ωα}i−1 ∈ Ωi−1, i = 1, k(α), α ∈ X0,
where
ϕ
0,α
i (s|{ωα}i−1) =
= gαi ({ωα}i−1, s)−
1
1− F α
i (s|{ωα}i−1)
∫
(s, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1).
The random process
ξt({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)fα(ξ
i,α
t ({ωα}i))
belongs to the subclass K0 if the family of functions fα(x) > 0, x ∈ R1, α ∈ X0,
are strictly monotonous, sup
x∈R1
|f ′
α(x)| = fα
1 <∞, inf
x∈R1
|f ′
α(x)| = fα
2 > 0, moreover,
sup
i
sup
{ωα}i−1∈Ωi−1
sup
s∈[aα
i
,aα
i+1)
∆F α
i (s|{ωα}i−1)
1− F α
i (s−|{ωα}i−1)
<
fα
2
fα
1
.
Proof. To prove the theorem 4 it is sufficient to verify the fulfillment of the condi-
tions of definition 2. The condition 1 is valid, because
ψα
i ({ωα}i−1, t) = fα(T
α
i (t|{ωα}i−1))
is continuous on the right, where
T α
i (t|{ωα}i−1) =
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1).
482
Mathematical model of a stock market
Moreover,
var
t∈[aα
i
, τ ]
ψα
i ({ωα}i−1, t) 6 fα
1
∫
[aα
i
, τ ]
|ϕ0,α
i (s|{ωα}i−1)|
F α
i (ds|{ωα}i−1)
1− F α
i (s−|{ωα}i−1)
<∞,
τ ∈ [aαi , a
α
i+1), {ωα}i−1 ∈ Ωi−1, α ∈ X0.
The Radon-Nicodym derivative of the charge ψα
i ({ωα}i−1, dt), generated by
ψα
i ({ωα}i−1, t), equals
ψα
i ({ωα}i−1, dt)
F α
i (dt|{ωα}i−1)
= −
f
′
α(T
α
i (t|{ωα}i−1))ϕ
0,α
i (t|{ωα}i−1)
1− F α
i (t−|{ωα}i−1)
.
Thus,
γi,α({ωα}i−1, t) =
∫
[aα
i
, t]
ψα
i ({ωα}i−1, dτ)
ψα
i ({ωα}i−1, τ)− fα(g
α
i ({ωα}i−1, τ))
=
∫
[aα
i
, t]
f
′
α(T
α
i (τ |{ωα}i−1))F
α
i (dτ |{ωα}i−1)
U({ωα}i−1, τ)[1− F α
i (τ−|{ωα}i−1)]
is non-negative and monotonously non-decreasing on [aαi , a
α
i+1), where
U({ωα}i−1, τ) =
1
∫
0
f
′
α(g
α
i ({ωα}i−1, τ) + z[T α
i (τ |{ωα}i−1)− gαi ({ωα}i−1, τ)])dz.
Further,
∆γi,α({ωα}i−1, t) 6
fα
1
fα
2
sup
i
sup
{ωα}i−1∈Ωi−1
sup
s∈[aα
i
,aα
i+1)
∆F α
i (s|{ωα}i−1)
1− F α
i (s−|{ωα}i−1)
< 1.
lim
t→aα
i+1
γi,α({ωα}i−1, t) >
fα
2
fα
1
lim
t→aα
i+1
∫
[aα
i
, t]
F α
i (dτ |{ωα}i−1)
1− F α
i (τ−|{ωα}i−1)
= ∞,
lim
t→aα
i
γi,α({ωα}i−1, t) = 0, {ωα}i−1 ∈ Ωi−1, α ∈ X0.
(c) is evident from (b) and boundedness of ψα
i ({ωα}i).
At last
∫
[aα
i
,aα
i+1)
|fα(g
α
i ({ωα}i−1, t))| exp{−γ
i,α({ωα}i−1, t−)}γ
i,α({ωα}i−1, dt) 6
6 e(βα
i f
α
1 + fα(0)).
The theorem 4 is proved.
483
N.S.Gonchar
5. Options and their pricing
We assume that {Ω,F , P} is a full probability space, generated by the family of
distribution functions F α
i (ω
α
i |{ωα}i−1), i = 1, k(α), α ∈ X0 and a measure dµ(α)
on the σ-algebra Σ. Further on we assume that X0 is a space of possible hypothesis
each of which may occur with probability µ(α), that is, an evolution of stock price
can come by one of the possible scenario. This scenario is determined by sequence
α and a probability space {Ωα,F
0
α, Pα}.
Theorem 5. Let φ({α, ωα}) = φα({ωα}k(α)) be a random value on the probability
space {Ω,F , P}, satisfying conditions:
1) |φα({ωα}k(α))| 6 Cα <∞, α ∈ X0,
∫
X0
Cαdµ(α) <∞;
2) there exists tαi ∈ [aαi , a
α
i+1) such that
|φα({ωα}i−1, s1, {ωα}[i+1,k(α)])− φα({ωα}i−1, s2, {ωα}[i+1,k(α)])| 6
6 Cα
i |F
α
i (s1|{ωα}i−1)− F α
i (s2|{ωα}i−1)|
εiα, εiα > 0, s1, s2 ∈ [tαi , a
α
i+1),
Cα
i <∞, i = 1, k(α), {ωα}i−1 ∈ Ωi−1, {ωα}[i+1,k(α)] ∈ Ωk(α)−i, α ∈ X0.
Further, let ξ0t ({α, ωα}) be a local non-singular martingale on {Ω,F , P},
ξ0t ({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)ξi,αt ({ωα}i),
ξ
i,α
t ({ωα}i}) = gαi ({ωα}i)χ[aα
i
, t](ω
α
i )
+
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)χ(t, aα
i+1)
(ωα
i ),
satisfying conditions
sup
{ωα}i∈Ωi
|gαi ({ωα}i)| = βα
i <∞, i = 1, k(α), α ∈ X0,
∫
[aα
i
, aα
i+1)
|ρ0,αi (s|{ωα}i−1)|
F α
i (ds|{ωα}i−1)
1− F α
i (s−|{ωα}i−1)
<∞, {ωα}i−1 ∈ Ωi−1, α ∈ X0,
ρ
0,α
i (s|{ωα}i−1) =
= gαi ({ωα}i−1, s)−
1
1− F α
i (s|{ωα}i−1)
∫
(s, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1),
If the random process has got the form
ξt({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)fα(ξ
i,α
t ({ωα}i)),
484
Mathematical model of a stock market
where a family of functions fα(x) > 0, x ∈ R1, α ∈ ∈ X0, is such that each of the
functions fα(x) is strictly fulfillment, sup
x∈R1
|f ′
α(x)| = fα
1 <∞, inf
x∈R1
|f ′
α(x)| = fα
2 > 0,
and
sup
i
sup
{ωα}i−1∈Ωi−1
sup
s∈[aα
i
,aα
i+1)
∆F α
i (s|{ωα}i−1)
1− F α
i (s−|{ωα}i−1)
<
fα
2
fα
1
,
then there exists a measure P1 on {Ω,F0}, generated by a certain family of dis-
tribution functions F α,1
i (ωα
i |{ωα}i−1), i = 1, k(α), α ∈ X0, a probability measure
dµ1(α) on the σ-algebra Σ and a modification ξ̄t({α, ωα}) of the the random process
ξt({α, ωα}) such that ξ̄t({α, ωα}) is a local non-singular martingale on the probability
space {Ω,F1, P1} with respect to the flow of the σ-algebras F1
t , where the σ-algebras
F1 and F1
t are the completion of the σ-algebras F0 and F0
t correspondingly with
respect to the measure P1. Moreover, for the regular martingale M1{φ({α, ωα})|F
1
t }
on the probability space {Ω,F1, P1} the representation
M1{φ({α, ωα})|Ft} =M1
αφ
α({ωα}k(α)) +
∫
[a,t]
ψk(α)(s|ωα)dξ̄s({α, ωα}), t ∈ [a, b)
is valid, where
ψk(α)(s|ωα) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(s)
ϕα
i (s|{ωα}i−1)
ϕ
0,α
i (s|{ωα}i−1)
,
ϕ
0,α
i (s|{ωα}i−1) =
= fα(g
α
i ({ωα}i−1, s))−
1
1− F
α,1
i (s|{ωα}i−1)
∫
(s, aα
i+1)
fα(g
α
i ({ωα}i))F
α,1
i (dωα
i |{ωα}i−1),
ϕα
i (s|{ωα}i−1) =
= φα
i ({ωα}i−1, s)−
1
1− F
α,1
i (s|{ωα}i−1)
∫
(s, aα
i+1)
φα
i ({ωα}i)F
α,1
i (dωα
i |{ωα}i−1),
φα
i ({ωα}i) =
∫
Ωi+1
. . .
∫
Ωk(α)
φα({ωα}i, {ωα}[i+1,k(α)])
×F α,1
i+1(dω
α
i+1|{ωα}i)× . . .× F
α,1
k(α)(dω
α
k(α)|{ωα}k(α)−1).
Proof. The conditions of the theorem 5 guarantee the monotonous of the conditions
of the theorem 4. Therefore the random process ξt({α, ωα}) belongs to the subclass
K0. Moreover,
ψα
i ({ωα}i−1, a
α
i ) = fα(ξ
i,α
aα
i
({ωα}i)) = fα
∫
(aα
i
, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)
=
485
N.S.Gonchar
= fα(g
α
i−1({ωα}i−1)) = fα
i−1({ωα}i−1)
with probability 1 on the probability space {Ω,F , P}, since
gαi−1({ωα}i−1) =
∫
(aα
i
, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)
with probability 1. Further,
∫
X0
fα(v(α))dµ1(α) <∞,
where dµ1(α) = u(α)dµ(α),
v(α) =
∫
(aα
i
, aα
i+1)
gα1 (ω
α
1 )F
α
1 (dω
α
1 ),
u(α) =
χA(α) + [fα(v(α))]
−1χX0\A(α)
D
,
D =
∫
X0
{χA(α) + [fα(v(α))]
−1χX0\A(α)}dµ(α),
χA(α) is the characteristic function of the set A = {α, fα(v(α)) 6 1}. Based on
the lemma 7 there exists a measure P1 on the σ-algebra F 0, generated by a certain
family of distribution functions F α,1
i (ωα
i |{ωα}i−1), i = 1, k(α), α ∈ X0 and the
measure dµ1(α) on σ-algebra Σ such that the random process
ξ̄t({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)ξ̄i,αt ({ωα}i),
ξ̄
i,α
t ({ωα}i) = fα(g
α
i ({ωα}i))χ[aα
i
, t](ω
α
i )
+
1
1− F
α,1
i (t|{ωα}i−1)
∫
(t, aα
i+1)
fα(g
α
i ({ωα}i))F
α,1
i (dωα
i |{ωα}i−1)χ(t, aα
i+1)
(ωα
i ),
i = 1, k(α), α ∈ X0,
is a modification of the random process ξt({α, ωα}) on the probability space
{Ω,F1, P1}.
Since ξ0t ({α, ωα}) is also a local non-singular martingale, then ξ̄t({α, ωα}) is also
a local non-singular martingale because
{t ∈ [a, b), ξ̄i,αt,a ({ωα}i−1) = 0, i = 1, k(α)} ⊆
⊆ {t ∈ [a, b), ξi,αt,a ({ωα}i−1) = 0, i = 1, k(α)},
486
Mathematical model of a stock market
due to the strict monotony of fα(x), where
ξ̄
i,α
t,a ({ωα}i−1) =
= fα(g
α
i ({ωα}i))−
1
1− F
α,1
i (t|{ωα}i−1)
∫
(t, aα
i+1)
fα(g
α
i ({ωα}i))F
α,1
i (dωα
i |{ωα}i−1)
= fα(g
α
i ({ωα}i))− fα(T
α
i (t|{ωα}i−1)),
T α
i (t|{ωα}i−1) =
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1),
ξ
i,α
t,a ({ωα}i−1) = gαi ({ωα}i)− T α
i (t|{ωα}i−1).
To finish the proof of the theorem 5 it is sufficient to verify the monotonous of the
conditions of the theorem 3. Really,
∫
[aα
i
, aα
i+1)
|ϕ0,α
i (s|{ωα}i−1)|
F
α,1
i (ds|{ωα}i−1)
1− F
α,1
i (s−|{ωα}i−1)
6
6
[fα
1 ]
2
fα
2
∫
[aα
i
, aα
i+1)
|ρ0,αi (s|{ωα}i−1)|
F α
i (ds|{ωα}i−1)
1− F α
i (s−|{ωα}i−1)
<∞,
{ωα}i−1 ∈ Ωi−1, α ∈ X0.
∫
[aα
i
, aα
i+1)
|ϕα
i (s|{ωα}i−1)|
F
α,1
i (ds|{ωα}i−1)
1− F
α,1
i (s−|{ωα}i−1)
6
6 2Cα
∫
[aα
i
, tα
i
]
F
α,1
i (ds|{ωα}i−1)
1− F
α,1
i (s−|{ωα}i−1)
+Cα
i
∫
[tα
i
, aα
i+1)
(1− F α
i (s|{ωα}i−1))
εiα
F
α,1
i (ds|{ωα}i−1)
1− F
α,1
i (s−|{ωα}i−1)
<∞,
because the first integral is finite and the second integral is finite since
γi,α({ωα}i−1, t) =
∫
[aα
i
, t]
ψα
i ({ωα}i−1, dτ)
ψα
i ({ωα}i−1, τ)− fα(gαi ({ωα}i−1, τ))
=
∫
[aα
i
, t]
F
α,1
i (dτ |{ωα}i−1)
1− F
α,1
i (τ−|{ωα}i−1)
=
∫
[aα
i
, t]
f
′
α(T
α
i (τ |{ωα}i−1))F
α
i (dτ |{ωα}i−1)
U({ωα}i−1, τ)[1− F α
i (τ−|{ωα}i−1)]
.
487
N.S.Gonchar
Therefore
∫
[tα
i
, aα
i+1)
(1− F α
i (s|{ωα}i−1))
εiα
F
α,1
i (ds|{ωα}i−1)
1− F
α,1
i (s−|{ωα}i−1)
6
fα
1
εiαf
α
2
.
The theorem 5 is proved. Then we assume that interval [a, b) coincides with the
interval [0, T ), that is a = 0, b = T. The time T is the terminal time of monotonous
of the option.
Definition 5. A stock market is effective on the time interval [0, T ), if there is a cer-
tain probability space {Ω,F , P}, constructed above, a random process ξ0t ({α, ωα}) on
it, describing the evolution of the average price of stocks such that ξ0t ({α, ωα})e
−rt
is a non-negative uniformly integrable and non-singular martingale on {Ω,F , P}
with respect to the flow of the σ-algebras Ft, where the σ-algebras F and Ft are the
completion of the σ-algebras F0 and F0
t with respect to the measure P on F0, gen-
erated by the family of distribution functions F α
i (ω
α
i |{ωα}i−1). The random process
ξ0t ({α, ωα}) has the form
ξ0t ({α, ωα}) = B0e
rt
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)ξi,αt ({ωα}i), (23)
ξ
i,α
t ({ωα}i}) = gαi ({ωα}i)χ[aα
i
, t](ω
α
i )
+
1
1− F α
i (t|{ωα}i−1)
∫
(t, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1)χ(t, aα
i+1)
(ωα
i ), (24)
where r is an interest rate, the evolution of price of a stock being described by a
certain random process
St({α, ωα}) =
S0e
rt
Vα
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)fα(ξ
i,α
t ({ωα}i)),
Vα = fα(Mαφ
α({ωα}k(α))) (25)
for a certain family of functions fα(x) > 0, x ∈ R1, α ∈ X0, which are strictly
fulfillment, sup
x∈R1
|f ′
α(x)| = fα
1 <∞, inf
x∈R1
|f ′
α(x)| = fα
2 > 0, moreover,
sup
i
sup
{ωα}i−1∈Ωi−1
sup
s∈[aα
i
,aα
i+1)
∆F α
i (s|{ωα}i−1)
1− F α
i (s−|{ωα}i−1)
<
fα
2
fα
1
. (26)
The limit
φα({ωα}k(α))) = lim
t→T
ξ0t ({α, ωα})B
−1
0 e−rt
satisfies the conditions:
1) |φα({ωα}k(α))| 6 Cα <∞, α ∈ X0,
∫
X0
Cαdµ(α) <∞;
2) there exists tαi ∈ [aαi , a
α
i+1) such that
|φα({ωα}i−1, s1, {ωα}[i+1,k(α)])− φα({ωα}i−1, s2, {ωα}[i+1,k(α)])| 6
488
Mathematical model of a stock market
6 Cα
i |F
α
i (s1|{ωα}i−1)− F α
i (s2|{ωα}i−1)|
εiα, εiα > 0, s1, s2 ∈ [tαi , a
α
i+1),
Cα
i <∞, i = 1, k(α), {ωα}i−1 ∈ Ωi−1, {ωα}[i+1,k(α)]) ∈ Ωk(α)−i, α ∈ X0.
Let us consider an economic agent on the stock market, who acts an as investor,
that is, he or she wants to multiply his or her capital using the possibilities of the
stock market. We assume that the stock market is effective and the evolution of a
stock price occurs according to the formula (25). We assume that the evolution of
non-risky active price occurs according to the law
B(t) = B0e
rt, (27)
where r is an interest rate, B0 is an initial capital of the investor on a deposit.
Definition 6. A stochastic process δt({α, ωα}) belongs to the class A0, if
δt({α, ωα}) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)δi,αt ({ωα}i),
δ
i,α
t ({ωα}i) = b
α,1
i ({ωα}i, t)χ[aα
i
, t](ω
α
i ) + b
α,2
i ({ωα}i−1, t)χ(t, aα
i+1)
(ωα
i ),
b
α,1
i ({ωα}i, t) is a measurable mapping from the measurable space {Ωi, F̄0
i } to the
measurable space {R1,B(R1)} at every fixed t from the interval [0, T ), bα,2i ({ωα}i−1, t)
is a measurable mapping from the measurable space {Ωi−1, F̄0
i−1} to the measurable
space {R1,B(R1)} at every fixed t ∈ [0, T ). Moreover, bα,1i ({ωα}i, t) is a bounded
measurable mapping from the measurable space {[0, T ),B([0, T ))} to the measurable
space {R1,B(R1)} at every fixed {ωα}i ∈ Ωi, b
α,2
i ({ωα}i−1, t) is a bounded measurable
mapping from {[0, T ),B([0, T ))} to {R1,B(R1)} at every fixed {ωα}i−1 ∈ Ωi−1.
Let the capital of an investor Xt({α, ωα}) at time t equal
Xt({α, ωα}) = B(t)βt({α, ωα}) + γt({α, ωα})St({α, ωα}), (28)
where the stochastic processes βt({α, ωα}) and γt({α, ωα}) belong to the class A0.
The pair πt = {βt({α, ωα}), γt({α, ωα})} is called the financial strategy of the in-
vestor. The capital of the investor with the financial strategy πt will be denoted by
Xπ
t ({α, ωα}).
Definition 7. A financial strategy πt = {βt({α, ωα}), γt({α, ωα})} of an investor is
called self-financing if the random processes βt({α, ωα}) and γt({α, ωα}) belong to
the class A0, for the investor capital Xπ
t ({α, ωα}) the representation
Xπ
t ({α, ωα}) = Xπ
0 (α) +
∫
[0,t]
βτ ({α, ωα})dB(τ) +
∫
[0,t]
γτ({α, ωα})dSτ ({α, ωα}) (29)
is valid, the discounted capital
Y π
t ({α, ωα}) =
Xπ
t ({α, ωα})
B(t)
489
N.S.Gonchar
belongs to the class of local martingale on the probability space {Ω,F1, P1} with
respect to the flow of the σ-algebras F1
t , M
1|Xπ
t ({α, ωα})| < ∞, where F1, P1 and
F1
t are constructed in the theorem 5.
A class of self-financing strategy is denoted by SF.
Lemma 10. Let a financial strategy πt = {βt({α, ωα}), γt({α, ωα})} be self-finan-
cing, then for the investor capital the representations
Xπ
t ({α, ωα}) = Xπ
0 (α) +
∫
[0,t]
βτ ({α, ωα})dB(τ) +
∫
[0,t]
γτ ({α, ωα})dSτ ({α, ωα}) (30)
Xπ
t ({α, ωα}) = ertXπ
0 (α) +B0e
rt
∫
[0,t]
γτ ({α, ωα})dS
0
τ ({α, ωα}) (31)
are equivalent, where
S0
t ({α, ωα}) =
S0
B0Vα
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)fα(ξ
i,α
t ({ωα}i)). (32)
Proof. Since Xπ
t ({α, ωα}) is a process of a bounded variation on any interval [0, t],
therefore from (31) and lemma 1
Xπ
t ({α, ωα}) = Xπ
0 (α) +
∫
[0,t]
Xα
0 +B0
∫
[0,t]
γτ ({α, ωα})dS
0
τ ({α, ωα})
dert
+B0
∫
[0,t]
erτγτ ({α, ωα})dS
0
τ ({α, ωα})
= Xπ
0 (α) +
∫
[0,t]
Xτ ({α, ωα})
dB(τ)
B(τ)
+
∫
[0,t]
B(τ)γτ ({α, ωα})dS
0
τ ({α, ωα}).
Since
St({α, ωα}) = B(t)S0
t ({α, ωα}),
dSt({α, ωα}) = S0
t ({α, ωα})dB(t) +B(t)dS0
t ({α, ωα}), (33)
therefore, taking into account (28) and (33), we obtain
Xπ
t ({α, ωα}) = Xπ
0 (α) +
∫
[0,t]
βτ ({α, ωα})dB(τ) +
∫
[0,t]
dB(τ)
B(τ)
γτ({α, ωα})Sτ ({α, ωα})
+
∫
[0,t]
γτ ({α, ωα})dSτ ({α, ωα})−
∫
[0,t]
γτ ({α, ωα})S
0
τ ({α, ωα})dB(τ)
490
Mathematical model of a stock market
= Xπ
0 (α) +
∫
[0,t]
βτ ({α, ωα})dB(τ) +
∫
[0,t]
γτ ({α, ωα})dSτ ({α, ωα}).
This proves the lemma 10 in one direction. Applying the same argument in the
inverse direction we obtain the proof of the lemma 10.
Denote by SFR a set of self-financing strategies satisfying the conditions
M1{Y π
t ({α, ωα})|F
1
t } > −M1{R|F1
t }, M1R <∞,
where R is a non-negative random value on {Ω,F1, P1}.
Lemma 11. Let πt = {βt({α, ωα}), γt({α, ωα})} be a self-financing strategy, that
is, πt ∈ SFR, then {Y π
t ,F
1
t , t ∈ [0, T ]} is a supermartingale and for any stop time
τ1 and τ2 such that P1(τ1 6 τ2) = 1 the inequality
M1{Y π
τ2
({α, ωα})|F
1
τ1
} 6 Y π
τ1
({α, ωα})
is valid.
The proof is similar to the proof of the analogous lemma in [2].
Corollary 3. If πt ∈ SFR, then for any stop time τ > 0, P1(τ <∞) = 1
M1Y π
τ ({α, ωα}) 6 Y π
0 (α) =
Xπ
0 (α)
B0
.
Definition 8. A self-financing strategy πt is an arbitrage strategy on [0, T ], if from
that
Xπ
0 (α) 6 0, Xπ
T ({α, ωα}) > 0
it follows that Xπ
T ({α, ωα}) > 0 with a positive probability.
Lemma 12. Any strategy πt ∈ SFR, where R is non-negative and integrable random
value on probability space, is not arbitrage strategy.
The proof of the lemma is analogous to the proof of the similar lemma in [2]. Let
φT = φT ({α, ωα}) = φα
T ({ωα}k(α)) be F
0 measurable random value on the probability
space {Ω,F0, P}.
Definition 9. A self-financing strategy πt ∈ SFR is (xα, φT )-hedge for the European
type option if the capital Xπ
t ({α, ωα}), corresponding to this strategy is such that
Xπ
0 (α) = xα and with probability 1 with respect to the measure P1
Xπ
T ({α, ωα}) > φT ({α, ωα}).
(xα, φT )-hedge π
∗
t ∈ SFR is called minimal if for any (xα, φT )-hedge πt ∈ SFR the
inequality
Xπ
T ({α, ωα}) > Xπ∗
T ({α, ωα})
is valid.
491
N.S.Gonchar
Then we consider self-financing strategies, belonging to SF 0, that is, in this case
Xπ
t ({α, ωα}) > 0.
Definition 10. Let HT (x
α, φT ) be the set of (xα, φT )-hedges from SF 0. Investment
value is called the value
Cα
T (φT ) = inf{xα > 0, HT (x
α, φT ) 6= ∅}, α ∈ X0,
where ∅ is the empty set.
The main problem is to calculate C α
T (φT ) and to find an expression for the portfolio
of an investor π∗
t at every moment of time t the initial capital of which is xα.
Further on we assume that T <∞, then
lim
t→T
St({α, ωα}) = ST ({α, ωα}) =
S0e
rT
Vα
fα(φ({α, ωα})).
Theorem 6. Let a stock market be effective, the evolution of a risky active price
comes according to the formula (25) and the evolution of non-risky active price occur
by (27). If f(x) is a certain function such that |f(x1)− f(x2)| 6 C|x1 − x2| and the
paying function at terminal time T is given by the formula
fT ({α, ωα}) = f(ST ({α, ωα})),
moreover, the conditions
∫
X0
fα
1 Cα
Vα
dµ(α) <∞,
∫
X0
fα(0)
Vα
dµ(α) <∞,
are valid, then the minimal hedge π∗
t exists, evolution of the capital investor
X∗
t ({α, ωα}), option price X∗
0 (α) and self-financial strategy {β∗
t ({α, ωα}),
γ∗t ({α, ωα})} corresponding to the minimal hedge π∗
t are given by the formulas
X∗
t ({α, ωα}) = er(t−T )M1{f(ST ({α, ωα}))|F
1
t }, (34)
X∗
0 (α) = e−rTM1
αf(ST ({α, ωα})), γ∗t ({α, ωα}) = ψk(α)(t|{ω}α), (35)
β∗
t ({α, ωα}) =
X∗
t ({α, ωα})− γ∗t ({α, ωα})St({α, ωα})
B(t)
, (36)
where
ψk(α)(s|ωα) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(s)
ϕα
i (s|{ωα}i−1)
ϕ
0,α
i (s|{ωα}i−1)
,
ϕ
0,α
i (s|{ωα}i−1) =
= fα(g
α
i ({ωα}i−1, s))
−
1
1 − F
α,1
i (s|{ωα}i−1)
∫
(s, aα
i+1)
fα(g
α
i ({ωα}i))F
α,1
i (dωα
i |{ωα}i−1),
492
Mathematical model of a stock market
ϕα
i (s|{ωα}i−1) =
= φ̄α
i ({ωα}i−1, s)−
1
1− F
α,1
i (s|{ωα}i−1)
∫
(s, aα
i+1)
φ̄α
i ({ωα}i)F
α,1
i (dωα
i |{ωα}i−1),
φ̄α
i ({ωα}i) =
1
B0erT
∫
Ωi+1
. . .
∫
Ωk(α)
f(S0e
rTVα
−1fα(φ
α({ωα}i, {ωα}[i+1,k(α)])))
×F α,1
i+1(dω
α
i+1|{ωα}i)× . . .× F
α,1
k(α)(dω
α
k(α)|{ωα}k(α)−1).
Proof. To prove the theorem 6 it is sufficient to verify the monotonous of the
conditions of the theorem 5. Since
ST ({α, ωα}) =
S0e
rT
Vα
fα(φ({α, ωα})),
then
f(ST ({α, ωα}))
B0erT
6
6
1
B0erT
[
f(0) + CS0e
rT fα(0)
Vα
+ CS0e
rT f
α
1 Cα
Vα
]
= C
′
α,
∫
X0
C
′
αdµ(α) <∞.
|f(fα(φ
α({ωα}i−1, s1, {ωα}[i+1,k(α)])))− f(fα(φ
α({ωα}i−1, s2, {ωα}[i+1,k(α)])))| 6
6 Cfα
1 C
α
i |F
α
i (s1|{ωα}i−1)− F α
i (s2|{ωα}i−1)|
εiα, εiα > 0, s1, s2 ∈ [tαi , a
α
i+1),
i = 1, k(α), {ωα}i−1 ∈ Ωi−1, {ωα}[i+1,k(α)]) ∈ Ωk(α)−i, α ∈ X0.
Further, ξ0t ({α, ωα})e
rT is a non-negative martingale on {Ω,F , P} satisfying condi-
tions:
sup
{ωα}i∈Ωi
|gαi ({ωα}i)| = βα
i 6 Cα <∞, i = 1, k(α), α ∈ X0,
moreover, since
gαi ({ωα}i) =
∫
Ωi+1
. . .
∫
Ωk(α)
φα({ωα}i, {ωα}[i+1,k(α)])
×F α
i+1(dω
α
i+1|{ωα}i)× . . .× F α
k(α)(dω
α
k(α)|{ωα}k(α)−1).
|φα({ωα}i−1, s1, {ωα}[i+1,k(α)])− φα({ωα}i−1, s2, {ωα}[i+1,k(α)])| 6
6 Cα
i |F
α
i (s1|{ωα}i−1)− F α
i (s2|{ωα}i−1)|
εiα, εiα > 0, s1, s2 ∈ [tαi , a
α
i+1),
Cα
i <∞, i = 1, k(α), {ωα}i−1 ∈ Ωi−1, {ωα}[i+1,k(α)] ∈ Ωk(α)−i, α ∈ X0,
493
N.S.Gonchar
therefore
∫
[aα
i
, aα
i+1)
|ρ0,αi (s|{ωα}i−1)|
F α
i (ds|{ωα}i−1)
1− F α
i (s−|{ωα}i−1)
<∞, {ωα}i−1 ∈ Ωi−1, α ∈ X0,
ρ
0,α
i (s|{ωα}i−1) =
= gαi ({ωα}i−1, s)−
1
1− F α
i (s|{ωα}i−1)
∫
(s, aα
i+1)
gαi ({ωα}i)F
α
i (dω
α
i |{ωα}i−1).
Hence it follows that for the regular martingale
M1
{
f(ST ({α, ωα}))
B0erT
∣
∣
∣
∣
F1
t
}
the representation
M1
{
f(ST ({α, ωα}))
B0erT
∣
∣
∣
∣
F1
t
}
=
=M1
α
f(ST ({α, ωα}))
B0erT
+
∫
[a,t]
ψk(α)(τ |ωα)dS̄
0
τ ({α, ωα}), t ∈ [a, b)
is valid, where
ψk(α)(s|ωα) =
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(s)
ϕα
i (s|{ωα}i−1)
ϕ
0,α
i (s|{ωα}i−1)
,
ϕ
0,α
i (s|{ωα}i−1) =
= fα(g
α
i ({ωα}i−1, s))
−
1
1− F
α,1
i (s|{ωα}i−1)
∫
(s, aα
i+1)
fα(g
α
i ({ωα}i))F
α,1
i (dωα
i |{ωα}i−1),
ϕα
i (s|{ωα}i−1) =
= φ̄α
i ({ωα}i−1, s)−
1
1− F
α,1
i (s|{ωα}i−1)
∫
(s, aα
i+1)
φ̄α
i ({ωα}i)F
α,1
i (dωα
i |{ωα}i−1),
φ̄α
i ({ωα}i) =
1
B0erT
∫
Ωi+1
. . .
∫
Ωk(α)
f(S0e
rTVα
−1fα(φ
α({ωα}i, {ωα}[i+1,k(α)])))
×F α,1
i+1(dω
α
i+1|{ωα}i)× . . .× F
α,1
k(α)(dω
α
k(α)|{ωα}k(α)−1).
S̄0
t ({α, ωα}) is a modification of
S0
t ({α, ωα}) =
S0
B0Vα
k(α)
∑
i=1
χ[aα
i
, aα
i+1)
(t)fα(ξ
i,α
t ({ωα}i)).
494
Mathematical model of a stock market
such that S̄0
t ({α, ωα}) is a regular martingale on the probability space {Ω,F1, P1},
where F1 is the completion of F 0 with respect to the measure P1, generated by the
family of distributions F α,1
i (ωα
i |{ωα}i−1), i = 1, k(α), α ∈ X0 and
fα(ξ
i,α
t ({ωα}i})) = fα(g
α
i ({ωα}i))χ[aα
i
, t](ω
α
i )
+
1
1− F
α,1
i (t|{ωα}i−1)
∫
(t, aα
i+1)
fα(g
α
i ({ωα}i))F
α,1
i (dωα
i |{ωα}i−1)χ(t, aα
i+1)
(ωα
i ),
i = 1, k(α), α ∈ X0.
The latter means that for the discounted capital
Yt({α, ωα}) =M1
{
f(ST ({α, ωα}))
B0erT
|F1
t
}
the representation
Yt({α, ωα}) =
=M1
α
f(ST ({α, ωα}))
B0erT
+
∫
[a,t]
ψk(α)(τ |ωα)dS̄
0
τ ({α, ωα}), t ∈ [a, b)
is valid. Since
Xt({α, ωα}) = B0e
rtYt({α, ωα}),
then
Xt({α, ωα}) = erte−rTM1
αf(ST ({α, ωα}))
+B0e
rt
∫
[a,t]
ψk(α)(τ |ωα)dS
0
τ ({α, ωα}), t ∈ [a, b). (37)
Taking into account the lemma 10, the definition of self-financing strategy, we obtain
the proof of the theorem 6.
References
1. Neveu J. Bases Mathematiques du Calcul des Probabilites. Paris, Masson et Cie, 1964.
2. Gonchar N.S. Financial Mathematics, Economic Growth. Kyiv, Rada, 2000 (in Rus-
sian).
495
N.S.Gonchar
Математична модель фондового ринку
М.С. Гончар
Інститут теоретичної фізики ім. М.М.Боголюбова НАН України,
252143 Київ, вул. Метрологічна, 14б
Отримано 30 травня 2000 р.
В роботі побудовано математичну модель ринку цінних паперів. От-
римані результати є доброю основою для аналізу подій на фондово-
му ринку.
Ключові слова: випадковий процес, ефективний ринок цінних
паперів, оцінювання опціонів
PACS: 02.50.+s, 05.40.+j
496
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