Integral Option

Kramkov D.O.

Steklov Mathematical Institute, Moscow, Russia,

Mordecki E.

Universidad de la República. Montevideo, Uruguay. ^{∗}

October 5, 1993

Abstract

In the framework of the Black and Sholes model of the security market we present the closed form solution for the American option with the “integral type” reward f = (f_{t})_{t≥0} :

f_{t} = e^{−λt}[ ∫_{0}^{t}S_{u} du + sψ_{0}],

where S = (S_{t})_{t≥0} is the price process of a stock, s = S_{0}, and λ > 0, ψ_{0} ≥ 0 are given constants .

Keywords : Black and Sholes model, American option, Asian option, optimal stopping problem, Kummer’s functions.

1 Introduction. Main results

1. We consider the Black and Sholes model of the security market, consisting of two assets: bonds (or bank accounts) and stocks (see [2], [11], [12], [13], [16], [17]). The price process of the bonds B = (B_{t})_{t≥0} represents the time value of money and appreciates at a constant rate r ≥ 0, the interest rate:

B_{t} = B_{0}e^{rt},  B_{0} > 0.  (1)

∗The paper was written in Steklov Mathematical Institute

The price process of the stocks S = (S_{t})_{t≥0} has random character and is modeled as a geometric Brownian motion, defined on a filtered probability space (Ω, F, (F_{t})_{t≥0}, P) with constant drift μ ∈ R, the appreciation rate and constant variance σ > 0, the volatility coefficient:

S_{t} = S_{0}e^{μt} · e^{σW_{t}− \frac{σ^{2}}{2} t}.  (2)

Here W = (W_{t})_{t≥0} is a standard Brownian motion and we suppose that the filtration (F_{t})_{t≥0} is generated by W.

Let P_{t} = P|F_{t} be the restriction of the measure P on F_{t}, and P^{μ−r} be the probability on (Ω, F) such that

P_{t}^{μ−r}(dω) = Z_{t}^{μ−r}(ω)P_{t}(dω),  (3)

where P_{t}^{μ−r} = P^{μ−r}|F_{t} and

Z_{t}^{μ−r}(ω) = exp \{ − \frac{μ − r}{σ}W_{t}(ω) − \frac{1}{2}(\frac{μ − r}{σ} )^{2}t\}  (4)

(see [13]).

By Girsanov’s Theorem the process W^{μ−r} = (W_{t}^{μ−r})_{t≥0} with

W_{t}^{μ−r} = W_{t} + \frac{μ − r}{σ}t  (5)

is a Wiener process with respect to the measure P^{μ−r}:

Law (W^{μ−r}|P^{μ−r}) = Law (W |P)  (6)

Remark that from (2) and Ito’s formula

dS_{t} = S_{t}(μdt + σdW_{t}).  (7)

and by (5) we can also rewrite that as

dS_{t} = S_{t}(rdt + σdW_{t}^{μ−r}).  (8)

2. Let f = (f_{t}(ω))_{t≥0} be a nonnegative progressively measurable process with the function f_{t} = f_{t}(ω) interpreted as a payment of an option seller to an option buyer if the option is exercised at time t (see [11], [12], [13], [16], [17]). We consider the American type option which can be exercised at arbitrary stopping time τ = τ(ω). According to the general pricing theory for the American options the fair price C^{∗} or premium, which the option buyer pays to the seller at time t = 0 is given by the formula

C^{∗} = B_{0} sup E^{μ−r} \frac{f_{τ}}{B_{τ}} ,  (9)

where sup is taking over the set of all finite (P^{μ−r}-a.s.) stopping times τ = τ(ω) and E^{μ−r} is the expectation with respect to the probability P^{μ−r}.

Remark that from (6), (8), (9) it follows that the fair option price does not depend on the appreciation rate μ. So without the loss of generality we can assume that μ = r.

In this paper we suppose that the reward process f has the form

f_{t} = e^{−λt}[ ∫_{0}^{t}S_{u} du + sψ_{0}],  (10)

where s = S_{0} and ψ_{0} ≥ 0, λ > 0 are some constants.

We baptized this option as “Integral Option” taking into account the formula (10) for the reward process. As a matter of fact it is a particular case of Asian options. This option may be looking attractive for “careful” investor because it keeps track of past events in a smooth “integral” way. As a consequence it provides its holder a psychological comfort by reducing regrets. By appearance it is similar to the “Russian Option” introduced by Shepp and Shiryaev ([4], see also [18]); that is also inscribed in the American type and has a reward process

f_{t} = e^{−λt} max(sup_{u≤t} S_{u}, sψ_{0}).

So the problem of the calculation of C^{∗} can be formulated as an optimal stopping problem for the process S = (S_{t})_{t≥0} satisfying to the stochastic differential equation

dS_{t} = S_{t}(rdt + σdW_{t}), S_{0} = s,

with price

C^{∗} = B_{0} sup_{τ} E \frac{f_{τ}}{B_{τ}} ,  (11)

where f = (f_{t})_{t≥0} is given in (10) and E is the expectation with respect to the measure P.

From (10) and (11) it follows that

C^{∗} = sup_{τ} Ee^{−(λ+r)τ}g( ∫_{0}^{τ}S_{u} du),  (12)

where

g( ∫_{0}^{t}S_{u} du) = ∫_{0}^{t}S_{u} du + sψ_{0}.  (13)

The two-dimensional process

(S_{t}, ∫_{0}^{t}S_{u} du)_{t≥0}

is a Markov process with respect to the measure P. So the “problem (12)” belongs to the set of optimal stopping problems for two-dimensional Markov processes and in principle can be solved using general methods from [14].

3. It turns out, however, that the problem (12) can be reduced to some optimal stopping problem for a one-dimensional Markov process. The idea of such reduction is based on the introduction of a dual martingale measure (see §7 in [13]) and consists in the following.

Let us introduce the probability measures

P̃_{t}(dω) = e^{σW_{t}(ω)− \frac{σ^{2}}{2} t}P_{t}(dω), t ≥ 0.  (14)

The family of measures (P̃_{t})_{t≥0} is consistent and there is a measure P̃ such that P̃|F_{t} = P̃_{t}, t ≥ 0 (compare with §7 in [13]). It can be shown that measures P̃ and P are locally equivalent (i.e. P̃_{t} ∼ P_{t}, t ≥ 0), moreover if τ is a finite (P̃ and P a.s.) stopping time then the restrictions P̃_{τ} = P̃|F_{τ} and P_{τ} = P|F_{τ} of measures P̃ and P on σ-field F_{τ} are equivalent and

P̃_{τ}(dω) = e^{σW_{τ} − \frac{σ^{2}}{2} τ}P_{τ}(dω)  (15)

(the definition of σ-field F_{τ} can be found for example in [7]).

Remark now that since

S_{t} = S_{0}e^{rt} · e^{σW_{t}− \frac{σ^{2}}{2} t},

then

\frac{S_{t}}{B_{t}} = \frac{S_{0}}{B_{0}} e^{σW_{t}− \frac{σ^{2}}{2} t}.  (16)

Therefore from (15) we have for finite τ

B_{0}E \frac{f_{τ}}{B_{τ}} = S_{0}E (\frac{B_{0}}{B_{τ}} · \frac{S_{τ}}{S_{0}} · \frac{f_{τ}}{S_{τ}} ) = S_{0}E (e^{σW_{τ} − \frac{σ^{2}}{2} τ} \frac{f_{τ}}{S_{τ}} ) = S_{0}Ẽ \frac{f_{τ}}{S_{τ}} .  (17)

So from (9) it follows that the fair price C^{∗} can be also calculated as

C^{∗} = S_{0}Ẽ \frac{f_{τ}}{S_{τ}} ,  (18)

where Ẽ is the expectation with respect to the measure P̃.

Let W̃_{t} = W_{t} − σt, t ≥ 0. It is easy to see that the process W̃ = (W̃_{t})_{t≥0} is a Wiener process with respect to P̃:

Law (W̃ |P̃) = Law (W |P).

So with respect to P̃ the process S = (S_{t})_{t≥0} with S_{t} = S_{0}e^{rt} · e^{σW_{t}− \frac{σ^{2}}{2} t} can be represented in the form

S_{t} = S_{0}e^{(r+ \frac{σ^{2}}{2} )t+σW̃_{t}},  (19)

and by Ito’s formula

dS_{t} = S_{t}[(r + σ^{2})dt + σdW̃_{t}].  (20)

Remark, that according to (16) the process \frac{S}{B} = (\frac{S_{t}}{B_{t}})_{t≥0} is a P-martingale. At the same time

\frac{B_{t}}{S_{t}} = \frac{B_{0}}{S_{0}} e^{−σW̃_{t}− \frac{σ^{2}}{2} t}

and so the process (\frac{B}{S}) = (\frac{B_{t}}{S_{t}})_{t≥0} is a P̃-martingale. That explains the name “dual martingale measure” for P̃.

4. Let us consider the process \frac{f}{S} = (\frac{f_{t}}{S_{t}} )_{t≥0}, with respect to P̃, where f_{t} is given in (10).

If we denote

ψ_{t} = \frac{1}{S_{t}} [ ∫_{0}^{t}S_{u} du + sψ_{0}],  ψ_{0} ≥ 0.  (21)

then

\frac{f_{t}}{S_{t}} = e^{−λt}ψ_{t}  (22)

and therefore the fair price is equaled to

C^{∗} = S_{0} sup_{τ} Ẽe^{−λτ}ψ_{τ}.  (23)

Remark, that with respect to the dual measure P̃ the process ψ = (ψ_{t})_{t≥0} is a Markov one and so (compare with (12)) the problem of the calculation of C^{∗} is the typical optimal stopping problem for the one-dimensional Markov process (ψ_{t})_{t≥0} with

C^{∗} = S_{0} sup_{τ} Ẽe^{−λτ}g(ψ_{τ}),  (24)

where g(ψ) = ψ (compare with the corresponding problems for “Russian” options in [4], [10] (continuous time) and [6] (discrete time)). In accordance with the explanations given in [12], [13] the optimal stopping time τ^{∗}, i.e. the stopping time for which Ẽe^{−λτ^{∗}}g(ψ_{τ^{∗}}) = C^{∗}, will be referred as rational.

The main result of the paper is the following

Theorem For the Integral option of American type with the reward f = (f_{t})_{t≥0} given in (10) the fair price equals

C^{∗} = S_{0} · \{ \begin{matrix}c^{∗}u(ψ_{0}), & 0 ≤ ψ_{0} ≤ ψ^{∗}, \\ ψ_{0}, & ψ_{0} > ψ^{∗}, \end{matrix}  (25)

where

u(ψ) = ∫_{0}^{∞}exp ( − \frac{2y}{σ^{2}} )y^{−(γ_{1}+1)}(1 + ψy)^{γ_{2}} dy,

constants γ_{1} and γ_{2} are defined in (31), ψ^{∗} is the root of the equation

ψu′(ψ) = u(ψ)

and

c^{∗} = \frac{ψ^{∗}}{u(ψ^{∗})}.

The rational stopping time in the problem (24) is

τ^{∗} = inf{t ≥ 0 : ψ_{t} ≥ ψ^{∗}}.  (26)

The proof of the theorem is given in the section 3. In the next section we present some facts about Kummer’s functions. These functions are used for another description of the function u = u(ψ) and the constants ψ^{∗}, c^{∗} in (25).

The idea to look for the closed form solution for the American option with the integral type reward (10) belongs to A.N. Shiryaev whom we would like to express our gratitude for fruitful discussions. We also thank A.A. Afanasiev for useful remarks.

2 Auxiliary results.

1. For the proof of the main result we need the confluent hypergeometric Kummer’s functions M(a, b, z) and U(a, b, z) (see [1], p. 504):

M(a, b, z) = 1 + \frac{az}{b} + \frac{(a)_{2}z^{2}}{(b)_{2}2!} + … + \frac{(a)_{n}z^{n}}{(b)_{n}n!} + …  (27)

with (a)_{n} = a(a + 1) … (a + n − 1), (a)_{0} = 1 and

U(a, b, z) = \frac{π}{sin πb}\{\frac{M(a, b, z)}{Γ(1 + a − b)Γ(b)} − z^{1−b} \frac{M(1 + a − b, 2 − b, z)}{Γ(a)Γ(2 − b)}\}, (28)

where Γ(a) is the Gamma function, a ∈ R.

From the theory of the Kummer’s functions (see [1], 13.1.29 and 13.2.5, p. 505) it is known that if 1 + a − b > 0 and 2 − b > 0 then U(a, b, z) admits the following representation:

U(a, b, z) = \frac{1}{Γ(1 + a − b)} ∫_{0}^{∞}e^{−zt}t^{a−b}(1 + t)^{1−a} dt.  (29)

Let us denote by γ_{1} and γ_{2} (γ_{1} < γ_{2}) the roots of the following quadratic equation

\frac{σ^{2}}{2} γ^{2} − (\frac{σ^{2}}{2} + r)γ − λ = 0,  (30)

i.e.

γ_{k} = \frac{(\frac{σ^{2}}{2} + r) + (−1)^{k}\sqrt{(\frac{σ^{2}}{2} + r)^{2} + 2λσ^{2}}}{σ^{2}} ,  k = 1, 2.  (31)

For x ≥ 0 define the function

u(x) = Γ(−γ_{1})x^{γ_{2}}U( − γ_{2}, 1 − γ_{2} + γ_{1}, \frac{2}{xσ^{2}} ).  (32)

Notice that since λ > 0, then γ_{1} < 0 and γ_{2} > 1. That permits to use for the function U(−γ_{2}, 1 − γ_{2} + γ_{1}, \frac{2}{xσ^{2}} ) the integral representation (29), that after the substitution zt = 2y/σ^{2} gives for u(x) the following expression:

u(x) = ∫_{0}^{∞}exp ( − \frac{2y}{σ^{2}} )y^{−(γ_{1}+1)}(1 + yx)^{γ_{2}} dy.  (33)

Since γ_{1} < 0 and γ_{2} > 1 it follows that u = u(x) is an increasing strictly convex function such that \frac{u(x)}{x} → ∞, x → ∞.

2. In the sequel we will use the fact that the function u = u(x), x ≥ 0, satisfies some second order differential equation (see (36) ).

First we remark that the functions M(a, b, z) and U(a, b, z) defined in (27) and (28) with a = −γ_{2} and b = 1 + γ_{1} − γ_{2} (with γ_{1} and γ_{2} from (31)) are the independent solutions of Kummer’s equation

zω′′(z) + (1 + γ_{1} − γ_{2} − z)ω′(z) + γ_{2}ω(z) = 0,  (34)

that is equivalent (since γ_{1} + γ_{2} = 1 + \frac{2r}{σ^{2}} ) to the equation

zω′′(z) + (\frac{2r}{σ^{2}} − 2 − 2γ_{2} − z)ω′(z) + γ_{2}ω(z) = 0.  (35)

It follows that the function u(x) = x^{γ_{2}}ω(\frac{2}{xσ^{2}} ) with ω(z) = Γ(−γ_{1})z^{γ_{2}}U(−γ_{2}, 1 − γ_{2} + γ_{1}, z) satisfies the following equation

\frac{σ^{2}}{2} x^{2}u′′(x) + (1 − rx)u′(x) − λu(x) = 0,  x ≥ 0.  (36)

3. Let us define the family of functions {u_{c}(x), c ≥ 0}, where u_{c}(x) = cu(x).

From the properties of the function u = u(x) it follows that there are unique constants c^{∗} > 0 and ψ^{∗} > 0 such that

u_{c^{∗}}(ψ^{∗}) = ψ^{∗},  u′_{c^{∗}}(ψ^{∗}) = 1,  (37)

i.e. c^{∗} > 0 and ψ^{∗} > 0 are the solutions of the system

c^{∗}u(ψ^{∗}) = ψ^{∗},  c^{∗}u′(ψ^{∗}) = 1,  (38)

or the equivalent system

c^{∗}u(ψ^{∗}) = ψ^{∗},  c^{∗}u(x) ≥ x,  x ≥ 0.  (39)

From (37) it follows that ψ^{∗} is the root of the equation

ψu′(ψ) = u(ψ)  (40)

and the constant c^{∗} is equal to

c^{∗} = \frac{ψ^{∗}}{u(ψ^{∗})}.  (41)

Finally let us denote

v(x) = u_{c^{∗}}(x) (= c^{∗}u(x)).  (42)

3 The proof of the theorem

1. From (21), (20) and by Ito’s formula we find that the process (ψ_{t})_{t≥0} is the solution of the stochastic differential equation

dψ_{t} = (1 − rψ_{t})dt − σψ_{t}dW̃_{t}.  (43)

This equation arises in the change point problem (see [15], formula (27)) and is sometimes referred as “Shiryaev-Robbins equation”(see [3], p. 168).

Let us define

Ṽ (ψ) = sup_{τ} Ẽ_{ψ}e^{−λτ}ψ_{τ},  ψ ≥ 0,  (44)

where Ẽ_{ψ} denotes the expectation with respect to the measure P̃ under the assumption that the process (ψ_{t})_{t≥0}, defined as the strong solution of the stochastic differential equation (43) starts from ψ_{0} = ψ ≥ 0.

By (24)

C^{∗} = S_{0}Ṽ (ψ_{0})  (45)

with the constant ψ_{0} from (10).

Along with Ṽ (ψ) we define a function

V (ψ) = \{ \begin{matrix}v(ψ), & 0 ≤ ψ ≤ ψ^{∗}, \\ ψ, & ψ > ψ^{∗}, \end{matrix}  (46)

where the function v = v(ψ), ψ ≥ 0, is given in (42) and ψ^{∗} is the solution of the equation (40).

We assert that

Ṽ (ψ) = V (ψ),  (47)

i.e. that the function V (ψ) defined in (46) is the “price” for the problem (44), and that the stopping time

τ^{∗} = inf{t ≥ 0 : ψ_{t} ≥ ψ^{∗}}  (48)

is optimal (rational):

Ẽ_{ψ}e^{−λτ^{∗}}ψ_{τ^{∗}} = Ṽ (ψ),  ψ ≥ 0.  (49)

2. To prove these assertions it is sufficient to check the following “verification” properties:

(A_{1}) Ẽ_{ψ}e^{−λτ}V (ψ_{τ}) ≤ V (ψ), ψ ≥ 0, for all finite (P̃-a.s.) stopping times τ and (A_{2}) stopping time τ^{∗} defined in (48) is (P̃-a.s.) finite and

Ẽ_{ψ}e^{−λτ^{∗}}V (ψ_{τ^{∗}}) = V (ψ), ψ ≥ 0.

Indeed if these conditions are fulfilled then from (46) we have V (ψ) ≥ ψ and so (A_{1}) applied gives

sup_{τ} Ẽ_{ψ}e^{−λτ}ψ_{τ} ≤ sup_{τ} Ẽ_{ψ}e^{−λτ}V (ψ_{τ}) ≤ V (ψ).  (50)

This fact together with the definition of τ^{∗} and the property (A_{2}) gives the optimality of τ^{∗} as well as the equality (47).

3. We begin with the proof of the property (A_{1}).

As the function V = V (ψ) is two times continuously differentiable we can apply the Ito’s formula to (e^{−λt}V (ψ_{t}))_{t≥0}, that gives

d(e^{−λt}V (ψ_{t})) = e^{−λt}[LV (ψ_{t}) − λV (ψ_{t})]dt − σe^{−λt}V′(ψ_{t})ψ_{t}dW̃_{t},  (51)

where

LV (ψ) = (1 − rψ)V′(ψ) + \frac{σ^{2}}{2} ψ^{2}V′′(ψ)  (52)

is the infinitesimal operator of the Markov process (ψ_{t})_{t≥0}. In the integral form the equation (51) becomes

e^{−λt}V (ψ_{t}) = V (ψ_{0}) + ∫_{0}^{t}e^{−λs}[LV (ψ_{s}) − λV (ψ_{s})] ds − ∫_{0}^{t}σe^{−λs}V′(ψ_{s})ψ_{s} dW̃_{s}.

(53)

Remark, that if ψ > ψ^{∗} then V (ψ) = ψ (see (46)) and therefore

LV (ψ) − λV (ψ) = 1 − (r + λ)ψ ≤ 1 − (r + λ)ψ^{∗} ≤ 0.  (54)

The last inequality 1 − (r + λ)ψ^{∗} ≤ 0 follows from the fact that according to (36) and (37) at point x = ψ^{∗}

\frac{σ^{2}}{2} ψ^{∗}^{2}rv′′(ψ^{∗}) + (1 − rψ^{∗}) − λψ^{∗} = 0,  (55)

whereas from (33) we have v′′(ψ) ≥ 0, ψ ≥ 0. Hence from (55) 1 − (r + λ)ψ^{∗} ≤ 0.

In the domain ψ ≤ ψ^{∗} we have V (ψ) = v(ψ) and therefore by (36)

Lv(ψ) − λv(ψ) = 0.  (56)

So LV (ψ_{s}) − λV (ψ_{s}) ≤ 0 and therefore from (53)

V (ψ_{0}) − ∫_{0}^{t}σe^{−λs}V′(ψ_{s})ψ_{s} dW̃_{s} ≥ e^{−λt}V (ψ_{t}).  (57)

The stochastic integral here (as the process) is a local martingale ([8], ch. 2, §2). Moreover by (57) it is bounded from bellow and therefore is a supermartingale.

So for each finite stopping time τ

Ẽ_{ψ} ∫_{0}^{τ}σe^{−λs}V′(ψ_{s})ψ_{s} dW̃_{s} = 0  (58)

and therefore from (57)

Ẽ_{ψ}e^{−λτ}V (ψ_{τ}) ≤ V (ψ_{0}), ψ ≥ 0,

proving the property (A_{1}). Similarly it may be shown that the process (e^{−λt}V (ψ_{t}))_{t≥0} is a P̃_{ψ}-supermartingale.

Now let us check the property (A_{2}). To start with we assume that P̃_{ψ}(τ^{∗} < ∞) = 1, ψ ≥ 0. This fact will be proved below in item 4.

From (53)

e^{−λ(t∧τ^{∗})}V (ψ_{t∧τ^{∗}}) = V (ψ_{0}) + ∫_{0}^{t∧τ^{∗}}e^{−λs}[LV (ψ_{s}) − λV (ψ_{s})] ds−

∫_{0}^{t∧τ^{∗}}σe^{−λs}V′(ψ_{s})ψ_{s} dW̃_{s}.  (59)

The property (A_{2}) is obvious if ψ ≥ ψ^{∗}. Therefore we suppose in (59) that ψ_{0} = ψ ≤ ψ^{∗}. Then for s ≤ t ∧ τ^{∗} we have V (ψ_{s}) = v(ψ_{s}) and hence from (56) (P̃_{ψ}-a.s., ψ ≤ ψ^{∗})

∫_{0}^{t∧τ^{∗}}e^{−λs}[LV (ψ_{s}) − λV (ψ_{s})] ds = 0.  (60)

Let

I_{t}^{∗} = − ∫_{0}^{t∧τ^{∗}}σe^{−λs}V′(ψ_{s})ψ_{s} dW̃_{s}.

Then from (59) and (60)

I_{t}^{∗} = e^{−λ(t∧τ^{∗})}V (ψ_{t∧τ^{∗}}) − V (ψ_{0}).  (61)

It follows that for all t ≥ 0

−V (ψ_{0}) ≤ I_{t}^{∗} ≤ V (ψ^{∗}) − V (ψ_{0})  (62)

and therefore the local martingale (I_{t}^{∗})_{t≥0} is uniformly integrable. Therefore by Doob theorem Ẽ_{ψ}I_{τ}^{∗} = 0 for each (P̃_{ψ}-a.s.) finite stopping time τ and in particularly for the stopping time τ^{∗}. Hence the property (A_{2}) (under the assumption that τ^{∗} is finite) is proved.

4. Let as prove that the stopping time τ^{∗} is finite, i.e. that P̃_{ψ}(τ^{∗} < ∞) = 1 for all ψ ≥ 0.

With this purpose we consider the harmonic function K = K(ψ) satisfying to the equation

LK(ψ) = 0,  ψ > 0, K(1) = 0,  (63)

where the differential operator L was defined in (52) . The solution of (63) is a strictly increasing function

K(x) = ∫_{1}^{x}y^{\frac{2r}{σ^{2}}} exp (\frac{2}{yσ^{2}} ) dy,  (64)

where as usually ∫_{1}^{x} = − ∫_{x}^{1} for 0 < x ≤ 1.

From (63) and (53) (with λ = 0) we have

K(ψ_{t}) = K(ψ_{0}) − ∫_{0}^{t}σψ_{s}K′(ψ_{s}) dW̃_{s}.  (65)

As

τ^{∗} = inf{t ≥ 0 : ψ_{t} ≥ ψ^{∗}} = inf{t ≥ 0 : K(ψ_{t}) ≥ K(ψ^{∗})},

then it is sufficient to show that the process (K(ψ_{t}))_{t≥0} reaches any level A almost surely:

P̃_{ψ}(sup_{t} K(ψ_{t}) < A) = 0,  ψ ≥ 0.  (66)

The process (K(ψ_{t}))_{t≥0} is the diffusion Markov process such that

dK(ψ_{t}) = −σ(ψ_{t})dW̃_{t},  (67)

where

σ(ψ) = σψK′(ψ).

As

c = inf_{ψ>0} σ(ψ) = inf_{ψ>0} (σψ^{\frac{2r}{σ^{2}} +1} exp (\frac{2}{ψσ^{2}} )) == σ(\frac{2}{2r + σ^{2}} )^{\frac{2r}{σ^{2}} +1} exp (\frac{2r}{σ^{2}} ) > 0,

then the quadratic variation (⟨K(ψ)⟩_{t})_{t≥0} (see [8], ch. 1, §8) of the continuous martingale (K(ψ_{t}))_{t≥0} is equal to

⟨K(ψ)⟩_{t} = ∫_{0}^{t}σ^{2}(ψ_{s}) ds ≥ c^{2}t.

Hence P̃_{ψ}(⟨K(ψ)⟩_{t} → ∞) = 1 and the stopping time (“random time change”)

κ(t) = inf{s ≥ 0 : ⟨K(ψ)⟩_{s} ≥ t}

is finite with P̃_{ψ}-probability one.

Remark now that the process \overline{W} = (\overline{W_{t}})_{t≥0} with \overline{W_{t}} = K(ψ_{κ(t)}) is a continuous local martingale with quadratic variation ⟨\overline{W} ⟩_{t} = t, t ≥ 0. Therefore by Levy’s theorem ([7, theorem 4.1]) this process is a Wiener process. Hence with probability one it reaches any level. It follows that the process (K(ψ_{t}))_{t≥0} also possesses such property, proving (66).

The theorem is proved.

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