Long-term returns in stochastic interest rate models: Convergence in law.

G. Deelstra F. Delbaen Vrije Universiteit Brussel Departement Wiskunde Pleinlaan 2 B-1050 Brussel Belgium

Abstract

Using an extension of the Cox-Ingersoll-Ross [1] stochastic model of the short interest rate r, we study the convergence in law of the longterm return in order to make some approximations. We use the theory of Bessel processes and observe the convergence in law of the sequence (\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt}(X_{u} + \frac{δ_{u}}{2β})du)_{t≥0} with the X a generalized Besselsquare process with drift with stochastic reversion level. By Aldous’ criterion, we are able to prove that this sequence converges in law to a Brownian motion.

Key Words: Generalized Besselsquare processes, convergence in law, Aldous’ criterion, stochastic interest rates.

AMS subject classification: 60Gxx, 60H10, 60H30.

1 Introduction.

Controlling the risk induced by interest rate fluctuations is of crucial importance for banks and insurance companies. Interest rate models can be used to obtain explicit formulae for pricing interest rate derivative securities and to construct a hedging strategy. They are also a necessary tool in managing long-term life insurance contracts.

As in “Long-term returns in stochastic interest rate models.” [2], we study the long-term return using an extension of the Cox, Ingersoll and Ross [1] stochastic model of the short interest rate r. Cox, Ingersoll and Ross express the short interest rate dynamics as

dr_{t} = κ(γ − r_{t})dt + σ\sqrt{r_{t}}dB_{t}.

with (B_{t})_{t≥0} a Brownian motion and κ, γ and σ positive constants. This is a fairly good model since r cannot become negative and the randomly moving interest rate is elastically pulled towards a reversion level. However, this is a constant level, namely the long-term value γ. It is more reasonable to conjecture that the market will influence this level. Schaefer and Schwartz [3], Hull and White [4] and Longstaff and Schwartz [5], proposed time-dependent parameters. We extend the CIR model by assuming a stochastic reversion level. In this way, we can treat more factor models.

If we define X by a transformation of the CIR square root process r, namely X = \frac{4}{σ^{2}} r, then X is a Besselsquare process with drift −κ/2 and dimension \frac{4κγ}{σ^{2}} . The many results obtained by Yor [6,7], convinced us that these processes are very tractable. Therefore, we consider a family of stochastic processes X, which contains the Besselsquare processes with drift.

More precisely, we study processes X satisfying the stochastic differential equation:

dX_{s} = (2βX_{s} + δ_{s})ds + 2\sqrt{X_{s}}dB_{s} ∀s ∈ IR^{+}

with δ a non-negative adapted stochastic process and β < 0. In ”Existence of Solutions of Stochastic Differential Equations related to the Bessel process” [8], we have shown that this stochastic differential equation has a unique (nonnegative) strong solution as soon as ∫_{0}^{t} δ_{u} du < ∞ a.e. for all t ∈ IR^{+}. In this paper, we will assume that \frac{1}{t} ∫_{0}^{t} δ_{u}du →^{a.e.} \overline{δ} with \overline{δ} > 0.

The following generalized CIR two-factor model is an element of this family:

dr_{t} = κ(γ_{t} − r_{t})dt + σ\sqrt{r_{t}}dB_{t}  (1)
dγ_{t} = κ̃(γ^{∗} − γ_{t})dt + σ̃\sqrt{γ_{t}}dB̃_{t}

with κ, κ̃ > 0; γ^{∗}, σ and σ̃ positive constants and (B′_{t})_{t≥0} and (B̃_{t})_{t≥0} two Brownian motions. These Brownian motions are correlated in an arbitrarily way. Most authors suppose for technical reasons that the Brownian motions are uncorrelated or have a constant correlation. We do not need this assumption. The Brownian motions even may have a random correlation!

In “Long-term returns in stochastic interest rate models.” [2], we found a convergence theorem. Under some conditions such as \frac{1}{s} ∫_{0}^{s} δ_{u}du →^{a.e.} \overline{δ} with \overline{δ} : Ω → IR^{+}, the following convergence almost everywhere holds:

\frac{1}{s} ∫_{0}^{s} X_{u}du ⟶^{a.e.} \frac{−\overline{δ}}{2β} .

In this paper, we study the convergence in law. By convergence in law or in distribution we mean weak convergence of probability measures on the space C(IR^{+}, IR) of continuous sample paths. This space endowed with the topology of uniform convergence on compact sets of IR^{+}, is a Polish space. (See Revuz–Yor [7] p. 472).

We are interested in the convergence in law since it is always useful to know how the long-term return is distributed in the limit so that one can find approximations. In some earlier papers, authors have modeled interest rates by Wiener models. On long-term, the Central Limit Theorems are indeed applicable.

We suppose the same hypothesis as in [2] so that we can apply the previous convergence theorem. Using the theory of Bessel processes, we prove the following theorem:

Theorem 1 Suppose that a probability space (Ω, (F_{t})_{t≥0}, IP) is given and that a stochastic process X : Ω × IR^{+} → IR^{+} is defined by the stochastic differential equation

dX_{s} = (2βX_{s} + δ_{s})ds + 2\sqrt{X_{s}}dB_{s} ∀s ∈ IR^{+}

with (B_{s})_{s≥0} a Brownian motion with respect to (F_{t})_{t≥0} and β < 0. Let us make the following assumptions about the adapted and measurable process δ : Ω × IR^{+} → IR^{+}:

• \frac{1}{s} ∫_{0}^{s} δ_{u}du →^{a.e.} \overline{δ} where \overline{δ} is a real number, \overline{δ} > 0

• There is a constant k such that sup_{t≥1} \frac{1}{t} ∫_{0}^{t} IE [δ_{u}^{2}] du ≤ k

• For all a ∈ IR^{+} lim_{t→∞} \frac{1}{t} ∫_{t−a}^{t} IE[δ_{u}^{2}]du = 0.

Under these conditions, the following convergence in distribution holds:

(\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} (X_{u} + \frac{δ_{u}}{2β} ) du)_{t≥0} ⟶^{L} (B′_{t})_{t≥0}

where (B′_{t})_{t≥0} is a Brownian motion and where ’⟶^{L}’ denotes convergence in law.

Remark that there is no assumption about the correlation between the process X and the process δ.

The organisation of the paper is as follows: Section 2 contains some technical lemmas. We show that, under the conditions of theorem 1, the second moment of X_{t} is bounded by a constant and that for a given ε > 0, there exists a constant c such that IP [sup_{u≤t} \frac{X_{u}}{\sqrt{t}} > c] ≤ ε. In section 3, the main theorem is proved. We also state an alternative version of theorem 1, whose conditions are measure-invariant. Since in Finance the measure is often transformed, measure-invariance is an important property. Section 4 gives some applications of theorem 1. We explain the usefulness of theorem 1 by the two-factor model (1).

2 Technical lemmas.

In this section, we prove a technical lemma and its corollary, both needed in the proof of the convergence result. We assume without further notice that B is a continuous Brownian motion with respect to the filtration (F_{s})_{s≥0}.We consider a family of stochastic processes X, which contains the Besselsquare processes with drift.

Lemma 1 Suppose the stochastic process X : Ω × IR^{+} → IR^{+} is defined by the stochastic differential equation

dX_{s} = (2βX_{s} + δ_{s}) ds + g(X_{s})dB_{s} ∀s ∈ IR^{+}  (2)

with

• β < 0

• g : IR → IR^{+} is a function, vanishing at zero and such that there is a constant b with |g(x) − g(y)| ≤ b\sqrt{|x − y|}.

• δ : Ω × IR^{+} → IR^{+} is an adapted and measurable process such that \frac{1}{t} ∫_{0}^{t} IE [δ_{u}^{2}] du ≤ k for all t ≥ 1 with k a constant.

Then for all t ≥ 1

IE [sup_{u≤t} X_{u}^{2}] ≤ K t

with K a constant independent of t and in fact only depending on β, k and b.

Remark: Before proving the lemma, we recall from [8] that a (continuous) adapted process X, given by the stochastic differential equation (2) has a unique strong non-negative solution as soon as ∫_{0}^{t} δ_{u} du < ∞ a.e. for all t ∈ IR^{+}. In [2], we obtained the expectation of X_{s}, namely:

IE[X_{s}] = e^{2βs}X_{0} + e^{2βs} ∫_{0}^{s} IE [δ_{u}] e^{−2βu} du.  (3)

Proof. By Itô’s formula,

X_{t}^{2} = X_{0}^{2} + 4β ∫_{0}^{t} X_{u}^{2}du + 2 ∫_{0}^{t} X_{u}δ_{u}du + ∫_{0}^{t} g^{2}(X_{u})du + 2 ∫_{0}^{t} X_{u}g(X_{u})dB_{u}.

Let us denote the continuous local martingale 2 ∫_{0}^{t} X_{u}g(X_{u})dB_{u} by M_{t}. We recall the notation M_{t}^{∗} = sup_{u≤t} M_{u}. For all u ≤ t, the positivity of X and δ imply the following inequality:

X_{u}^{2} − 4β ∫_{0}^{u} X_{s}^{2}ds ≤ X_{0}^{2} + 2 ∫_{0}^{t} X_{u}δ_{u}du + ∫_{0}^{t} g^{2}(X_{u})du + M_{t}^{∗}.

Since this is true for all u ≤ t, we conclude that the supremum of the left hand side is smaller than the right hand side.

sup_{u≤t} (X_{u}^{2} − 4β ∫_{0}^{u} X_{s}^{2}ds) ≤ X_{0}^{2} + 2 ∫_{0}^{t} X_{u}δ_{u}du + ∫_{0}^{t} g^{2}(X_{u})du + M_{t}^{∗}.

Since β < 0,

sup_{u≤t} X_{u}^{2} − 4β ∫_{0}^{t} X_{s}^{2}ds ≤ 2 sup_{u≤t} (X_{u}^{2} − 4β ∫_{0}^{u} X_{s}^{2}ds) .

Consequently,

IE [sup_{u≤t} X_{u}^{2}] ≤ 2X_{0}^{2} + 4βIE [∫_{0}^{t} X_{u}^{2}du] + 4IE [∫_{0}^{t} X_{u}δ_{u}du]

+ 2IE [∫_{0}^{t} g^{2}(X_{u})du] + 2IE [M_{t}^{∗}] .

Applying Cauchy–Schwarz’ inequality in the third term and the inequality |g(x)| ≤ b\sqrt{x} in the fourth term, we obtain:

IE [sup_{u≤t} X_{u}^{2}] ≤ 2X_{0}^{2} + 4βIE [∫_{0}^{t} X_{u}^{2}du]

+ 4IE [∫_{0}^{t} X_{u}^{2}du]^{1/2} IE [∫_{0}^{t} δ_{u}^{2}du]^{1/2} + 2b^{2}IE [∫_{0}^{t} X_{u}du] + 2IE [M_{t}^{∗}] .

We remark that X_{u} ≤ \frac{b^{2}}{−4β} + \frac{−β}{b^{2}} X_{u}^{2} since 0 ≤ \frac{−β}{b^{2}} (X_{u} + \frac{b^{2}}{2β} )^{2}. Substituting this inequality, we find:

IE [sup_{u≤t} X_{u}^{2}] − 2βIE [∫_{0}^{t} X_{u}^{2}du] ≤ 2X_{0}^{2} + \frac{b^{4}}{−2β} t

+ 4IE [∫_{0}^{t} X_{u}^{2}du]^{1/2} IE [∫_{0}^{t} δ_{u}^{2}du]^{1/2} + 2IE [M_{t}^{∗}] .

By the Burkholder–Davis–Gundy inequality (see Revuz–Yor [7] p. 151), we can rewrite the last term:

IE [M_{t}^{∗}] ≤ cIE [< M, M >_{t}^{1/2}]
≤ c b 2IE [(∫_{0}^{t} X_{u}^{2}X_{u}du)^{1/2}]

≤ c b 2IE [(∫_{0}^{t} X_{u}du)^{1/2} (sup_{u≤t} X_{u}^{2})^{1/2}]

≤ c b 2IE [∫_{0}^{t} X_{u}du]^{1/2} IE [sup_{u≤t} X_{u}^{2}]^{1/2}

≤ c b 2 t^{1/4}IE [∫_{0}^{t} X_{u}^{2}du]^{1/4} IE [sup_{u≤t} X_{u}^{2}]^{1/2}

where we applied the inequality of Cauchy–Schwarz in the last inequalities. Summarising, we conclude that:

IE [sup_{u≤t} X_{u}^{2}] − 2βIE [∫_{0}^{t} X_{u}^{2}du] ≤ 2X_{0}^{2} + \frac{b^{4}}{−2β} t

+ 4IE [∫_{0}^{t}X_{u}^{2}du]^{1/2}IE[∫_{0}^{t} δ_{u}^{2}du]^{1/2}+ cb4t^{1/4}IE[∫_{0}^{t} X_{u}^{2}du]^{1/4}IE[sup_{u≤t} X_{u}^{2}]^{1/2}.

Remark that by hypothesis IE[∫_{0}^{t} δ_{u}^{2}du] ≤ k t for t ≥ 1.

If we denote IE [sup_{u≤t} X_{u}^{2}]^{1/2} by A, \sqrt{−2β}IE[∫_{0}^{t} X_{u}^{2}du]^{1/2} by B and the constants by C_{i} (i ≥ 1), then we have to find the solution of the inequality:

A^{2} + B^{2} ≤ C_{1} + C_{2}t + C_{3}t^{1/2}B + C_{4}t^{1/4}AB^{1/2}.

We remark that B^{1/2} ≤ \frac{1}{C_{4}t^{1/4}} B + \frac{C_{4}t^{1/4}}{4} since \frac{1}{C_{4}t^{1/4}} (B^{1/2} − \frac{C_{4}t^{1/4}}{2} )^{2} ≥ 0. Substitution yields:

A^{2} + B^{2} ≤ C_{1} + C_{2}t + C_{3}t^{1/2}B + AB + C_{5}t^{1/2}A.

Since AB ≤ \frac{1}{2}(A^{2} + B^{2}), we obtain:

A^{2} + B^{2} ≤ 2C_{1} + 2C_{2}t + 2C_{3}t^{1/2}B + 2C_{5}t^{1/2}A.

This is equivalent with:

(A − C_{5}t^{1/2})^{2} + (B − C_{3}t^{1/2})^{2} ≤ 2C_{1} + 2C_{2}t + C_{3}^{2}t + C_{5}^{2}t.

Thus (A − C_{5}t^{1/2})^{2} as well as (B − C_{5}t^{1/2})^{2} are smaller than the right hand side. Substituting the expression for A, we find for all t ≥ 1

IE [sup_{u≤t} X_{u}^{2}] ≤ K t

with K a constant, only depending on k, β and b.

q.e.d.

Corollary 1. Suppose the stochastic process X : Ω × IR^{+} → IR^{+} is defined by the stochastic differential equation

dX_{s} = (2βX_{s} + δ_{s})ds + g(X_{s})dB_{s} ∀s ∈ IR^{+}

with

• β < 0

• g : IR → IR^{+} is a function, vanishing at zero and such that there is a constant b with |g(x) − g(y)| ≤ b\sqrt{|x − y|}.

• δ : Ω × IR^{+} → IR^{+} is an adapted and measurable process such that \frac{1}{t} ∫_{0}^{t} IE [δ_{u}^{2}] du ≤ k for all t ≥ 1 with k a constant.

Then, for ε > 0, there exists a real number c such that for all t ≥ 1

IP [sup_{u≤t} \frac{X_{u}}{\sqrt{t}} > c] ≤ ε.

Proof. By Chebyshev’s inequality

IP [sup_{u≤t} \frac{X_{u}}{\sqrt{t}} > c] ≤ \frac{1}{c^{2}t}IE [sup_{u≤t} X_{u}^{2}] .

Since by lemma 1, IE [sup_{u≤t} X_{u}^{2}] ≤ K t with K a constant, we obtain for all c > 0 and all t ≥ 1

IP [sup_{u≤t} \frac{X_{u}}{\sqrt{t}} > c] ≤ \frac{K}{c^{2}} .

Thus, we conclude that for ε > 0, there exists a real number c such that for all t ≥ 1

IP [sup_{u≤t} \frac{X_{u}}{\sqrt{t}} > c] ≤ ε.

q.e.d.

3 Convergence in law to a Brownian motion.

In this section, we give the proof of the main result of this paper, namely the convergence in law to a Brownian motion. We suppose the same hypothesis as in [2] so that we can apply the convergence a.e. result in our proof of the convergence in law.

Theorem 1 Suppose that a probability space (Ω, (F_{t})_{t≥0}, IP) is given and that a stochastic process X : Ω × IR^{+} → IR^{+} is defined by the stochastic differential equation

dX_{s} = (2βX_{s} + δ_{s})ds + 2\sqrt{X_{s}}dB_{s} ∀s ∈ IR^{+}

with (B_{s})_{s≥0} a Brownian motion and β < 0. Let us make the following assumptions about the adapted and measurable process δ : Ω × IR^{+} → IR^{+}:

• \frac{1}{s} ∫_{0}^{s} δ_{u}du →^{a.e.} \overline{δ} where \overline{δ} is a real number, \overline{δ} > 0

• sup_{t≥1} \frac{1}{t} ∫_{0}^{t} IE [δ_{u}^{2}] du ≤ k with k a constant independent of t.

• For all a ∈ IR^{+} lim_{t→∞} \frac{1}{t} ∫_{t−a}^{t} IE [δ_{u}^{2}] du = 0.

Under these conditions, the following convergence in distribution is true:

(\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} (X_{u} + \frac{δ_{u}}{2β} ) du)_{t≥0} ⟶^{L} (B′_{t})_{t≥0}

with (B′_{t})_{t≥0} a Brownian motion.

Remark: The statements of theorem 1 imply the conditions of the theorem in the previous paper [2] and hence, we may conclude that the long-term return converges a.e.:

\frac{1}{s} ∫_{0}^{s} X_{u}du ⟶^{a.e.} \frac{−\overline{δ}}{2β} .  (4)

Proof: First, we will check that the sequence (Y^{n}), defined by

Y_{t}^{n} = \sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} (X_{u} + \frac{δ_{u}}{2β} ) du n ≥ 1

converges to the Brownian motion Bt^{′} in the sense of finite distributions. Afterwards, we will show that the sequence is weakly relatively compact by using Aldous’ criterion for tightness.

To prove the convergence in the sense of finite distributions, we show that for any finite collection (t_{1}, ⋯, t_{k}) of times, the random variables (Y_{t_{1}}^{n}, ⋯, Y_{t_{k}}^{n}) converge in law to (Bt_{1}^{′}, ⋯, Bt_{k}^{′}). Thus, we have to check that:

(\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt_{1}} (X_{u} + \frac{δ_{u}}{2β} ) du, ⋯, \sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt_{k}} (X_{u} + \frac{δ_{u}}{2β} ) du)

⟶^{L} (Bt_{1}^{′}, ⋯, Bt_{k}^{′}) .

We recall the following well known theorem of probability theory (See Feller [9] p. 247):

Theorem A Suppose that a sequence Z_{n} : Ω ⟶ IR^{k} is given. If Z_{n} = U_{n} +V_{n} where

• U_{n} ⟶^{L} U with U ∼ μ a probability measure

• V_{n} ⟶^{IP} 0

then

Z_{n} ⟶^{L} U.

From the stochastic differential equation

dX_{s} = (2βX_{s} + δ_{s})ds + 2\sqrt{X_{s}}dB_{s}

we arrive at the following formula:

\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} (X_{u} + \frac{δ_{u}}{2β} ) du = \sqrt{\frac{−2β^{3}}{\overline{δ}n}} \frac{X_{nt} − X_{0}}{2β} − \sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} \frac{\sqrt{X_{s}}}{β} dB_{s}.

Consequently, we can rewrite the random variables:

(Y_{t_{1}}^{n}, ⋯, Y_{t_{k}}^{n})
= (\sqrt{\frac{−2β^{3}}{\overline{δ}n}} \frac{X_{nt_{1}} − X_{0}}{2β} , ⋯, \sqrt{\frac{−2β^{3}}{\overline{δ}n}} \frac{X_{nt_{k}} − X_{0}}{2β} )

− (\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt_{1}} \frac{\sqrt{X_{s}}}{β} dB_{s}, ⋯, \sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt_{k}} \frac{\sqrt{X_{s}}}{β} dB_{s}) .

The first vector converges in distribution to zero because each component converges to zero in probability. Namely, for all ε > 0, there exists a t_{0} such that for all t ≥ t_{0} \frac{IE[X_{t}]}{\sqrt{t+1}} < ε. Indeed,

\frac{IE[X_{t}]}{\sqrt{t + 1}} = \frac{e^{2βt}X_{0}}{\sqrt{t + 1}} + \frac{1}{\sqrt{t + 1}}IE [e^{2βt} ∫_{0}^{t} e^{−2βu}δ_{u}du]

by remark (3) in section 2.

Trivially, the first term converges to zero. Let us treat the second term:

\frac{1}{\sqrt{t + 1}}IE [e^{2βt} ∫_{0}^{t} e^{−2βu}δ_{u}du]

= \frac{1}{\sqrt{t + 1}}IE [e^{2βt} ∫_{0}^{t−a} e^{−2βu}δ_{u}du] + \frac{1}{\sqrt{t + 1}}IE [e^{2βt} ∫_{t−a}^{t} e^{−2βu}δ_{u}du]

≤ IE [(∫_{0}^{t−a} e^{4β(t−u)}du)^{1/2} (\frac{1}{t + 1} ∫_{0}^{t−a} δ_{u}^{2}du)^{1/2}]

+ IE [(∫_{t−a}^{t} e^{4β(t−u)}du)^{1/2} (\frac{1}{t + 1} ∫_{t−a}^{t} δ_{u}^{2}du)^{1/2}]

≤ \frac{e^{2βa}}{2\sqrt{−β}} (\frac{1}{t + 1} ∫_{0}^{t−a} IE[δ_{u}^{2}]du)^{1/2} + \frac{1}{2\sqrt{−β}} (\frac{1}{t + 1} ∫_{t−a}^{t} IE[δ_{u}^{2}]du)^{1/2} .

Since by hypothesis sup_{t≥1} \frac{1}{t} ∫_{0}^{t} IE [δ_{u}^{2}] du ≤ k with k a constant, the first term is smaller than Ke^{2βa} with K a constant. We can choose a such that Ke^{2βa} < ε/2. Since by hypothesis for all a ∈ IR^{+}, lim_{t→∞} \frac{1}{t} ∫_{t−a}^{t} IE[δ_{u}^{2}]du = 0, there exists a t_{0} ≥ a such that for all t ≥ t_{0}: \frac{1}{t+1} ∫_{t−a}^{t} IE[δ_{u}^{2}]du < −βε^{2}. Thus, we proved that \frac{IE[X_{t}]}{\sqrt{t+1}} ⟶ 0. Consequently,

\frac{X_{nt_{i}}}{\sqrt{n}} ⟶^{IP} 0 i = 1, ⋯, k.

By theorem A, it remains to be shown that

(\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt_{1}} \frac{\sqrt{X_{s}}}{β} dB_{s}, ⋯, \sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt_{k}} \frac{\sqrt{X_{s}}}{β} dB_{s})

converges in distribution to a k-dimensional Brownian motion.

Let us define the stopping times

τ_{t} = inf \{s | ∫_{0}^{s} X_{u}du ≥ \frac{−\overline{δ}t}{2β}\} .

Let us remark that two properties hold for these stopping times:

lim_{t→∞} τ_{t} = ∞

and

lim_{t→∞} \frac{τ_{t}}{t} = 1.  (5)

Let us give a short proof of the second statement: By theorem 1, \frac{1}{u} ∫_{0}^{u} X_{s}ds ⟶^{a.e.} \frac{−\overline{δ}}{2β}. Thus, for an arbitrary ε > 0, we have for all u large enough that:

(1 − ε)\frac{−\overline{δ}u}{2β} ≤ ∫_{0}^{u} X_{s}ds ≤ (1 + ε)\frac{−\overline{δ}u}{2β}.

This implies that lim_{t→∞} τ_{t} = ∞ and hence, we have for all ε and u large enough that:

(1 − ε)\frac{−\overline{δ}τ_{u}}{2β} ≤ ∫_{0}^{τ_{u}} X_{s}ds ≤ (1 + ε)\frac{−\overline{δ}τ_{u}}{2β} .

But by definition of τ_{u}, ∫_{0}^{τ_{u}} X_{s}ds = \frac{−\overline{δ}u}{2β}. After a simplification, we obtain for u large enough:

\frac{1}{1 + ε} ≤ \frac{τ_{u}}{u} ≤ \frac{1}{1 − ε}.

Since this is true for any ε, we have proved that lim_{t→∞} \frac{τ_{t}}{t} = 1. With the help of the theorem of Dambis, Dubins and Schwarz (see Revuz–Yor [7], p. 170), it is easy to verify that (\sqrt{\frac{−2β}{\overline{δ}n}} ∫_{0}^{τ_{nt}} \sqrt{X_{u}}dB_{u})_{t≥0} is a Brownian

motion B′.

Since trivially,

(\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt_{1}} \frac{\sqrt{X_{s}}}{β} dB_{s}, ⋯, \sqrt{\frac{−2β^{3}}{_{\overline{lineδ}} n}} ∫_{0}^{nt_{k}} \frac{\sqrt{X_{s}}}{β} dB_{s})

= (\sqrt{\frac{−2β}{\overline{δ}n}} ∫_{0}^{τ_{nt_{1}}} \sqrt{X_{u}}dB_{u}, ⋯, \sqrt{\frac{−2β}{\overline{δ}n}} ∫_{0}^{τ_{nt_{k}}} \sqrt{X_{u}}dB_{u})

+ (\sqrt{\frac{−2β}{\overline{δ}n}} ∫ (11_{[[ 0,nt_{1} ]]} − 11_{[[ 0,τ_{nt_{1}} ]]}) \sqrt{X_{u}}dB_{u}, ⋯

⋯, \sqrt{\frac{−2β}{\overline{δ}n}} ∫ (11_{[[ 0,nt_{k} ]]} − 11_{[[ 0,τ_{nt_{k}} ]]}) \sqrt{X_{u}}dB_{u})

By theorem A, we only need to show the convergence in distribution or in probability to zero of the second vector. We will show that each component converges in probability to zero. This result follows from stochastic integration theory. We cite the theorem that we will use, see e.g. Karatzas-Shreve [10] p. 147, proposition 2.26 which remains valid for T ≡ ∞:

Theorem B Let us denote L^{2,0} = { H | H predictable with ∫_{0}^{∞} H_{u}^{2}du < ∞ } and let us define

L^{2,0} ⟶ L^{0}(Ω, F_{∞}, IP) H ⟼ (H · B)_{∞}

If ∫_{0}^{∞}(H_{u}^{n})^{2}du ⟶^{IP} 0, then (H^{n} · B)_{∞} ⟶^{IP} 0.

We apply this theorem for i = 1, ⋯, k and for the processes:

H_{·}^{n,i} ≡ (11_{[[ 0,nt_{i} ]]} − 11_{[[ 0,τ_{nt_{i}} ]]}) \begin{matrix}√ & \overline{X_{·}} \\ √ & \overline{nt_{i}} \end{matrix} i = 1, ⋯, k.

In this case, the condition of theorem B is fulfilled:

∫_{0}^{∞}(H_{u}^{n,i})^{2}du = \frac{1}{nt_{i}} ∫ (11_{[[ 0,nt_{i} ]]} − 11_{[[ 0,τ_{nt_{i}} ]]}) X_{u}du ⟶^{IP} 0.

Indeed

\frac{1}{nt_{i}} ∫ (11_{[[ 0,nt_{i} ]]} − 11_{[[ 0,τ_{nt_{i}} ]]}) X_{u}du

= \frac{1}{nt_{i}} (∫_{0}^{nt_{i}} X_{u}du) − \frac{τ_{nt_{i}}}{nt_{i}} (∫_{0}^{τ_{nt_{i}}} \frac{X_{u}}{τ_{nt_{i}}} du) ⟶^{a.e.} 0

because

• \frac{1}{s} ∫_{0}^{s} X_{u}du ⟶^{a.e.} \frac{−\overline{δ}}{2β} by remark (4)

• \frac{τ_{nt_{i}}}{nt_{i}} ⟶^{a.e.} 1 by remark (5).

Thus, we conclude that for all i = 1, ⋯, k

(H^{n,i} · B)_{∞} = \frac{1}{\sqrt{nt_{i}}} ∫ (11_{[[ 0,nt_{i} ]]} − 11_{[[ 0,τ_{nt_{i}} ]]}) \sqrt{X_{u}}dB_{u} ⟶^{IP} 0

or equivalently that for all i = 1, ⋯, k

\sqrt{\frac{−2βt_{i}}{\overline{δ}n}} (H^{n,i} · B)_{∞} = \sqrt{\frac{−2β}{\overline{δ}n}} ∫ (11_{[[ 0,nt_{i} ]]} − 11_{[[ 0,τ_{nt_{i}} ]]}) \sqrt{X_{u}}dB_{u} ⟶^{IP} 0.

Thus, we have shown that for any finite collection (t_{1}, ⋯, t_{k}) of times, the random variables (\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt_{1}} (X_{u} + \frac{δ_{u}}{2β} ) du, ⋯, \sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt_{k}} (X_{u} + \frac{δ_{u}}{2β} ) du) converge in law to (Bt_{1}^{′}, ⋯, Bt_{k}^{′}).

We will now prove that the laws of (Y^{n}) form a weakly relatively compact sequence. We have to check two conditions (see e.g. Jacod–Shiryaev [11], p. 320):

For all N ∈ IN^{∗}, ε > 0, there exists n_{0} ∈ IN^{∗} and K ∈ IR^{+}

such that for all n ≥ n_{0} : IP (sup_{t≤N} |Y_{t}^{n}| > K) ≤ ε.

and Aldous’ criterion, namely

For all N ∈ IN^{∗}, ε > 0 :

lim_{θ→0} lim sup_{n} sup_{S≤T ≤S+θ} IP (|Y_{T}^{n} − Y_{S}^{n}| ≥ ε) = 0

where S and T are restricted to the set of F^{n}-stopping times that are bounded by N.

Let us start with Aldous’ criterion. This is equivalent to:

For all N ∈ IN^{∗}, ε > 0 :

lim_{θ→0} lim sup_{n} sup_{S} IP ( sup_{0≤u≤θ} |Y_{S+u}^{n} − Y_{S}^{n}| ≥ ε) = 0

where S is restricted to the set of F^{n}-stopping times bounded by N. We will therefore search a bound for

IP [ sup_{0≤u≤θ} |Y_{S+u}^{n} − Y_{S}^{n}| ≥ 2ε]

or if we substitute the expression of Y^{n}, for:

IP [ sup_{0≤u≤θ} |\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{Sn}^{(S+u)n} (X_{t} + \frac{δ_{t}}{2β} ) dt| ≥ 2ε] .

From the stochastic differential equation, we easily find

∫ \frac{dX_{u}}{2β} = ∫ (X_{u} + \frac{δ_{u}}{2β})du + ∫ \frac{\sqrt{X_{u}}}{β} dB_{u}.

We replace ∫ (X_{u} + \frac{δ_{u}}{2β})du and obtain:

IP [ sup_{0≤u≤θ} |−\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{Sn}^{(S+u)n}\frac{\sqrt{X_{t}}}{β} dB_{t} + \sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{Sn}^{(S+u)n} \frac{dX_{t}}{2β} | ≥ 2ε] .

Trivially, this probability is smaller than

IP [ sup_{0≤u≤θ} |−\sqrt{\frac{−2β}{\overline{δ}n}} ∫_{Sn}^{(S+u)n} \sqrt{X_{t}}dB_{t}| ≥ ε]  (6)

+ IP [ sup_{0≤u≤θ} |\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{Sn}^{(S+u)n} \frac{dX_{t}}{2β} | ≥ ε] .

Let us concentrate on the first term. We know that (∫_{0}^{s} \sqrt{X_{t}}dB_{t})_{s≥0} is a martingale. If S is a F^{n}-stopping time, then nS and n(S +u) are F-stopping times. Consequently (∫_{Sn}^{(S+u)n} \sqrt{X_{t}}dB_{t})_{u≥0} is a (F_{Sn+nu})_{u≥0}-martingale.

Therefore, we can apply the martingale inequality

IP [ sup_{0≤u≤θ} |−\sqrt{\frac{−2β}{\overline{δ}n}} ∫_{Sn}^{(S+u)n} \sqrt{X_{t}}dB_{t}| ≥ ε]

≤ \frac{−2β}{\overline{δ}nε^{2}} ∥∫_{Sn}^{(S+θ)n} \sqrt{X_{t}}dB_{t}∥_{2}^{2} .

By stochastic calculus, this is equal to

\frac{−2β}{\overline{δ}nε^{2}} IE [∫_{Sn}^{(S+θ)n} X_{u}du] .

We can replace ∫_{Sn}^{(S+θ)n} X_{u}du, using the integrated stochastic differential equation:

∫_{Sn}^{(S+θ)n} dX_{u} = 2β ∫_{Sn}^{(S+θ)n} X_{u} du + ∫_{Sn}^{(S+θ)n} δ_{u} du + 2 ∫_{Sn}^{(S+θ)n} \sqrt{X_{u}}dB_{u}.

Since ∫_{Sn}^{(S+θ)n} \sqrt{X_{u}}dB_{u} is a martingale, its expected value is equal to zero and we obtain that

IP [ sup_{0≤u≤θ} |−\sqrt{\frac{−2β}{\overline{δ}n}} ∫_{Sn}^{(S+u)n} \sqrt{X_{t}}dB_{t}| ≥ ε]

≤ \frac{1}{\overline{δ}nε^{2}} IE [∫_{Sn}^{(S+θ)n} δ_{u}du] − \frac{1}{\overline{δ}nε^{2}} IE [∫_{Sn}^{(S+θ)n} dX_{u}] .  (7)

We will handle both terms in the right hand side separately. Let us start with the first term. By an application of the Cauchy-Schwarz inequality, we obtain for all N ∈ IN^{∗} and ε > 0:

lim_{θ→0} lim sup_{n} sup_{S} \frac{1}{\overline{δ}nε^{2}} IE [∫_{Sn}^{(S+θ)n} δ_{u}du]

≤ lim_{θ→0} lim sup_{n} sup_{S} \frac{1}{\overline{δ}ε^{2}} IE [(θn)^{1/2} (\frac{1}{n^{2}} ∫_{Sn}^{(S+θ)n} δ_{u}^{2}du)^{1/2}]

≤ lim_{θ→0} lim sup_{n} \frac{1}{\overline{δ}ε^{2}} (θn)^{1/2} (\frac{1}{n^{2}} ∫_{0}^{(N+θ)n} IE [δ_{u}^{2}] du)^{1/2}

≤ lim_{θ→0} lim sup_{n} \frac{1}{\overline{δ}ε^{2}} \sqrt{θ} ((N + 1)\frac{1}{(N + 1)n} ∫_{0}^{(N+1)n} IE [δ_{u}^{2}] du)^{1/2}

≤ lim_{θ→0} \sqrt{θ} \frac{\sqrt{k(N + 1)}}{\overline{δ}ε^{2}} = 0

where we have used the hypothesis that sup_{t≥1} \frac{1}{t} ∫_{0}^{t} IE [δ_{u}^{2}] du ≤ k withk a constant.In the case of the second term of inequality (7), we remark that

lim_{θ→0} lim sup_{n} sup_{S} \frac{−1}{\overline{δ}nε^{2}} IE [∫_{Sn}^{(S+θ)n} dX_{u}]

= lim_{θ→0} lim sup_{n} sup_{S} \frac{IE[X_{Sn}] − IE[X_{(S+θ)n}]}{\overline{δ}nε^{2}}
≤ lim_{θ→0} lim sup_{n} sup_{S} \frac{IE[X_{Sn}]}{\overline{δ}nε^{2}}
≤ lim_{θ→0} lim sup_{n} \frac{IE [sup_{u≤Nn} X_{u}]}{nN} \frac{N}{\overline{δ}ε^{2}} .

If we use Jensen’s inequality, we obtain:

\frac{IE [sup_{u≤nN} X_{u}]}{nN} ≤ \frac{IE [sup_{u≤nN} X_{u}^{2}]^{1/2}}{nN}

By corollary 1, we find

\frac{IE [sup_{u≤nN} X_{u}]}{nN} ≤ \frac{K\sqrt{nN}}{nN} .

Thus,

\frac{sup_{u≤nN} X_{u}}{nN} ⟶^{L^{1}} 0.

We conclude that the second term in inequality (7) also converges:

lim_{θ→0} lim sup_{n} sup_{S} \frac{−1}{\overline{δ}nε^{2}} IE [∫_{Sn}^{(S+θ)n} dX_{u}] = 0.

Let us now look at the second probability in inequality (6) and let us check if for all N ∈ IN^{∗} and ε > 0:

lim_{θ→0} lim sup_{n} sup_{S} IP [ sup_{0≤u≤θ} |\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{Sn}^{(S+u)n} \frac{dX_{t}}{2β} | ≥ ε] = 0

where S is restricted to the set of F^{n}-stopping times bounded by N. We start the calculations:

sup_{S} IP [ sup_{0≤u≤θ} |∫_{Sn}^{(S+u)n} \frac{dX_{t}}{\sqrt{n}} | ≥ 2ε]

≤ sup_{S} IP [( sup_{0≤u≤θ} \frac{X_{(S+u)n}}{\sqrt{n}} + \frac{X_{Sn}}{\sqrt{n}} ) ≥ 2ε]
≤ sup_{S} IP [ sup_{0≤u≤θ} \frac{X_{(S+u)n}}{\sqrt{n}} ≥ ε] + sup_{S} IP [\frac{X_{Sn}}{\sqrt{n}} ≥ ε]

≤ IP [ sup_{0≤v≤(N+θ)n} \frac{X_{v}}{\sqrt{n(N + θ)}} ≥ ε\sqrt{N + θ}] + IP [ sup_{0≤v≤Nn} \frac{X_{v}}{\sqrt{nN}} ≥ ε\sqrt{N}] .

And by lemma 2, Aldous’ criterion is fulfilled.

To obtain tightness, we also have to check the remaining condition:

For all N ∈ IN^{∗}, ε > 0, there exists a n_{0} ∈ IN^{∗} and a K ∈ IR^{+}

such that for all n ≥ n_{0} IP (sup_{t≤N} |Y_{t}^{n}| > K) ≤ ε.

By an analogous reasoning:

IP (sup_{t≤N} |\sqrt{\frac{−2β^{3}}{\overline{δ}n}} (∫_{0}^{nt} (X_{u} + \frac{δ_{u}}{2β} ) du)| > 2K)

≤ IP ((sup_{t≤N} |−\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} \frac{\sqrt{X_{u}}}{β} dB_{u}| + sup_{t≤N} |\sqrt{\frac{−2β^{3}}{\overline{δ}n}} \frac{X_{tn} − X_{0}}{2β} |) > 2K)

≤ IP (sup_{t≤N} |−\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} \frac{\sqrt{X_{u}}}{β} dB_{u}| > K)  (8)

+ IP (sup_{t≤N} |\sqrt{\frac{−2β^{3}}{\overline{δ}n}} \frac{X_{tn} − X_{0}}{2β} | > K) .  (9)

For the first term (8), we can apply the martingale inequality:

IP (sup_{t≤N} |−\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} \frac{\sqrt{X_{u}}}{β} dB_{u}| > K)

≤ \frac{−2β}{\overline{δ}nK^{2}} ∥∫_{0}^{Nn} \sqrt{X_{u}}dB_{u}∥_{2}^{2}

≤ \frac{−2β}{\overline{δ}nK^{2}} ∫_{0}^{Nn} IE[X_{u}]du
≤ \frac{−2β}{\overline{δ}nK^{2}} ∫_{0}^{Nn} e^{2βs}X_{0}ds − \frac{2β}{\overline{δ}nK^{2}} ∫_{0}^{Nn} e^{2βs} ∫_{0}^{s} e^{−2βu}IE[δ_{u}]du

where we have applied remark (4) of section 2. By Fubini’s theorem, we find:

IP (sup_{t≤N} |−\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} \frac{\sqrt{X_{u}}}{β} dB_{u}| > K)

≤ \frac{−2βNX_{0}}{\overline{δ}K^{2}} − \frac{2β}{\overline{δ}nK^{2}} ∫_{0}^{Nn} e^{−2βu}IE[δ_{u}]du ∫_{s}^{nN} e^{2βs}ds
≤ \frac{−2βNX_{0}}{\overline{δ}K^{2}} − \frac{2β}{\overline{δ}nK^{2}} ∫_{0}^{Nn} IE[δ_{u}]du

≤ \frac{−2βNX_{0}}{\overline{δ}K^{2}} − \frac{2βN}{\overline{δ}K^{2}} (\frac{1}{nN} ∫_{0}^{Nn} IE[δ_{u}^{2}]du)^{1/2}

≤ \frac{−2βN(X_{0} + \sqrt{k})}{\overline{δ}K^{2}}

since by hypothesis sup_{t≥1} \frac{1}{t} ∫_{0}^{t} IE [δ_{u}^{2}] du ≤ k with k a constant.

We conclude that for all N ∈ IN^{∗}, ε > 0, there exists a n_{0} ∈ IN^{∗} and a K ∈ IR^{+} such that for all n ≥ n_{0}:

IP (sup_{t≤N} |−\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} \frac{\sqrt{X_{u}}}{β} dB_{u}| > K) ≤ ε/2.

As regards the second term (9), it remains to be shown that for all N ∈ IN^{∗}, ε > 0, there exists a n_{0} ∈ IN^{∗} and a K ∈ IR^{+} such that for all n ≥ n_{0}:

IP (sup_{t≤N} |\sqrt{\frac{−2β^{3}}{\overline{δ}n}} \frac{X_{tn} − X_{0}}{2β} | > K) < ε/2.

We will transform this condition to a previous result:

IP (sup_{t≤N} |\sqrt{\frac{−2β^{3}}{\overline{δ}n}} \frac{X_{tn} − X_{0}}{2β} | > K)

≤ IP ( sup_{t≤nN} \frac{X_{t}}{\sqrt{nN}} > K′) + IP (\frac{X_{0}}{\sqrt{n}} > K”) .

Again by lemma 2, the first term is smaller than ε/4 for K′ large enough. The convergence of the second term is a triviality.

q.e.d.

This theorem is very useful in deducing results about the limit-distribution of the long-term return. In the following section, we will give some applications.

First, we want to improve theorem 1 in the sense that we search for conditions which are measure-invariant. Since in Finance the measure is often transformed to obtain risk-neutral measures, measure-invariant hypothesis are important. Therefore, we will give an alternative version of theorem 1, in which the assumptions are not expressed in function of moments but in which the boundedness in L^{0} of a convex hull is needed. As boundedness in L^{0} is a measure-invariant property, this is an improvement.

Before stating theorem 2, we give a direct corollary of theorem 1, which is needed in the proof of theorem 2:

Corollary 2 Suppose that a probability space (Ω, (F_{t})_{t≥0}, IP) is given and that a stochastic process X : Ω × IR^{+} → IR^{+} is defined by the stochastic differential equation

dX_{s} = (2βX_{s} + δ_{s})ds + 2\sqrt{X_{s}}dB_{s} ∀s ∈ IR^{+}

with (B_{s})_{s≥0} a Brownian motion with respect to (F_{t})_{t≥0} and β < 0. Let us make the following assumptions about the adapted and measurable process δ : Ω × IR^{+} → IR^{+}:

• \frac{1}{s} ∫_{0}^{s} δ_{u}du →^{a.e.} \overline{δ} with \overline{δ} ∈ IR_{0}^{+}

• There exists a function ψ such that lim_{x→∞} ψ(x)/x^{2} = ∞ and such that there exists a constant K so that

sup_{t≥1} \frac{1}{t} ∫_{0}^{t} IE [ψ(δ_{u})] du ≤ K.  (10)

Under these conditions, the following convergence in distribution holds:

(\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} (X_{u} + \frac{δ_{u}}{2β} ) du)_{t≥0} ⟶^{L} (B′_{t})_{t≥0}

with (B′_{t})_{t≥0} a Brownian motion.

Proof: We will check that the conditions of theorem 1 are fulfilled.

First, we prove that sup_{t≥1} \frac{1}{t} ∫_{0}^{t} IE [δ_{u}^{2}] du ≤ k with k a constant independent of t. Since lim_{x→∞} ψ(x)/x^{2} = ∞, there exists a c large enough such that x^{2} ≤ ψ(x)ε_{c} holds for all x ≥ c, with ε_{c} a constant. Consequently:

\frac{1}{t} ∫_{0}^{t} IE [δ_{u}^{2}] du = \frac{1}{t} ∫_{0}^{t} IE [δ_{u}^{2}11_{(δ_{u}<c)}] du + \frac{1}{t} ∫_{0}^{t} IE [δ_{u}^{2}11_{(δ_{u}≥c)}] du

≤ c^{2} + \frac{1}{t} ∫_{0}^{t} IE [ψ(δ_{u})ε_{c}] du
≤ c^{2} + ε_{c}K = k

where k is a constant independent of t.

Analogously, we prove that for all a ∈ IR^{+} lim_{t→∞} \frac{1}{t} ∫_{t−a}^{t} IE [δ_{u}^{2}] du = 0. Indeed,

\frac{1}{t} ∫_{t−a}^{t} IE [δ_{u}^{2}] du = \frac{1}{t} ∫_{t−a}^{t} IE [δ_{u}^{2}11_{(δ_{u}<c)}] du + \frac{1}{t} ∫_{t−a}^{t} IE [δ_{u}^{2}11_{(δ_{u}≥c)}] du

≤ \frac{c^{2}a}{t} + \frac{1}{t} ∫_{t−a}^{t} IE [ψ(δ_{u})ε_{c}] du

For a given ε, we choose c such that ε_{c} \frac{1}{t} ∫_{0}^{t} IE [ψ(δ_{u})] du ≤ ε/2 and for this fixed c, we can choose t large enough such that \frac{c^{2}a}{t} ≤ ε/2.

q.e.d.

By using techniques from functional analysis and by stopping and localisation, we find theorem 2 which has measure-invariant conditions.

Theorem 2 Suppose that a probability space (Ω, (F_{t})_{t≥0}, IP) is given and that all martingales with respect to the filtration (F_{t})_{t≥0} are continuous. Assume that a stochastic process X : Ω × IR^{+} → IR^{+} is defined by the stochastic differential equation on IR^{+}

dX_{s} = (2βX_{s} + δ_{s})ds + 2\sqrt{X_{s}}dB_{s}

with (B_{s})_{s≥0} a Brownian motion with respect to (F_{t})_{t≥0} and β < 0. Let us make the following assumptions about the adapted and measurable process δ : Ω × IR^{+} → IR^{+}:

• \frac{1}{s} ∫_{0}^{s} δ_{u}du →^{a.e.} \overline{δ} with \overline{δ} ∈ IR_{0}^{+}

• there exists a function ψ such that

lim_{x→∞} \frac{ψ(x)}{x^{2}} = ∞

and such that the convex hull of the set \{\frac{1}{t} ∫_{0}^{t} ψ(δ_{u})du | t ≥ 1\}, namely

conv(\frac{1}{t} ∫_{0}^{t} ψ(δ_{u})du | t ≥ 1), is bounded in L^{0}.

Under these conditions, the following convergence in distribution holds:

(\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} (X_{u} + \frac{δ_{u}}{2β} ) du)_{t≥0} ⟶^{L} (B′_{t})_{t≥0}  (11)

with (B′_{t})_{t≥0} a Brownian motion.

Proof. Since conv(\frac{1}{t} ∫_{0}^{t} ψ(δ_{u})du | t ≥ 1) is bounded in L^{0}, there exist a α > 0 and a F_{∞}-measurable function h with 0 < h ≤ 1 such that for all t:

IE [\frac{1}{t} ∫_{0}^{t} ψ (δ_{u}) du h] ≤ α.

Let us fix this function h and let us define the stopping times T_{m} by

T_{m} = inf \{t | h_{t} = IE[h | F_{t}] ≤ \frac{1}{m}\} .

Since h > 0, the probabilities IP [T_{m} = ∞] are increasing to 1. Let us define the stochastic process

δ_{u}^{m} = \{ \begin{matrix}δ_{u} & u ≤ T_{m} \\ \overline{δ} & u > T_{m}. \end{matrix}

For the stochastic process X^{m}, defined by the stochastic differential equation

dX_{s}^{m} = (2βX_{s}^{m} + δ_{s}^{m})ds + 2\sqrt{X_{s}^{m}}dB_{s},

the hypotheses of corollary 2 are fulfilled. Indeed, there is a constant k such that for all t ≥ 0

\frac{1}{t} ∫_{0}^{t} IE [ψ (δ_{u}^{m}) du] du ≤ k.

This is shown as follows:

IE [\frac{1}{t} ∫_{0}^{t} ψ (δ_{u}^{m}) du]

= IE[\frac{1}{t} ∫_{0}^{T_{m}} ψ (δ_{u}) du 11_{(T_{m}<t)}] + IE [\frac{1}{t} ∫_{T_{m}}^{t} ψ (\overline{δ}) du 11_{(T_{m}<t)}]
+ IE [\frac{1}{t} ∫_{0}^{t} ψ (δ_{u}) du 11_{(T_{m}≥qt)}] .

Since mh_{T_{m}} ≥ 1, we obtain

IE [\frac{1}{t} ∫_{0}^{t} ψ (δ_{u}^{m}) du]

≤ IE[\frac{1}{t} ∫_{0}^{T_{m}} ψ (δ_{u}) du mh_{T_{m}} 11_{(T_{m}<t)}] + IE [\frac{1}{t} ∫_{T_{m}}^{t} ψ (\overline{δ}) du 11_{(T_{m}<t)}]
+ IE [\frac{1}{t} ∫_{0}^{t} ψ (δ_{u}) du mh_{T_{m}} 11_{(T_{m}≥t)}] .

If we take the conditional expectation and recall that h_{T_{m}} = IE[h | F_{T_{m}}], we find

IE [\frac{1}{t} ∫_{0}^{t} ψ (δ_{u}^{m}) du]

≤ ψ (\overline{δ}) IP [T_{m} < t] + IE[mh \frac{1}{t} ∫_{0}^{T_{m}∧t} ψ (δ_{u}) du].

Since also IE[\frac{1}{t} ∫_{0}^{t} ψ (δ_{u}) du h] ≤ α, we may conclude that for all t ≥ 1:

IE [\frac{1}{t} ∫_{0}^{t} ψ (δ_{u}^{m}) du] ≤ ψ (\overline{δ}) IP [T_{m} < t] + mα ≤ K(m)

where K is a constant depending on m but not on t. Since the hypotheses of corollary 2 are fulfilled in case of the process X^{m}, we can apply corollary 2:

(Y_{t}^{n,m})_{t≥0} = (\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} (X_{u}^{m} + \frac{δ_{u}^{m}}{2β} ) du)_{t≥0} ⟶^{L} (B′_{t})_{t≥0}

with (B′_{t})_{t≥0} a Brownian motion. Thus, for every continuous and bounded function f : C(IR^{+}, IR) ⟶ IR:

IE [f(Y^{n,m})] ⟶ ∫ fdμ

where μ denotes the Wiener measure.

In order to prove that

(Y_{t}^{n})_{t≥0} = (\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} (X_{u} + \frac{δ_{u}}{2β} ) du)_{t≥0} ⟶^{L} (B′_{t})_{t≥0}

with (B′_{t})_{t≥0} a Brownian motion, we show that for every bounded and continuous function f : C(IR^{+}, IR) ⟶ IR:

IE [f(Y^{n})] ⟶ ∫ fdμ.

This is quite standard:

|IE [f(Y^{n})] − ∫ fdμ|
≤ |IE [f(Y^{n,m})] − ∫ fdμ| + |IE [f(Y^{n})] − IE [f(Y^{n,m})]|
≤ |IE [f(Y^{n,m})] − ∫ fdμ| + IE [|f(Y^{n}) − f(Y^{n,m})|]
≤ |IE [f(Y^{n,m})] − ∫ fdμ| + 2∥f∥_{∞}IP [∃u Y_{u}^{n,m} ≠ Y_{u}^{n}]

Since IP [T_{m} = ∞] ↑ 1 for m going to infinity, also IP [∃u X_{u}^{n,m} ≠ X_{u}^{n}] → 0 and consequently IP [∃u Y_{u}^{n,m} ≠ Y_{u}^{n}] → 0. Thus, for every ε > 0, we can fix m such that the second term is smaller than ε/2. Since (Y_{t}^{n,m})_{t≥0} converges to a Brownian motion, there exist n large enough that the first term is bounded by ε/2.

q.e.d.

An immediate consequence of this corollary is that the theorem holds if there exists ε > 0 such that conv(\frac{1}{t} ∫_{0}^{t} δ_{u}^{2+ε}du | t ≥ 1) is bounded in L^{0}.

4 Approximations.

In this section, we propose some applications of theorem 1. We consider the possibility of approximating ∫_{0}^{t} r_{u}du. It is very interesting to know an approximation of ∫_{0}^{t} r_{u}du since this integral appears in bond prices, accumulation and discounting factors, etcetera.

As an example, we apply theorem 1 to the generalized Cox–Ingersoll–Ross two-factor model (1) and we approximate ∫_{0}^{t} r_{u}du by the sum of two terms: the constant reversion level γ^{∗} multiplied by t and a scaled Brownian motion which represents random changes. Afterwards, we evaluate the approximation in case of the Cox–Ingersoll–Ross model.

For a stochastic process (X_{u})_{u≥0} given by dX_{u} = (2βX_{u}+δ_{u})du+2\sqrt{X_{u}}dB̃_{u} and satisfying the assumptions of theorem 1, the following convergence in law to a Brownian motion (B_{t})_{t≥0} was shown:

(\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} (X_{u} + \frac{δ_{u}}{2β} ) du)_{t≥0} ⟶^{L} (B_{t})_{t≥0}

This result inspires us to approximate

\sqrt{\frac{−2β^{3}}{\overline{δ}n}} ∫_{0}^{nt} (X_{u} + \frac{δ_{u}}{2β} ) du

by B_{t} for n large enough.

Using the scaling property of Brownian motion, namely \sqrt{n}B_{t/n} =^{d} B_{t}, we can estimate

∫_{0}^{t} X_{u}du by − ∫_{0}^{t} \frac{δ_{u}}{2β} du + \sqrt{\frac{\overline{δ}}{−2β^{3}}} B_{t}.

As \frac{1}{t} ∫_{0}^{t} δ_{u}du ⟶^{a.e.} \overline{δ}, we obtain for t large enough

∫_{0}^{t} X_{u}du ≈ −\frac{\overline{δ}}{2β} t + \sqrt{\frac{\overline{δ}}{−2β^{3}}} B_{t}.

In “Long-term returns in stochastic interest rate models.” [2], it is proved that under the hypothesis of theorem 1

\frac{1}{t} ∫_{0}^{t} X_{u}du ⟶^{a.e.} −\frac{\overline{δ}}{2β} .

Therefore, we approximate ∫_{0}^{t} X_{u}du by the constant convergence a.e. limit of the long-term return times t plus a scaled Brownian motion.

Let us now study the two-factor model (1), which is a generalisation of the Cox–Ingersoll–Ross model (see also [2]):

dr_{t} = κ(γ_{t} − r_{t})dt + σ\sqrt{r_{t}}dB′_{t}
dγ_{t} = κ̃(γ^{∗} − γ_{t})dt + σ̃\sqrt{γ_{t}}dB̃_{t}

with κ, κ̃ > 0; γ^{∗}, σ and σ̃ positive constants and (B′_{t})_{t≥0} and (B̃_{t})_{t≥0} two Brownian motions. These Brownian motions may be correlated in an arbitrarily way, they even may have a random correlation. As mentioned in the introduction, this is in contrast with the assumptions of most papers: most of the authors suppose for technical reasons that the Brownian motions are uncorrelated or have a constant correlation.

Since we are interested in the convergence of the long-term return \frac{1}{t} ∫_{0}^{t} r_{u}du, we verify if theorem 1 is applicable:

If we make the transformation X_{u} = \frac{4}{σ^{2}} r_{u}, then X_{u} satisfies the stochastic differential equation

dX_{u} = (\frac{4κγ_{u}}{σ^{2}} + 2 (−\frac{κ}{2}) X_{u})du + 2\sqrt{X_{u}}dBu^{′}.

In terms of theorem 1:

dX_{u} = (δ_{u} + 2βX_{u})du + 2\sqrt{X_{u}}dBu^{′}

with

• β = −\frac{κ}{2} < 0

• δ_{u} = \frac{4κγ_{u}}{σ^{2}} ∀u ∈ IR^{+}, which is measurable and adapted since γ is a CIR square root process.

We check the hypothesis of theorem 1:

It is easy to show that (see e.g. [2]):

\frac{1}{s} ∫_{0}^{s} δ_{u}du ⟶^{a.e.} \frac{4κγ^{∗}}{σ^{2}} with \frac{4κγ^{∗}}{σ^{2}} > 0.

Since γ follows a CIR square root process, its second moment is given by:

IE [γ_{s}^{2}] = (2κ̃γ̃ + σ̃^{2}) [\frac{γ_{0} − γ̃}{κ̃} e^{−κ̃s} + \frac{γ̃}{2κ̃}] + e^{−2κ̃s} [γ_{0}^{2} + \frac{γ̃ − γ_{0}^{2}}{2κ̃} (2κ̃γ̃ + σ̃^{2})] .

As δ_{s} equals \frac{4κγ_{s}}{σ^{2}} , a technical calculation leads to the results that for all a ∈ IR^{+}

lim_{t→∞} \frac{1}{t} ∫_{t−a}^{t} IE [δ_{s}^{2}] ds = 0

and that

sup_{t≥1} \frac{1}{t} ∫_{0}^{t} IE [δ_{s}^{2}] ds ≤ k

with k a constant independent of t. Consequently, theorem 1 is applicable and one finds that

(\frac{κ}{σ\sqrt{γ^{∗}n}} ∫_{0}^{nt}(r_{u} − γ_{u})du)_{t≥0} ⟶^{L} (B_{t})_{t≥0}.

This is not a trivial result since r and γ may have an arbitrary, random correlation.

Repeating the reasoning above, we approximate

∫_{0}^{t} r_{u}du ≈ ∫_{0}^{t} γ_{u}du + \sqrt{\frac{σ^{2}γ^{∗}}{κ^{2}}} B_{t}

or

∫_{0}^{t} r_{u}du ≈ γ^{∗}t + \sqrt{\frac{σ^{2}γ^{∗}}{κ^{2}}} B_{t}

for t large enough.

This is a very easy approximation. We approximate the surface under the instantaneous interest rate curve by the rectangle with height the constant reversion level γ^{∗} plus a factor which represents random changes by using a scaled Brownian motion.

In order to evaluate this estimator, we have a look at the moments of the estimator. They do not equal those of ∫_{0}^{t} r_{u}du, but they are the same asymptotically. Moreover, if we assume that r_{t} follows the CIR square root process

dr_{t} = κ(γ^{∗} − r_{t})dt + σ\sqrt{r_{t}}dBt^{′}

with γ^{∗} as well as κ and σ constants and if we assume that r_{0} is distributed according to its stationary distribution Γ(\frac{2κγ^{∗}}{σ^{2}} , \frac{2κ}{σ^{2}}) , we find that the expectation values are equal for all t.

Indeed, by Fubini and the expectation value IE[r_{u}] we obtain

IE [∫_{0}^{t} r_{u}du] = ∫_{0}^{∞} IE_{x} [∫_{0}^{t} r_{u}du] f_{R_{0}}(x)dx
= ∫_{0}^{∞} [∫_{0}^{t} (xe^{−κs} + γ^{∗}(1 − e^{−κs})) ds] f_{R_{0}}(x)dx
= \frac{1 − e^{−κt}}{κ} ∫_{0}^{∞} xf_{R_{0}}(x)dx + γ^{∗}t − γ^{∗} \frac{1 − e^{−κt}}{κ}

where we have used that f_{R_{0}} = Γ(\frac{2κγ^{∗}}{σ^{2}} , \frac{2κ}{σ^{2}}) is a density function. If we recall that ∫_{0}^{∞} xf_{R_{0}}(x)dx = IE[R_{0}] = \frac{2κγ^{∗}/σ^{2}}{2κ/σ^{2}} = γ^{∗}, the result follows:

IE [∫_{0}^{t} r_{u}du] = γ^{∗}t = IE [γ^{∗}t + \frac{σ\sqrt{γ^{∗}}}{κ} B_{t}] .

In case of the CIR square root process, an explicit formula for the bond price is given by Pitman–Yor [6] and Cox–Ingersoll–Ross [1]. From Pitman-Yor [6], we recall that the bond price has been given by

IE_{x} [exp (− ∫_{0}^{t} r_{u}du)]

= \frac{exp \{−\frac{x}{σ^{2}} w \frac{1+κ/w coth(wt/2)}{coth(wt/2)+κ/w}\} e^{κx/σ^{2}}e^{κ^{2}γ^{∗}t/σ^{2}}}{(cosh(wt/2) + κ/w sinh(wt/2))^{\frac{2κγ^{∗}}{σ^{2}}}}

with w = \sqrt{κ^{2} + 2σ^{2}} and x = r_{0}.

By our approximation, we obtain

IE_{0} [exp (−γ^{∗}t − \sqrt{\frac{σ^{2}γ^{∗}}{κ^{2}}} B_{t})] = exp (−γ^{∗}t + \frac{σ^{2}γ^{∗}}{2κ^{2}} t)

If we make the quotient of the two results to the power 1/t, for t going to infinity, we should obtain the value 1. But:

lim_{t→∞} \frac{IE_{0} [exp (−γ^{∗}t − \sqrt{\frac{σ^{2}γ^{∗}}{κ^{2}}} B_{t})]^{1/t}}{IE_{x} [exp (− ∫_{0}^{t} r_{u}du)]^{1/t}}

= lim_{t→∞} \frac{exp \{γ^{∗}(\frac{σ^{2}}{2κ^{2}} − 1)\}}{exp (\frac{κ^{2}γ^{∗}}{σ^{2}} ) /2 cosh (\frac{κγ^{∗}ω}{σ^{2}} )} ≠ 1

The approximation to a Brownian motion is too slow.

In practice however, the period of interest usually is shorter than 40 or 50 years. Using the parameters estimated within the empirical work of Chan, Karolyi, Longstaff and Sanders [12], namely κ = 0.2339, γ^{∗} = 0.0808 and σ = 0.0854, we have calculated the exact bond prices and the proposed approximation.

As the approximation does not depend on the present interest rate r_{0}, we do not need the knowledge of r_{0} to calculate them. The approximations for the prices of bonds with duration t year can be found in the first column of table 1. The other columns collect the quotients of the exact bond prices by the approximations for different values of the present interest rate r_{0}.

Clearly, three situations are possible: for r_{0} < γ^{∗}, the approximation underestimates the bond price; for r_{0} > γ^{∗} there is an overestimation and for r_{0} ≈ γ^{∗}, the fit is fairly satisfying. In general, the quality of the approximation of a long-term zero coupon depends on the value of the parameters.

However, if the objective is to approximate the distribution of the long-term return of an investment made at time 0, it is appropriate to approximate ∫_{0}^{t} r_{u}du by a scaled Brownian motion with drift for t going to infinity. A lot of authors previously proposed Wiener models. The argumentation of using Wiener models is that Central Limit Theorems are applicable on long-term.

As an illustration, we have simulated γ^{∗} + \sqrt{\frac{κ^{2}γ^{∗}}{40σ^{2}}} B_{1} and \frac{1}{40} ∫_{0}^{40} r_{u}du where (r_{u})_{u≥0} specifies the CIR process from above with the parameters taken from [12]. For both the long-term return and the approximation, we have calculated the probabilities of being in the intervals [ln (1 + (i − 1)/100) , ln (1 + i/100)] for 1 ≤ i ≤ 18. Figure 1 shows the histogram of the approximation ( ) which is independent of r_{0} and the histograms of the long-term return for r_{0} = 0.07 (- - -), for r_{0} = 0.04 (....) and for r_{0} = 0.1 ( . .).

Table 1: Bond prices: The exact values and the approximations.

t Approximation  Exact/Approximation
r_{0} = 0.07  r_{0} = 0.04  r_{0} = 0.1
1  0.92735916  1.00423575  1.03152406  0.97784472
5  0.68586677  1.00988853  1.10162997  0.92578721
10  0.47041327  1.00885379  1.12789547  0.90237617
15  0.32264084  1.00619245  1.13247085  0.89399487
20  0.22128864  1.00320458  1.13114345  0.88973624
25  0.15177453  1.00014389  1.12823772  0.88659304
30  0.10409711  0.99707115  1.12491739  0.88375461
35  0.07139675  0.99400234  1.12149405  0.88100392
40  0.04896866  0.99094135  1.11805093  0.87828267

References

[1] J.C. Cox, J.E. Ingersoll and S.A. Ross, A theory of the term structure of interest rates, Econometrica 58 (1985), 385-407.

[2] G. Deelstra and F. Delbaen, Long-term interest rate models, forthcoming.

[3] S.M. Schaefer and E.S. Schwartz, A two-factor model of the term structure: An approximate analytical solution, Journal of Financial and Quantitive Analysis 19 (1984), 413-424.

[4] J. Hull and A. White, Pricing Interest-Rate-Derivative Securities, The Review of Financial Studies, Vol. 3, nr 4 (1990), 573-592.

[5] F.A. Longstaff and E.S. Schwartz, Interest rate volatility and the term structure: a two-factor general equilibrium model, The Journal of Finance 47, nr 4 (1992), 1259-1282.

[6] J. Pitman and M. Yor, A decomposition of Bessel Bridges, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 59 (1982), 425-457.

[7] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer-Verlag Berlin, Heidelberg, New York, 1991.

[8] G. Deelstra and F. Delbaen, Existence of Solutions of Stochastic Differential Equations related to the Bessel process, forthcoming.

[9] W. Feller, An Introduction to Probability Theory and its Applications, Volume II, John Wiley and Sons, 1971.

[10] I. Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus, Springer-Verlag New York, Berlin, Heidelberg, 1988.

[11] J. Jacod and A.N.Shiryaev, Limit Theorems for Stochastic Processes., Springer-Verlag, Berlin-Heidelberg, New-York, 1987.

[12] K.C. Chan, G.A. Karolyi, F.A. Longstaff and A.B. Sanders, An Empirical Comparison of Altermative Models of the Short-Term Interest Rate, Journal of Finance, 47 (1992), 1209-1227.

[13] G. Deelstra, Ph.D., (in preparation).