Finance and Stochastics Manuscript-Nr. (will be inserted by hand later)

Invariant measures for the Musiela equation with deterministic diffusion term^{⋆}

Tiziano Vargiolu

Dipartimento di Matematica Pura ed Applicata, Universitá di Padova, via Belzoni 7, I-35131

Padova, Italy, (e-mail: vargiolu@galileo.math.unipd.it)

Abstract. In this article the forward rates equation of the Musiela model is analysed. The equation is studied in the Sobolev spaces H_{γ}^{1}(ℝ^{+}) and H^{1}(ℝ^{+}). Explicit mild solutions and equivalent conditions for the existence and uniqueness of invariant measures are presented.

Key words: term structure of interest rates, stochastic partial differential equations, mild solutions, invariant measures, C^{0}-semigroups in Hilbert spaces

JEL classification: E43

Mathematics Subject Classification (1991): 60G15, 60H15, 60H30, 90A09

1. Introduction

The aim of this work is to study in a rigorous way some asymptotic properties of the Musiela equation in the Gaussian case. Our methods are based on the theory of stochastic equations in separable Hilbert spaces by Da Prato and Zabczyk [5], and we study invariant measures for the equation and weak convergence of the solution to such measures.

The Musiela model, developed by Musiela himself and other authors in [3], [4], [8], [9] and [12], is based on the family of rates (r(t, x))_{t,x≥0}, where r(t, x)

⋆ The author wishes to thank Franco Flandoli and Jerzy Zabczyk for their invaluable advice, anonymous referees for useful remarks and Frank Holzwarth of the Editorial/Book Production of Springer Verlag for his assistance in using Springer’s macro package. The author also thanks the Scuola Normale Superiore of Pisa, which was the former affiliation of the author, and where this work was written and revised up to the final version, and the Laboratory of Probability of the University of Paris VI, which hosted him during the first revision of this work. Manuscript received: June 1996; final revision received: November 1998

represents the forward rate prevailing in t for the time t + x. This model is a reparametrization of the well-known Heath-Jarrow-Morton (HJM) model (see [10]), such that r(t, x) = f(t, t + x), where f(t, T) are the forward rates analysed in [10]. The Musiela reparametrization is coherent with other forward rates models (see [3] and [14]) and allows us to consider the forward curve r(t, ·) as a Markov process in a suitable function space, while in the HJM model the state space changes with time.

Let us now introduce more in detail the Musiela model. We suppose that we have a probability space (Ω, F, ℙ) endowed with the filtration (F_{t})_{t≥0}, and that the price at time t of a bond expiring at time T is represented by the process (B(t, T))_{t∈[0,T]}. Furthermore, we suppose that we have a random field (r(t, x))_{t,x≥0} such that r(t, x) is F_{t}-measurable and B(t, T) = exp(− ∫_{0}^{T −t} r(t, u) du). The quantity r(t, x) is called instantaneous forward rate at time t for the maturity t + x and represents the rate at time t at which one can enter a forward contract for the time t + x for a short (infinitesimal) period of time. The actualized price at time t of a bond expiring at time T is given by B̃(t, T) = B(t, T)/β(t), where β(t) = exp(∫_{0}^{t} r(u, 0) du) is the actualizing factor. Now we add the hypotheses that there exists a probability measure ℚ equivalent to ℙ, called risk-neutral probability under which the processes (B̃(t, T))_{t∈[0,T]} are martingales for all T > 0, and that there exists a k-dimensional standard ℚ-Brownian motion (W_{t})_{t≥0} adapted to (F_{t})_{t≥0} such that (r(t, x))_{t,x≥0} satisfies the stochastic partial differential equation

\{ \begin{matrix}dr(t, x) = (\frac{∂}{∂x}r(t, x) + ∑_{n=1}^{k} τ_{n}(t, x) ∫_{0}^{x} τ_{n}(t, u) du) dt + ∑_{n=1}^{k} τ_{n}(t, x) dW_{t}^{n} , \\ r(0, ·) ∈ AC(ℝ^{+}) , \end{matrix}

(1)

where the (τ_{n}(t, x))_{t,x≥0}, n = 1, …, k are random fields such that τ_{n}(t, x) is F_{t}-measurable and such that Eq. (1) is well defined for all maturities x ≥ 0. This model is justified by the following result that can be found both in [4] as in [12].

Theorem 1. If ∀t ∈ ℝ^{+} the application x → r(t, x) belongs to AC(ℝ^{+}) ℚ-a.s. and r satisfies Eq. (1), then for all T > 0 the process (B̃(t, T))_{t∈[0,T]} is a local martingale with respect to ℚ.

Our aim is to study asymptotic properties of Eq. (1) in the simple case when the diffusion term τ is deterministic and the process r takes values in a separable Hilbert space H contained in AC(ℝ^{+}). This leads to a model that is Gaussian in the forward rates r(t, x) and it is analytically tractable. In fact, in this case we can find an explicit solution to Eq. (1), given by Eq. (5) and usually called mild solution. First we show that this mild solution, under technical conditions, is well defined in some particular Hilbert spaces H ⊆ AC(ℝ^{+}), and is unique in the class of weak solutions of Eq. (1) (see [5]). Then we will study (when possible) the characterization of the invariant measures for the process of the solution. The study of invariant measures is a first step in studying asymptotic properties of a system; in this case, the existence of invariant measures means that the distribution of the forward curve converges, as t → +∞, to an invariant measure on H (see Theorem 7).

Let us restate the problem in a more systematic way. We rewrite Eq. (1) as a Langevin equation in a separable Hilbert space H (that we will specify later):

\{ \begin{matrix}dr_{t} = (Ar_{t} + c) dt + dW_{t} , \\ r_{0} ∈ L^{2}(Ω, F_{0}, ℚ; H) , \end{matrix}  (2)

where A = \frac{∂}{∂x}, c(x) = ∑_{n=1}^{∞} τ_{n}(x) ∫_{0}^{x} τ_{n}(u) du and dW_{t} stands for ∑_{n=1}^{∞} τ_{n} dW_{t}^{n} and is an H-valued Brownian motion. This means that, if we indicate with Q = ∑_{n=1}^{∞} τ_{n} ⊗ τ_{n} the covariance operator of W, we have that Tr Q = ∑_{n=1}^{∞} ∥τ_{n}∥_{H}^{2} < +∞. In practical terms, this situation corresponds to the well accepted idea that there are infinitely many sources of randomness in the model, but only a few “principal components” are significant, because the intensity of the noises decreases rather quickly (see [11] for another kind of genuine infinite dimensional dynamics of the forward rates). In order to obtain the usual case of a k-dimensional driving Brownian motion (see for examples [1], [2], [3], [4]), it is sufficient to suppose that τ_{n} = 0 for all n > k. By a solution of Eq. (1) (or better of Eq. (2)), we will mean, as in [4] and [12], a so called mild solution (see [5]), given by

r_{t} = S_{t}r_{0} + ∫_{0}^{t} S_{t−u}c du + ∫_{0}^{t} S_{t−u} dW_{u} ,  (3)

where (S_{t})_{t} is the C^{0}-semigroup in H generated by A (see [13]), which in this case is the translation semigroup defined by

(S_{t}f)(x) = f(x + t) ∀t, x ≥ 0 .

To specify exactly in which space H we will study Eq. (1), we define the Sobolev spaces

H_{γ}^{k}(ℝ^{+}) = {u : ℝ^{+} → ℝ | ∃u′, …, u^{(k)} and u, u′, …, u^{(k)} ∈ L_{γ}^{2}(ℝ^{+})} ,

where u′, u′′, …, u^{(k)} indicate respectively the first, second, …, k-th weak derivative, and

L_{γ}^{p}(ℝ^{+}) = {u : ℝ^{+} → ℝ | u meas. and s.t. ∫_{0}^{+∞} u^{p}(x)e^{−γx} dx < +∞} .

We notice that H_{0}^{k} and L_{0}^{p} coincide with the usual definitions of H^{k} and L^{p}, and we recall that H_{γ}^{k} is a Hilbert space with respect to the scalar product

⟨f, g⟩_{H_{γ}^{k}} = ∑_{i=0}^{k}⟨f^{(i)}, g^{(i)}⟩_{L_{γ}^{2}} = ∑_{i=0}^{k} ∫_{0}^{+∞} f^{(i)}(x)g^{(i)}(x)e^{−γx} dx .  (4)

Furthermore H_{γ}^{1} ⊆ H_{γ′}^{1} ∀γ < γ′, and H_{γ}^{1} ⊆ AC(ℝ^{+}) ∀γ ≥ 0.

In view of Theorem 1, we will study Eq. (1) in the spaces H_{γ}^{1}(ℝ^{+}) for γ ≥ 0, which are contained in the space AC(ℝ^{+}). If we studied the equation in L_{γ}^{2}(ℝ^{+}) (as [8] and [9] do), we would obtain an invariant measure in the whole space L_{γ}^{2}(ℝ^{+}), and it would be difficult to prove that the measure is concentrated on AC(ℝ^{+}). We start studying the Musiela equation in the spaces H_{γ}^{1}(ℝ^{+}) for γ > 0. In these spaces we find infinitely many Gaussian invariant measures (among which there are the ones found in [12]), but also other non-Gaussian ones, and we show that if the initial r_{0} is deterministic, then the solution r_{t} converges weakly as t → +∞ to a Gaussian invariant measure, which depends on the initial forward curve r_{0}. The properties of the equation in H^{1}(ℝ^{+}) are completely different, and for this reason we study this specific situation separately. In H^{1}(ℝ^{+}) flat term structures r(x) = const. ≠ 0 are not allowed and the forward curves must converge to 0 for x → +∞. This means that H^{1}(ℝ^{+}) does not contain the simplest forward curves. Moreover, in this space we find only one invariant measure, that is Gaussian. This is another drawback, because the invariant measure does not depend on the initial r_{0}. We conclude our work by summarizing these and other financial remarks and giving some final comments in a specific section.

Our work is organized as follows. In Section 2 we study the Musiela equation in the space H_{γ}^{1}(ℝ^{+}) for γ > 0. Section 3 is specifically devoted to the Musiela equation in the space H^{1}(ℝ^{+}). Section 4 contains the conclusions of our work.

2. The forward rate equation in the space H_{γ}^{1}

We first study the case when the Hilbert space is H_{γ}^{1}(ℝ^{+}), and start by presenting some sufficient conditions under which the mild solution of Eq. (1) is well defined in H_{γ}^{1}(ℝ^{+}). These conditions will be also necessary to obtain the existence of invariant measures in the same space.

Theorem 2. If ∑_{n=1}^{∞} ∥τ_{n}∥_{H_{γ}^{1}}^{2} < +∞, ∑_{n=1}^{∞} ∥τ_{n}∥_{H^{1}}^{4} < +∞, ∑_{n=1}^{∞} ∥τ_{n}∥_{L_{γ}^{4}}^{4} < +∞, and r_{0} ∈ L^{2}(Ω, F_{0}, ℚ; H), then the mild solution of Eq. (1) in H_{γ}^{1}(ℝ^{+}), given by

r_{t}(x) = r_{0}(x + t) + ∑_{n=1}^{∞} ∫_{0}^{t} τ_{n}(x + t − u) (∫_{0}^{x+t−u} τ_{n}(v) dv) du +

+ ∫_{0}^{t} τ_{n}(x + t − u) dW_{u}^{n} =

= r_{0}(x + t) + \frac{1}{2} ∑_{n=1}^{∞} ((∫_{0}^{x+t} τ_{n}(u) du)^{2} − (∫_{0}^{x} τ_{n}(u) du)^{2}) +

+ ∑_{n=1}^{∞} ∫_{0}^{t} τ_{n}(x + t − u) dW_{u}^{n} ,  (5)

is well defined. Moreover, it is the unique solution in the class of weak solutions (see [5]). The mild solution is a Gaussian process with mean

𝔼[r_{t}(x)] = 𝔼[r_{0}(x + t)] + \frac{1}{2} ∑_{n=1}^{∞} ( (∫_{0}^{x+t} τ_{n}(u) du)^{2} − (∫_{0}^{x} τ_{n}(u) du)^{2} ) (6)

and covariance

Cov (r_{t}(x), r_{v}(y)) = Cov (r_{0}(t + x), r_{0}(v + y)) +
+ ∑_{n=1}^{∞} ∫_{0}^{t} τ_{n}(x + t − u)τ_{n}(y + v − u) du .  (7)

Proof. Following [5], Theorem 5.4, p. 121, it is sufficient to check that Tr Q < +∞, c ∈ H, r_{0} ∈ L^{2}(Ω, F_{0}, ℚ; H) and that A generates a C^{0}-semigroup (S_{t})_{t} in H_{γ}^{1}. The first condition is equivalent to ∑_{n=1}^{∞} ∥τ_{n}∥_{H_{γ}^{1}}^{2} < +∞. Now we observe that the weak derivative of c is

c′(x) = ∑_{n=1}^{∞} (τ′_{n}(x) ∫_{0}^{x} τ_{n}(u) du + τ_{n}^{2}(x)) .  (8)

If we impose ∑_{n=1}^{∞} ∥τ_{n}∥_{L^{2}}^{4} < +∞, then, from Jensen’s and Hölder’s inequalities, it follows that c ∈ H_{γ}^{1}. To prove the last condition, we will use the following well known fact, whose proof we omit:

Lemma 3. For all γ ≥ 0, the translation semigroup (S_{t})_{t} is a C^{0}-semigroup in H_{γ}^{1}(ℝ^{+}), having as infinitesimal generator A, which has domain equal to H_{γ}^{2}(ℝ^{+}).

In view of Lemma 3 and Eq. (3), the theorem is proved. □

Now we analyse the problem of finding invariant measures. In the following, N(b, Q) will indicate the Gaussian measure in H_{γ}^{1}(ℝ^{+}) having as mean the function b ∈ H_{γ}^{1}(ℝ^{+}) and as covariance the operator Q : H_{γ}^{1}(ℝ^{+})×H_{γ}^{1}(ℝ^{+}) → ℝ.

Theorem 4. There exist invariant measures for Eq. (2) in the Hilbert space H_{γ}^{1} if and only if ∑_{n=1}^{∞} ∥τ_{n}∥_{H^{1}}^{2} < +∞ and ∑_{n=1}^{∞} ∥τ_{n}∥_{L_{γ}^{4}}^{4} < +∞. If these conditions hold there exists an infinite number of invariant measures. In particular, the measures N(b^{∗}(·) + b_{0}, Q_{∞}) are invariant, where b_{0} is an arbitrary real number, b^{∗} is the function given by Eq. (11), and

Q_{∞} = ∑_{n=1}^{∞} ∫_{0}^{+∞} τ_{n}(· + u) ⊗ τ_{n}(· + u) du .  (9)

In the proof we will need the following result from [6]:

Theorem 5. Let the following conditions be satisfied:

(i) sup_{t≥0} ∫_{0}^{t} Tr S_{u}QS_{u}^{∗} du < +∞ (ii) ∃b ∈ D(A) such that Ab + c = 0, and there exists an invariant measure ν

for the equation

dZ_{t} = AZ_{t} dt .  (10)

Then there exists an invariant measure for Eq. (2). In particular every invariant measure is of the form ν ∗ N(b, Q_{∞}), where Q_{∞} = ∫_{0}^{+∞} S_{u}QS_{u}^{∗} du.

Now we prove Theorem 4.

Proof. It is sufficient to check points (i) and (ii) of Theorem 5. First of all we calculate

sup_{t≥0} ∫_{0}^{t} Tr Q_{u} du = ∫_{0}^{+∞} ∑_{n=1}^{∞} ∫_{0}^{+∞} (τ_{n}^{2}(x + u) + τ_{n}^{′2}(x + u)) e^{−γx} dx du =

= ∫_{0}^{+∞} ∑_{n=1}^{∞} ∫_{u}^{+∞} (τ_{n}^{2}(x) + τ_{n}^{′2}(x)) e^{−γ(x−u)} dx du =

= ∫_{0}^{+∞} ∑_{n=1}^{∞} (τ_{n}^{2}(x) + τ_{n}^{′2}(x)) e^{−γx} ∫_{0}^{x} e^{γu} du dx =

= ∫_{0}^{+∞} ∑_{n=1}^{∞} (τ_{n}^{2}(x) + τ_{n}^{′2}(x)) \frac{1 − e^{−γx}}{γ} dx .

We notice that the condition ∑_{n=1}^{∞} ∥τ_{n}∥_{H^{1}}^{2} < +∞ is necessary and sufficient to guarantee that point (i) in Theorem 5 holds. We now check (ii). An invariant measure for Eq. (10) is given by the measure δ_{0}. Moreover, a solution of the equation Ab + c = 0 is:

b^{∗}(x) = − ∫_{0}^{x} c(u) du = −\frac{1}{2} ∑_{n=1}^{∞} (∫_{0}^{x} τ_{n}(u) du)^{2} .  (11)

To check that b^{∗}(x) ∈ D(A), we first verify that b^{∗}(x) ∈ L_{γ}^{2}:

|b^{∗}(x)| ≤ ∑_{n=1}^{∞} ∫_{0}^{x} |τ_{n}(u)| ∫_{0}^{u} |τ_{n}(v)| dv du =

= \frac{1}{2} ∑_{n=1}^{∞} (∫_{0}^{x} |τ_{n}(u)| du)^{2} ≤ \frac{1}{2}x ∑_{n=1}^{∞} ∫_{0}^{x} |τ_{n}(u)|^{2} du ≤ x ∑_{n=1}^{∞} ∥τ_{n}∥_{L^{2}}^{2} ,

so ∥b^{∗}∥_{L_{γ}^{2}} ≤ ∥x∥_{L_{γ}^{2}} ∑_{n=1}^{∞} ∥τ_{n}∥_{L^{2}}^{2}, and b^{∗} ∈ L_{γ}^{2}. Since b^{∗} has the weak derivative equal to −c, which is in H^{1}, then b^{∗} ∈ H_{γ}^{2}. We also notice that constant functions belong to H_{γ}^{1}, so every function b^{∗}(·) + b_{0}, b_{0} ∈ ℝ is a solution of Ab + c = 0 and we have the result. □

Remark 6. There exist also non-Gaussian invariant measures for Eq. (1). In fact, if in Eq. (10) we choose an initial datum Z_{0} ≡ f ∈ H_{γ}^{1} periodical with period T, then the solution would be Z_{t}(x) ≡ f(t + x). Now we define the random variable π : [0, T] → H_{γ}^{1} such that π(t) = f(· + t) and we set the uniform density on [0, T]; then π induces a measure μ_{f} on H_{γ}^{1}. So, if in Eq. (10) we choose an initial datum Z_{0} with law μ_{f}, then we obtain a solution (Z_{t})_{t} with Z_{t} having law μ_{f} ∀t ≥ 0. This means that μ_{f} ∗ N(b^{∗}(·) + b_{0}, Q_{∞}) is an invariant measure for Eq. (1), and it is not Gaussian.

We can see that the ergodic behaviour of Eq. (1) could be more intricate than a simply Gaussian one, and a characterization of all the invariant measures would be very difficult. However, in the next theorem we see that if we start from a deterministic initial datum r_{0}, then the asymptotic law is one of the Gaussian invariant measures found in Theorem 4. Besides, the theorem will show also that all the invariant measures found before can be reached starting from a suitable r_{0}.

Theorem 7. If ∑_{n=1}^{∞} ∥τ_{n}∥_{L^{1}}^{2} < +∞, the assumptions of Theorem 4 hold, and the initial datum is a deterministic r_{0} ∈ H_{γ}^{1} such that lim_{x→+∞} r_{0}(x) = r_{0}(∞) ∈ ℝ, lim_{x→+∞} r′_{0}(x) = 0, then r_{t} converges weakly for t → +∞ to the invariant measure N (r_{0}(∞) + b_{0} + b(x), Q_{∞}), where b and Q_{∞} are given respectively by (11) and (9), and b_{0} is given by

b_{0} = ∫_{0}^{+∞} c(u) du = \frac{1}{2} ∑_{n=1}^{∞} (∫_{0}^{+∞} τ_{n}(u) du)^{2} .  (12)

Proof. Let us consider an initial datum with a degenerate law concentrated on r_{0}, and see what happens to the marginal law of the solution r_{t}. The solution r_{t} has functional mean

𝔼[r_{t}](·) = r_{0}(· + t) + ∫_{0}^{t} c(· + u) du

and functional variance

Var [r_{t}] = ∑_{n=1}^{∞} ∫_{0}^{t} τ_{n}(· + u) ⊗ τ_{n}(· + u) du .

Then for all G ∈ C_{b}(H_{γ}^{1}) we have that

lim_{t→+∞} 𝔼[G(r_{t})] = lim_{t→+∞} 𝔼 [G (𝔼[r_{t}] + ∑_{n=1}^{∞} ∫_{0}^{t} τ_{n}(· + t − s) dW_{s})] =

= ∫_{H_{γ}^{1}} G(r)N(r_{0}(∞) + b_{0} + b(·), Q_{∞})(dr)

In fact:

lim_{t→+∞} ∥𝔼[r_{t}] − r_{0}(∞) − b_{0} − b(·)∥_{H_{γ}^{1}} ≤ lim_{t→+∞} ∥r_{0}(t + ·) − r_{0}(∞)∥_{H_{γ}^{1}}+

+ lim_{t→+∞} ∥∫_{0}^{·+t} c(u) du − ∫_{0}^{+∞} c(u) du + ∫_{0}^{·} c(u) du∥_{H_{γ}^{1}} =

= 0 + lim_{t→+∞} ∥∫_{·+t}^{+∞} c(u) du∥_{H_{γ}^{1}} = 0

where the two limits follow from Lebesgue’s theorem. Besides

∑_{n=1}^{∞} ∫_{0}^{t} τ_{n}(· + t − s) dW_{s}

has law N(0, Q_{t}), where Q_{t} = ∑_{n=1}^{∞} ∫_{0}^{t} τ_{n}(· + s) ⊗ τ_{n}(· + s) ds. Since Tr Q_{∞} < +∞, then lim_{t→+∞}(Q_{∞} − Q_{t}) = 0, so N(0, Q_{t}) converges weakly to N(0, Q_{∞}). □

Theorem 7 says that the solution r_{t} starting from a rather general deterministic initial forward curve r_{0}, under some regularity conditions, converges weakly to a Gaussian invariant measure. Moreover, the invariant measure to which the solution converges is determined by r_{0}(∞). From this we deduce that if we take a suitable initial datum r_{0}, it is possible to reach any invariant Gaussian measure we found in the last theorem. From a mathematical point of view this means that there are no “privileged” invariant measures, at least in the purely Gaussian case. From a practical point of view, this non-uniqueness is related to the deterministic part of the forward curve r_{0} known at time 0, and corresponds to the limit of the forward curve at infinity. This is intuitively quite appealing.

Remark 8. The quantity r_{t}(∞) is usually called long forward rate. In our case, it is easy to see from Eq. (5) that if r_{0}(∞) exists and τ_{n}(∞) exists and is equal to 0 for all n ∈ ℕ, then r_{t}(∞) = r_{0}(∞) ℚ-almost surely. This is a common behaviour of the long forward rate: in fact, while it can be shown by no-arbitrage techniques that this quantity is ℚ-almost surely increasing with t (see [7]), in our model (as in many models used in practice) the long forward rate is constant.

3. The forward rate equation in the space H^{1}

Now we study Eq. (1) in the space H^{1}(ℝ^{+}). As before, we first present some sufficient conditions for the mild solution of Eq. (1) to be well defined in this space. These conditions will be also necessary in order to obtain the existence of an invariant measure. Theorem 9. If ∑_{n=1}^{∞} ∥τ_{n}∥_{H^{1}}^{2} < +∞, ∑_{n=1}^{∞} ∥τ_{n}∥_{L^{4}}^{4} < +∞, the functions \sqrt{x}τ_{n}(x) and \sqrt{x}τ′_{n}(x) are uniformly bounded in L^{2}(ℝ^{+}) and r_{0} ∈ H^{1}(ℝ^{+}), then the mild solution of Eq. (2) in H^{1}(ℝ^{+}), given by (5), is well defined. Moreover, it is the unique solution in the class of weak solutions. The mild solution is a Gaussian process with mean (6) and covariance (7).

Proof. The proof is similar to the one of Theorem 2. □

Theorem 10. Given Eq. (2) in the Hilbert space H^{1}(ℝ^{+}), necessary and sufficient conditions to have an invariant measure are the following: ∑_{n=1}^{∞} ∥τ_{n}∥_{H^{1}}^{2} < +∞, ∑_{n=1}^{∞} ∥τ_{n}∥_{L^{4}}^{4} < +∞, ∑_{n=1}^{∞} ∥\sqrt{x}τ_{n}(x)∥_{L^{2}}^{2} < +∞, ∑_{n=1}^{∞} ∥\sqrt{x}τ′_{n}(x)∥_{L^{2}}^{2} < +∞, ∑_{n=1}^{∞} ∥τ_{n}∥_{L^{1}}^{2} < +∞, and that the functions ∫_{·}^{+∞} τ_{n}(u) du are uniformly bounded in L^{2}(ℝ^{+}). Under these hypotheses N(b^{∗}(·) + b_{0}, Q_{∞}) is the only invariant measure, where b^{∗} is given by Eq. (11), b_{0} is given by Eq. (12), and Q_{∞} is given by Eq. (9).

Proof. As before, we check if the conditions of Theorem 5 are satisfied. Since (i) must hold, and

sup_{t≥0} ∫_{0}^{t} Tr Q_{u} du = ∫_{0}^{+∞} ∑_{n=1}^{∞} ∫_{u}^{+∞} (τ_{n}^{2}(x) + τ_{n}^{′2}(x)) dx du =

= ∑_{n=1}^{∞} ∫_{0}^{+∞} (τ_{n}^{2}(x) + τ_{n}^{′2}(x)) ∫_{0}^{x} du dx =

= ∑_{n=1}^{∞} (∥\sqrt{x}τ_{n}(x)∥_{L^{2}}^{2} + ∥\sqrt{x}τ′_{n}(x)∥_{L^{2}}^{2}) ,

the conditions ∑_{n=1}^{∞} ∥\sqrt{x}τ_{n}(x)∥_{L^{2}}^{2} < +∞ and ∑_{n=1}^{∞} ∥\sqrt{x}τ′_{n}(x)∥_{L^{2}}^{2} < +∞ are necessary to have an invariant measure. We now check point (ii) of Theorem 5, and see if there exists b ∈ D(A) such that Ab + c = 0. A solution is b(x) = b^{∗}(x) + b_{0}, where b^{∗} is defined by Eq. (11) and b_{0} ∈ ℝ. Since b is a decreasing function, then b ∈ L^{2} only if lim_{x→+∞} b(x) = 0. This determines b_{0}:

0 = lim_{x→+∞} b(x) = b_{0} − lim_{x→+∞} \frac{1}{2} ∑_{n=1}^{∞} (∫_{0}^{x} τ_{n}(u) du)^{2} .

The necessary conditions for this to happen are that τ_{n} ∈ L^{1}, ∑_{n=1}^{∞} ∥τ_{n}∥_{L^{1}}^{2} < +∞ and that b_{0} is given by Eq. (12). Now let us see under which conditions b ∈ L^{2}. If we define T_{n}(x) = ∫_{x}^{+∞} τ_{n}(u) du then, applying the algebraic identity (c^{2}−d^{2})^{2} = (c−d)^{2}(c^{2}+2cd+d^{2}) to the function b, we can see that the conditions T_{n} ∈ L^{2} and T_{n} uniformly bounded in L^{2} are necessary and sufficient to have b ∈ L^{2}. Since b′ = −c ∈ H^{1}, we have b ∈ H^{2}, so b is a solution of Ab + c = 0.

Now we search for invariant measures for Eq. (10) of the kind δ_{z} with z ∈ H^{1}. This problem is equivalent to find solutions for the equation z′ = Az in H^{1} which are constant in t. This means that Az = 0 must hold, so the solutions must have the form z(x) = const. Since the only constant function in H^{1} is the null function, the only invariant measure of the kind δ_{z} is δ_{0}. Thus we have proved that N(b^{∗}(·) + b_{0}, Q_{∞}) is an invariant measure.

We claim that this is the only invariant measure for Eq. (1). To this aim, we refer to various results on asymptotic behaviour of solutions contained in [5]. We call Z_{t}(X) the solution of Eq. (10) at time t having initial datum Z_{0} = X, and L(Y) the law of a generic random variable Y under ℚ. Since lim_{t→+∞} S_{t}f = 0 for all f ∈ H^{1}, then L(Z_{t}(f)) = δ_{S_{t}f} converges weakly to δ_{0} for all f ∈ H^{1}. This means that δ_{0} is the only invariant measure for Eq. (10). Hence N(b^{∗}(·)+b_{0}, Q_{∞}) is the unique invariant measure for Eq. (1) in H^{1}(ℝ^{+}). □

We have just shown that in H^{1}(ℝ^{+}) there exists only one invariant measure. This means that the invariant measure is independent of the initial forward curve r_{0}. Moreover, a result like Theorem 6 in Section 2 would be useless in this case, because r_{0}(∞) is always equal to 0, and this is clearly inconsistent with the way the market extrapolates the yield curve to get an idea of levels of interest rates beyond the longest observable maturities. The conclusion seems to be that H^{1}(ℝ^{+}) is too poor to allow for a good financial interpretation of the results.

4. Concluding remarks

We analysed the Musiela forward rates equation in the Sobolev spaces H_{γ}^{1}(ℝ^{+}) for all γ ≥ 0 in the Gaussian case. For each of the spaces, we presented an explicit solution, called mild solution, and analysed its asymptotic behaviour. From our work it is clear that the asymptotic evolution of the solutions depends on the choice of the particular space H, and in our case it seems that working on H_{γ}^{1}(ℝ^{+}) for γ > 0 is better than working on H^{1}(ℝ^{+}) for the practical applications. This is because for γ > 0 we find a wide range of invariant measures, and the Gaussian ones are significant in the sense that if we start from a forward curve r_{0} that is known today, then the law of r_{t} converges to a Gaussian invariant measure for t → +∞. Moreover, the particular invariant measure to which the law of r_{t} converges is determined by the level of the long forward rate r_{0}(∞). Conversely, in H^{1}(ℝ^{+}) there exists only one invariant measure, that is independent of the particular initial forward curve r_{0}. Moreover in H^{1} flat term structures are not allowed and all the forward curves converge to 0 as x → +∞. These facts let us believe that H^{1} is not a suitable choice for practical purposes.

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