Singular perturbations and asymptotic expansions for SPDEs with an application to term structure models

We study the dependence of mild solutions to linear stochastic evolution equations on Hilbert space driven by Wiener noise, with drift having linear part of the type $A+\varepsilon G$, on the parameter $\varepsilon$. In particular, we study the limit and the asymptotic expansions in powers of $\varepsilon$ of these solutions, as well as of functionals thereof, as $\varepsilon \to 0$, with good control on the remainder. These convergence and series expansion results are then applied to a parabolic perturbation of the Musiela SPDE of mathematical finance modeling the dynamics of forward rates.


Introduction
Consider the family of stochastic evolution equations set in a Hilbert space H and indexed by ε > 0, where A and G are linear maximal dissipative operators on H such that A + εG is also maximal dissipative, α and B are coefficients satisfying suitable measurability, integrability and regularity conditions, and W is a cylindrical Wiener process.Precise assumptions on the data of the problem are given below.Our main goal is to obtain an expansion of the difference u ε − u in a (finite) series in powers of ε plus a remainder term, where u ε and u are the unique mild solutions to (1.1) with ε > 0 and ε = 0, respectively.Results in this sense are obtained assuming that the semigroups generated by A and G commute.As a first step, we show that u ε converges to u as ε → 0, also in the case where α and B are (random, time-dependent) Lipschitz continuous functions of the unknown, in suitable norms implying the convergence in probability uniformly on compact intervals in time.For such convergence result to hold it is enough that the resolvent of A + εG converges to the resolvent of A as ε → 0 in the strong operator topology, without any commutativity assumption.Sufficient conditions for the convergence of operators in the strong resolvent sense have been largely studied (see, e.g., [8,24] and references therein) and can be readily applied to obtain convergence results for solutions to stochastic evolution equations.On the other hand, expansions in power series of u ε −u are considerably harder to obtain.In fact, it is well known that solutions to singularly perturbed equations, also in the simpler setting of deterministic ODEs, do not admit series expansions in the perturbation parameter.This phenomenon appears also in the class of stochastic equations studied here, as it is quite obvious.This is essentially the reason behind the commutation assumption on the semigroups generated by A and G, as well as on the regularity conditions on the initial datum u 0 and on the coefficients α and B (see §4 below, where asymptotic expansion results are obtained also for functionals of u ε ).
As an application of the abstract results, we consider a singularly perturbed transport equation on R where, roughly speaking, A and G are the first and second derivative, respectively.This equation can be seen as a singular perturbation of an extension of Musiela's SPDE from a weighted Sobolev space on R + to the corresponding one on R. The motivation for considering this problem comes from the interesting article [6], where the author argues that second-order parabolic SPDEs reproduce many stylized statistical properties of forward curves.On the other hand, if forward rates satisfy a Heath-Jarrow-Morton dynamics, the differential operator in the drift of the corresponding SPDE must be of first order.It is then natural to consider singular perturbations of the (first-order) Musiela SPDE by second-order differential operators and to look for conditions implying uniform convergence of the "perturbed" forward rates, as well as of implied bond prices, to the corresponding "unperturbed" forward rates and bond prices, as well as a more precise description of the dependence of the pricing error on the "size" of the perturbation.Results in this sense are obtained in the form of asymptoptic expansions in ε of the solution u ε to a second-order perturbation of a suitable extension of the Musiela SPDE, as well as of functionals thereof.
The rest of the text is organized as follows.In §2 we introduce some notation, we recall basic results from semigroup theory, and we establish some inequalities and identities for classes of stochastic convolutions.In §3 we show that a commutation assumption between the semigroups generated by A and G implies that the closure of A + εG converges to A in the strong resolvent sense as ε → 0. This allows, thanks to a general convergence result for mild solutions to stochastic evolution equations, to deduce the convergence of u ε to u in a suitable norm.Under further regularity assumptions on u 0 , α and B, expansions of the difference u ε − u and of functionals thereof as power series in ε are obtained in §4, which is the core of the work.Finally, the applications described above to Musiela's SPDE are developed in §5.
Acknowledgments.The first-named author would like to thank the the Economics Department of the Università dell'Insubria, Varese and the Università degli Studi di Milano for warm hospitality.Large part of the work for this paper has been carried out while the second-named author was visiting the Interdisziplinäres Zentrum für Komplexe Systeme at the University of Bonn.The third-named author would like to thank the Institute of Applied Mathematics and the Hausdorff Center for Mathematics at the University of Bonn for warm hospitality.

Preliminaries
Throughout this section we shall use E and F to denote two Banach spaces.The domain of a linear operator L with graph in E × F will be denoted by D(L).The Banach space of continuous k-linear operators from E k to F , k ∈ N, is denoted by L k (E; F ) (without subscript, as usual, if k = 1).Given h ∈ E and k ∈ N, we shall set h ⊗k = (h, . . ., h) ∈ E k .If E and F are Hilbert spaces, L 2 (E; F ) will stand for the Hilbert space of Hilbert-Schmidt operators from E to F .An expression of the type a b means that there exists a positive constant N such that a ≤ N b, and a b stands for a b and b a.
For the purposes of this section only, we denote a strongly continuous semigroup on a Hilbert space H and its generator by S and A, respectively.As is well known, there exist M ≥ 1 and w ∈ R such that S(t) ≤ M e wt for all t ≥ 0. Let m ≥ 1 be an integer.If φ ∈ D(A m ), one has the Taylor-like formula (see, e.g., [4,Proposition 1.1.6]).We recall that A m is a closed operator and that D(A m ) is a Hilbert space with scalar product Let T be a further strongly continuous semigroup on H.We shall say that S and T commute if S(t)T (t) = T (t)S(t) for all t ∈ R + .It is immediate that the product semigroup ST is strongly continuous.It also follows that S(s)T (t) = T (t)S(s) for all t, s ≥ 0: first one proves it for rational s and t, hence the general case follows by density and continuity.For details see, e.g., [8, p. 44].Moreover T leaves invariant D(A): in fact, for any f ∈ D(A), one has This also implies, by uniqueness of the limit, that T (t)Af = AT (t)f .These observations in turn imply that the resolvent R λ of the generator of T commutes with A, in the sense that, if [12, p. 171]).
All stochastic elements will be defined on a fixed probability space (Ω, F , P), endowed with a filtration (F t ) t∈[0,T ] , with T a fixed positive number, that is assumed to satisfy the so-called usual assumptions.All equalities and inequalities between random variables are meant to hold outside a set of probability zero.We shall denote by W a cylindrical Wiener process on a real separable Hilbert space U .We shall denote the closed subspace of L p (Ω; C([0, T ]; H)), p > 0, of Hvalued adapted continuous processes by C p , which is hence a quasi-Banach space itself (with the induced quasi-norm).Given a progressively measurable process C ∈ L 0 (Ω; L 2 (0, T ; L 2 (U ; H))), the stochastic convolution S ⋄ C is the H-valued process defined by provided that the stochastic integral exists.Similarly, if f ∈ L 0 (Ω; L 1 (0, T ; H)), we shall define the H-valued process S * f by The stochastic integral of a process F with respect to W will be occasionally denoted by F • W for typographical convenience.
• W is a local martingale, the Burkholder-Davis-Gundy inequality (see, e.g., [19]) and the ideal property of Hilbert-Schmidt operators yield sup Setting for all δ > 0. Taking the limit as δ → 0 proves the claim.
The following recursive relation for certain nonlinear stochastic convolutions will be very useful in the sequel.Lemma 2.2.Let C ∈ L p (Ω; L 2 (0, T ; L 2 (U ; H))) be a progressively measurable process and define, for every k ∈ R + , the stochastic Fubini theorem, and the semigroup property, one has, for every t ∈ [0, T ],

Singular perturbations by commuting semigroups
Let us consider the stochastic evolution equation on the Hilbert space and the family of stochastic evolution equations on H indexed by a parameter ε ≥ 0 where (i) A and G are linear maximal dissipative operators on H such that the closure of A + εG, denoted by the same symbol, is maximal dissipative as well; (ii) the initial datum u 0 belongs to L 0 (Ω, F 0 ; H); (iii) the coefficients are Lipschitz continuous in the third variable, uniformly with respect to the other ones, and such that α(•, •, h) and B(•, •, h) are progressively measurable for every h ∈ H.It is well known that under these conditions the above stochastic equations admit unique mild solutions u and u ε , respectively, with continuous trajectories.Moreover, if u 0 ∈ L p (Ω, F 0 ; H) for some p > 0, then u and u ε belong to C p (see, e.g., [7,Chapter 7] for the case p ≥ 2 and [18] for the general case).The aim of this section is to provide sufficient conditions ensuring that u ε → u in C p .We rely on the following convergence result, which is a minor modification of [21,Theorem 2.4] (see also [13]).
We recall that a sequence of maximal dissipative operators (L n ) is said to converge to a maximal dissipative operator L in the strong resolvent sense if (λ The problem of the convergence of u ε to u is thus reduced to finding sufficient conditions for the convergence of A + εG to A in the strong resolvent sense as ε → 0. In view of the results on asymptotic expansions in the next sections, we limit ourselves to the special case where the semigroups generated by A and G, denoted respectively by S A and S G , commute.Lemma 3.2.Assume that S A and S G commute, i.e. that S A (t)S G (t) = S G (t)S A (t) for all t ≥ 0. Then A + εG converges to A in the strong resolvent sense as ε → 0.
Proof.One has, for any λ > 0 and f ∈ H, Remark 3.3.(i) Under the assumption that S A and S G commute, D(A) ∩ D(G) is a core for the generator of the product semigroup S A S εG , which is contractive and strongly continuous.Its generator is hence equal to the closure of A + εG.So the hypothesis of maximal dissipativity of (the closure of) A + εG is automatically satisfied here.
(ii) It is clear from the proof of the previous lemma that not even the assumption of dissipativity of A and G is needed, but just that the resolvent sets of A and G have non-empty intersection.
In particular, the statement of the lemma continues to hold if A and G are maximal quasidissipative, i.e. if there exist a and b ∈ R + such that A − aI and G − bI are maximal dissipative.In this respect, as long as one is concerned with applications to the stochastic equation, there is no loss of generality assuming that A and G are dissipative rather than quasi-dissipative, because the latter case reduces to the former by adding a linear term to the drift α.
(iii) The perturbation result in Lemma 3.2 strongly relies on the commutativity assumption between S A and S G .This assumption is essential also to derive the asymptotic expansion results in the next sections.For other assumptions on A and G leading to convergence of A + εG to A in the strong resolvent sense as ε → 0, see, e.g., [24] and references therein.

Asymptotic expansion of u ε
Our next goal is to obtain an expression of the difference u ε − u as a finite power series in ε plus a remainder.Once such an expression is obtained, the main issue is to prove estimates on the coefficients of the power series and on the remainder.Such estimates will crucially depend on suitable regularity assumptions on the coefficients α and B that will be assumed throughout the section to be random and time-dependent, but not explicitly dependent on u.In particular, let us consider the stochastic evolution equations and where A and G are maximal dissipative and generate commuting semigroups.As before, we denote the closure of A + εG, ε > 0, by the same symbol.Moreover, we assume that there exist p ∈ [1, ∞[ and an integer m ≥ 1 such that Then equations (4.1) and (4.2) admit unique mild solutions u and u ε in C p , respectively. 1 Just for convenience, we also assume that ε ∈ [0, 1].
All results in this section do not use the assumption that A and G are maximal dissipative, except in an indirect way in Proposition 4.10, namely through Theorem 3.1.In particular, all results except Proposition 4.10 continue to hold under the same assumptions on u 0 , α and B, commutativity of S A and S G , and the existence of of unique solutions u and u ε ∈ C p to (4.1) and (4.2), respectively.
We begin with a decomposition of u ε that is essentially of algebraic nature.Proposition 4.1.There exist adapted processes v 1 , . . ., v m−1 and R m,ε in C p such that Proof.It follows by commutativity of S A and S G that where, by the Taylor-like formula for strongly continuous semigroups of §2, for each k ∈ {1, . . ., m − 1}, and the family of H-valued processes (R m,ε ) ε∈]0,1] defined as hence the claim for m = 1 follows, choosing ε ∈ ]0, 1] arbitrarily, by the assumptions on u and u ε , and Lemma 4.7 below.By induction on m, the claim for general m is proved.
Remark 4.2.Estimates for the C p -norm of v k can be obtained in a more direct (and precise) way exploiting the dissipativity of A. In fact, one has and (in fact just assuming that A is only the generator of a strongly continuous semigroup), as well as, by maximal estimates for stochastic convolutions, Alternative assumptions on A yield similar estimates for the stochastic convolution, for instance if A generates an analytic semigroup.We shall not pursue this issue here.
We are going to estimate R m,ε in L p (Ω; H) pointwise with respect to the time variable as well as in C p .As mentioned above, such estimates do not use the dissipativity of A and G.For this reason, we shall prove them under the sole assumption that A and G are generators of strongly continuous semigroups S A and S G , respectively, with and introduce the function f ε : R + → R + defined as and Proof.Both estimates are immediate consequences of Minkowski's inequality and the definition of f .For instance, the second one is given by The running maximum of the function f ε defined in (4.5) will be denoted by Proof.The first estimate is evident.The second one follows by Proof.The first estimate is again evident, by Lemma 4.3.The second one follows by The estimates of R 3 m,ε are more delicate.The reason is that the double integral in (4.3) is not a stochastic convolution.In fact, while it can be written as the family of operators (R(t)) t∈R+ is not a semigroup.Unfortunately we are not aware of any maximal inequalities for such "nonlinear" stochastic convolutions.We shall nonetheless obtain estimates on the remainder term R 3 m,ε by different arguments.Lemma 4.6.One has, for every t ∈ [0, T ] and ε ∈ [0, 1], Proof.We shall use an argument analogous to the one used in the proof of Lemma 2.1.Write where and Moreover, setting we can write .
Since δ > 0 is arbitrary, taking the limit as δ → 0 yields Proof.Thanks to the stochastic Fubini theorem, R 3 m,ε (t) can be written as thus also, setting as S A+εG * Φ(t), with Minkowski's inequality yields where, by Lemma 2.1, Let us denote the constant in the Burkholder-Davis-Gundy inequality with power p > 0 by N p .Theorem 4.8.One has, for every t ∈ [0, T ], In particular, Proof.This is an immediate consequence of the previous propositions and lemmas in this section, upon observing that f ε and f * ε converge pointwise to a finite limit as ε → 0.
Remark 4.9.It seems interesting to remark that, without any dissipativity assumption on A and G, the previous theorem implies that, as soon as m ≥ 1, one has u ε → u in C p as ε → 0 without the need to appeal to Theorem 3.1.
We are now going to identify the process v k as the k-th derivative at zero of ε → u ε .We shall actually prove more than this, namely that u ε is m times continuously differentiable with respect to ε.
. We begin by establishing first-order continuous differentiability.One has where, recalling that S hG = S G (h •), in C p by dominated convergence.Similarly, one has for all s, t ∈ [0, T ] with s ≤ t, hence, again by dominated convergence, The stochastic convolution term cannot be treated the same way and requires more work.We shall write, for simplicity of notation, S ε in place of S A+εG .Introducing the processes y ε = y 0 ε and y 1 ε defined as we need to show that lim Duhamel's formula yields Since S ε+h ⋄ GB converges to S ε ⋄ GB in C p as h → 0 by Theorem 3.1, it follows by dominated convergence that lim h→0 Moreover, by Lemma 2.2, thus (4.6) is proved.Furthermore, it follows by the assumptions on B that the same argument also yields the stronger statement (with j integer).Let us turn to higher-order derivatives.We shall only consider the term involving the stochastic convolution, as the terms involving the initial datum and the deterministic convolution can be treated in a completely analogous (in fact easier) way.We need to show that the k-derivative of ε → y ε , denoted by y (k) , satisfies for all k ≥ 2, as the case k = 1 has just been proved.We begin with some preparations.Lemma 2.2 implies that for every k ∈ {1, . . ., m}, where S * k ε denotes the operation of k times convolution with S ε , i.e.
It follows by a repeated application of Theorem 3.1 that G j y k ε+h → G j y k ε in C p as h → 0 for all j, k ∈ N with j + k ≤ m.We shall now proceed by induction, i.e. we are going to prove that, for any ε ∈ [0, 1], y where, as discussed above, Gy k ε+h → Gy k ε in C p as h → 0, so that, by dominated convergence, . The assumptions on B and (4.8) imply that the inductive assumption also yields, in complete analogy to the argument leading to (4.7), that for every positive integer j such that j + k ≤ m.Therefore, again by dominated convergence, we have thus concluding the proof of the induction step.

Asymptotic expansion of functionals of u ε
We are now going to consider asymptotic expansions of processes of the type F (u ε ), where F is a functional taking values in a Banach space.All assumptions stated at the beginning of the sections are still in force.
We begin with a simple case.
Proposition 4.11.Let E be a Banach space and F : C p → E be of class C m−1 , m ≥ 2. Then there exist w 1 , . . ., w m−2 ∈ E and R m−1,ε ∈ E such that, for every ε ∈ ]0, δ], where and The expression for F (u ε ) then follows immediately by Taylor's theorem, and the expression for w n follows by the formula for higher derivatives of composite functions (sometimes called Faà di Bruno's formula -see, e.g., [2, p. 272]).Furthermore, denoting the map ε → u ε by ϕ, one has where D m−1 (F • ϕ) is bounded in E on the compact interval [0, δ] because it is continuous thereon.Denoting the maximum of the E-norm of this function on [0, δ] by M δ , we have where the right-hand side obviously tends to zero as ε → 0.
We shall now assume that F ∈ C m (C p ; E) and derive an expansion of u ε − u of order m − 1.Note that an argument based on Taylor's formula for ε → F (u ε ), as in the previous proposition, does not work because ε → u ε is only of class C m−1 , hence its composition with F is also of class C m−1 .We are going to use instead a construction based on composition of power series.Theorem 4.12.Let E be a Banach space and F : C p → E be of class C m , m ≥ 1.Then there exist w 1 , . . ., w m−1 ∈ E and R m,ε ∈ E such that, for every ε ∈ ]0, δ], where the (w n ) are defined as in (4.9) and Proof.Taylor's formula applied to F yields where, by Proposition 4.1, where Rm,ε ∈ C p by (4.4) and Lemmas 4.5 and 4.7.Multilinearity of the higher-order derivatives of F implies that where w n , n = 1, . . ., m − 1 are defined as in (4.9), and the a n are (finite) linear combinations of terms of the type , where j ∈ {1, . . ., m − 1} and k 1 , . . ., k n+1 ∈ N, Let us show that w n ∈ E for every n = 1, . . ., m − 1: by (4.9) it suffices to note that, for any j = 1, . . ., n and where the right-hand side is finite because u ∈ C p and F ∈ C m (C p ; E) by assumption, and v 1 , . . ., v m−1 ∈ C p by Proposition 4.1.The proof that a n ∈ E for all n = m, . . ., (m − 1)m, with norms bounded uniformly for ε ∈ [0, δ], is entirely similar, as it immediately follows by Lemmas 4.5 and 4.7.
Finally, by multilinearity of D m F , the integral on the right-hand side of (4.10) can be written as , where b n depends on ε.By a reasoning entirely similar to the previous ones, in order to prove that b n ∈ E for all n and that their E-norms are bounded as ε → 0, we proceed as follows: D m F is continuous, hence bounded on a neighborhood U of u.Since u ε → u in C p as ε → 0 by assumption, u ε ∈ U for ε sufficiently small, hence also tu  Then the conclusions of Theorem 4.12 hold with E := L q (Ω; E 0 ).
Proof.Taylor's theorem implies that (4.10), (4.11) and (4.12) still hold, as an identities of E 0valued random variables.As in the proof of the previous theorem, the integral on the right-hand side of (4.10) can be written as the finite series n=m b n ε n , with each b n possibly depending on ε.We have to show that w n , a n , b n ∈ L q (Ω; E 0 ) for every n, and that the elements in (a n ) and (b n ) that depends on ε remain bounded in L q (Ω; E 0 ) as ε → 0. To this purpose, denoting the norms of C([0, T ]; H) and L j (C([0, T ]; H); E 0 ) by • and • Lj , respectively, for simplicity of notation, note that one has, for any j ≤ m, where D j F (u) Lj 1 + u β by assumption and Applying Hölder's inequality with the exponents implied by this inequality yields where we have used the identity z β L p/β (Ω) = z β L p (Ω) , which holds for any positive random variable z.Recalling that Rm,ε is bounded in C p uniformly over ε ∈ [0, δ], the claim about (w n ) and (a n ) is proved.In order to show that (b n ) enjoys the same properties of (a n ), it is immediately seen that it suffices to bound the norm of D m F (tu ε + (1 − t)u) in L q/β (Ω; L m (C(0, T ; H); E 0 )), uniformly with respect to ε in a right neighborhood of zero.But where the norm in C p of u ε − u tends to zero as ε → 0. The proof is thus completed.Remark 4.14.One could have also approached the problem in a more abstract way, establishing conditions implying that the function F can be "lifted" to a function of class C m from C p to E = L q (Ω; E 0 ), and then applying the Theorem 4.12.We have preferred the above more direct way because it could also be applicable, mutatis mutandis, in situations where F admits a series representation not necessarily of Taylor's type.
5 Singular perturbations of a transport equation and the Musiela SPDE

A transport equation
Let w be a fixed strictly positive real number and set, for notational convenience, L 2 w := L 2 (R, e wx dx).Let H be the Hilbert space of absolutely continuous functions f ∈ L 1 loc (R) such that f ′ ∈ L 2 w , equipped with the scalar product The definition is well posed because f (+∞) := lim x→+∞ f (x) exists and is finite for every f ∈ H.
In fact, for any a ∈ R such that f (a) is finite, one has One has the following formula of integration by parts.
In particular, Proof.By definition, one has where f ′ (+∞) = 0 and g(+∞) is finite, hence, integrating by parts, where the two limits are equal to zero by the elementary estimate Let us now consider the operator A 2 , defined on its natural domain D(A 2 ) of elements f ∈ D(A) such that Af ∈ D(A).

Lemma 5.2. The operator
), and substitute g = Af in the integration by parts formula of the previous lemma.We get hence also Proof.The dissipativity of G has already been proved.Moreover, A 2 is closed, as is every integer positive power of the generator of a strongly continuous semigroup (see, e.g., [4, Proposition 1.1.6]).Hence we only have to show that there exist λ > 0 such that the image of λI − G is H.To this purpose, let f ∈ H and consider the equation λy −y ′′ = f , which yields λy ′ −y ′′′ = f ′ .Defining (formally, for the moment) z through y ′ (x) = z(x)e −wx/2 , one has We are thus led to consider the equation Let us introduce the bounded bilinear form a on It is immediately seen that, for any λ > w 2 /4, the bilinear form a is coercive on H 1 , hence the Lax-Milgram theorem yields the existence and uniqueness of a (weak) solution z ∈ H 1 .Moreover, the equation satisfied by z implies that, in fact, z ∈ H 2 .This immediately yields the existence of a solution y to λy − y ′′ = f .Moreover, by definition of z it is immediate that y ∈ H, the identity y ′′ (x)e wx/2 = z ′ (x) − wz(x)/2 implies that y ′′ ∈ L 2 w , and (5.2) implies that y ′′′ ∈ L 2 w , i.e. y ∈ D(A 2 ), thus completing the proof.
Since A is maximal dissipative, the transport equation on with α and B satisfying the measurability and Lipschitz continuity assumptions of §3 and u 0 ∈ L p (Ω; H), p > 0, admits a unique mild solution u ∈ C p (see, e.g., as already mentioned, [7, Chapter 7] and [18]).Under the same assumptions on α, B, and u 0 , the singularly perturbed equation admits a unique mild solution u ε ∈ C p , which converges to u in C p as ε → 0. Furthermore, if the coefficients α and B do not depend on u and there exists an integer number m ≥ 1 such that then we can construct a series expansion of u ε around u of order m − 1, applying the results of §4.

Parabolic approximation of Musiela's SPDE
Let u(t, x), t, x ≥ 0, denote the instantaneous forward rate at time t with maturity t + x.Musiela observed that the equation for forward rates in the Heath-Jarrow-Morton model can be written as (the mild form of) where (w k ) k∈N is a sequence of standard real Wiener processes, the volatilities σ k are possibly random, and α 0 is uniquely determined by (σ k ) if the reference probability measure is such that implied discounted bond prices are local martingales.In particular, in this case it must necessarily hold For details on the financial background we refer to, e.g., [5,9,11,22,23].There is a large literature on the well-posedness of (5.3) in the mild sense, also in the (more interesting) case where (σ k ), hence α 0 , depend explicitly on the unknown u, with different choices of state space as well as with more general noise (see, e.g., [1,3,9,14,17,28], [25, §20.3]).Here we limit ourselves to the case where (σ k ) are possibly random, but do not depend explicitly on u, and use as state space H(R + ), which we define as the space of locally integrable functions on R + such that f ′ ∈ L2 (R + , e wx dx), endowed with the inner product This choice of state space, introduced in [9] (cf.also [27]), to which we refer for further details, is standard and enjoys many good properties from the point of view of financial modeling.For instance, forward curves are continuous and can be "flat" at infinity without decaying to zero.
In order to give a precise notion of solution to (5.3), we recall that the semigroup of left translation on H(R + ) is strongly continuous and contractive, and that its generator is A 0 : φ → φ ′ on the domain D(A 0 ) = {φ : φ ′ ∈ H(R + )} (see [9]).Moreover, let us assume that there exists p > 0 such that so that the random time-dependent linear map one has the basic estimate φ Iφ H(R+) φ for every φ ∈ H(R + ) such that φ(+∞) = 0 (cf.[9], or see Lemma 5.6 below for a proof in a more general setting).This implies that the assumption on (σ k ) yields α 0 ∈ L p (Ω; L 1 (0, T ; H(R + ))).
We then have the following well-posedness result for (5.3), written in its abstract form as where W is a cylindrical Wiener process on U := ℓ 2 .
Musiela's equation (5.5) is closely related to the transport equation studied in §5.1 above, the main difference being the state space.In the following we shall denote the state space of the transport equation by H(R).
As mentioned in the introduction, it has been suggested (see [6] and references therein) that second-order parabolic SPDEs, with respect to the physical probability measure, capture several empirical features of observed forward rates.It seems reasonable to assume that such SPDEs would retain their parabolic character even after changing the reference probability measure to one with respect to which discounted bond prices are (local) martingales, thus excluding arbitrage.It is then natural to consider singular perturbations of the Musiela equation on H(R + ) adding a singular term εG to the drift A 0 in (5.5), with G = A 2 0 , which is, roughly speaking, a second derivative in the time to maturity.On the other hand, if forward rates satisfy the general assumptions of the Heath-Jarrow-Morton model, the HJM drift condition is sufficient and necessary for discounted bond prices to be local martingales.Therefore singular perturbations of the Musiela SPDE introduce arbitrage, in the sense that the implied discounted bond prices may not be local martingales.It is hence interesting to "quantify" and control the amount of arbitrage introduced by a parabolic perturbation of the Musiela SPDE (5.5).The arguments used in §5.1 for the transport equation, however, give rise to major problems, mainly because boundary terms (at zero) appear that seem impossible to control.To circumvent such issues, we "embed" the abstract Musiela equation (5.5) into a transport equation on H(R) of the type considered in §5.1, we perturb the equation thus obtained, get asymptotic expansions, and finally "translate" back the results, in a suitable sense, to the Musiela equation.
We need some technical results first.Let H 0 (R + ) be the Hilbert space of functions in H(R + ) that are zero at infinity.The following embeddings and estimates are rather straightforward (see [9] for a proof) and will be repeatedly used below: . Let H m w (R + ) be the set of functions in L 1 loc (R + ) that belong to L 2 w (R + ) together with all their derivatives up to order m, endowed with the norm defined by w (R+) .
Lemma 5.5.Let f ∈ L 1 loc (R + ) and m a positive integer.The following assertions are equivalent: where the implicit constant depends only on m and w.
Proof.The equivalence of (a) and (b) is immediate by the definition of A 0 and by an inequality completely analogous to (5.1).In particular, if f (+∞) = 0, the identity is a tautology.The other assertions follow by the identity ∀n ∈ {1, . . ., m}.
, where the implicit constant depends only on m and w.
Proof.Let f ∈ D(A m 0 ) with f (+∞) = 0.In view of the previous lemma, we will bound the One has, omitting the indication of R + in the notation, Let 1 ≤ n ≤ m be an integer.One has Similarly, The claim is then an immediate consequence of these estimates.
Lemma 5.8.There exists a linear continuous extension operator L : D(A 2m 0 ) → D(A 2m ).Proof.By an extension result due to Stein (see [26, p. 181]), there exists a linear continuous extension operator L 0 : H 2m (R + ) → H 2m (R).Since a locally integrable function f belongs to D(A 2m 0 ) if and only if x → f ′ (x)e wx/2 ∈ H 2m (R + ) by Lemma 5.5 and f ∈ H(R + ) implies that f (+∞) is finite, the map ) by an argument completely analogous to the proof of Lemma 5.5.Finally, Let L be the extension operator just introduced and set v 0 := Lu 0 , α := Lα 0 , and B := LB 0 , with α 0 and B 0 as in Proposition 5.7, so that and consider the following stochastic equation in H(R): where A is the generator of the semigroup of translation on H(R) and W is a cylindrical Wiener process on ℓ 2 .By the discussion at the beginning of this section, this equation admits a unique mild solution v ∈ C p (D(A 2m )), which is thus also a strong solution, i.e. such that where the equality is in the sense of indistinguishable H(R)-valued (hence also C(R)-valued) processes.In a more explicit form, one has for every x ∈ R, in particular for every x ∈ R + .Since the restriction of v 0 , α and B to R + are equal to u 0 , α 0 and B 0 , respectively, the restriction of v to R + must coincide with the unique strong solution in H(R + ) to the Musiela equation (5.5).Moreover, the equation in H(R) also admits a unique mild solution v ε ∈ C p (D(A 2m )), which converges to v in C p (D(A 2m )) as ε → 0, and satisfies an identity of the type where all H(R + ) norms involved are finite because they are dominated by the corresponding ones in H(R), that are finite.We have thus proved the following.
Theorem 5.9.Let p > 0 and m ≥ 1 be a positive integer such that Then equation (5.6) has a unique strong solution v in C p (H(R)) and its restriction to H(R + ) coincides with the unique strong solution u in C p (H(R + )) to the Musiela equation (5.5).Moreover, the restriction to H(R + ) of the mild solution v ε to the perturbed extended Musiela equation (5.7) converges to v in C p (D(A 2m )) and the estimate We shall now consider bond prices and their approximation in diffusive correction of Musiela's equation.The solutions v and v ε to the equations (5.6) and (5.7) have paths in H(R), hence their restrictions x → v(t, x) and x → v ε (t, x), x ∈ R + , belong to H(R + ) for every t ∈ [0, T ] and u(t, x) = v(t, x) for every (t, x) ∈ [0, T ] × R + .The price of a zero-coupon bond with face value equal to one at time t ≥ 0 with time to maturity x ≥ 0 is given by and the value at time t of the money market account is given by For fixed t ∈ [0, T ] and x ≥ 0, let us define the linear map thus proving (i).Raising both sides of both inequalities to the power p and taking expectations proves (ii).
More generally, it is easy to show that the linear map F defined as The operators F t,x and F , being linear and continuous, are automatically of class C ∞ (assuming p ≥ 1 when the norms in the domain and codomain depend on such a parameter), with F ′ (z) = F (z) for every z in the domain of F , and higher-order derivatives equal to zero (and completely analogously for F t,x ).
Given a series expansion of v ε around v of the type which can be considered as an identity in C p (H(R)), as well as in C p (H(R + )) by restriction, it follows immediately that as an identity in L p (Ω).Similar considerations can be made with F in place of F t,x .An alternative way to reach the same conclusion is to look at the composition of functions where ε → v ε is of class C m−1 from R to C p and F t,x is of class C ∞ from C p to L p (Ω), so that ε → F t,x v ε is of class C m−1 from R to L p (Ω), and the series expansion (5.8) follows by Taylor's theorem.
To obtain a series expansion for the difference B ε (t, x) − B(t, x) we need, however, to work pathwise, i.e. in L 0 (Ω), essentially because it seems difficult to find a (reasonable) Banach space E such that x → e −x is Fréchet differentiable from L p (Ω) to E, so that the chain rule could be applied to obtain a differentiability result for the map ε → B ε (t, x).We proceed instead as follows: Taylor's theorem yields = B(t, x) 1 + J m−1 F t,x v ε − F t,x v + r m F t,x v ε − F t,x v , so that the relative pricing error can be written as (5.9) Substituting the series expansion of F t,x v ε − F t,x v provided by (5.8), we obtain a series expansion of B ε (t, x) − B(t, x) in ε of order m − 1 with a rest of higher order, that has to be interpreted as an identity in L 0 (Ω).Note that if B 0 ∈ L 2mp (Ω; L 2 (0, T ; L 2 (ℓ 2 ; H(R + )))), then v ε and v ε belong to C mp (H(R)), which implies that v 1 , . . ., v m−1 and R m,ε in (5.8) belong to C mp (H(R)), hence all powers of F t,x v ε − F t,x v up to the exponent m produce series in ε whose coefficients belong to L p (Ω), by Hölder's inequality.We can thus write the first term on the right-hand side of (5.9) as a series of order m − 1 with coefficients in L p (Ω), plus a remainder of higher order.Estimating the second term on the right-hand side of (5.9) requires further assumptions.Let p ′ be the (Hölder) conjugate exponent to p.We have where (F v ε − F v) m is a series of order m or higher with coefficients in L p , hence its L p (Ω) norm is bounded.Since, by Hölder's inequality, exp −s(F t,x v ε − F t,x v) L p ′ (Ω) = E exp −p ′ s(F v ε − F v) it follows that if E exp −p ′ (F u ε − F u) is bounded for ε in a (right) neighborhood of zero, we have a series expansion of the relative pricing error with coefficients in L 1 (Ω).An alternative estimate can be obtained under a sign assumption, starting from the expression because F t,x is positivity preserving.This implies where the right hand side converges to zero in L p (Ω) faster than ε m−1 .Conditions ensuring the positivity of mild solutions to the Musiela SPDE are discussed, e.g., in [10,14], and in [16,20] in a more general context.

. 0 S 0 S
Since y k ε+h = kS ε+h * Gy k−1 ε+h , Duhamel's formula yields, setting z := y k ε+h /k, z(t) = h t ε (t − s)Gz(s) ds + t ε (t − s)Gy k−1 ε+h (s) ds, therefore, by the identity y uniformly over ε in a (right) neighborhood of zero and t ∈ [0, 1].Minkowski's inequality now implies that the E-norm of each b n can be estimated uniformly with respect to ε. SettingR m,ε := (m−1)m n=m a n ε n + m 2 n=m b n ε n ,the proof is completed.We now consider the case where F is defined only on C([0, T ]; H).

1 )
We shall denote the norm in H induced by the above scalar product as • .The following simple consequence of the definition of H will be repeatedly used below:if f ∈ H, then lim x→±∞ |f ′ (x)| 2 e wx = 0, in particular lim x→+∞ f ′ (x) = 0.Let S A be the strongly continuous semigroup on H defined by [S A (t)f ](x) := f (t + x).The elementary identityR |f ′ (t + x)| 2 e wx dx = e −wt R |f ′ (x)| 2 e wx dximplies that S A is a contraction semigroup.Its generator is the maximal dissipative operator A , 0) ds , hence the corresponding discounted price of the zero-coupon bond isB(t, x)Let us define the discounted price of the (fictitious) zero coupon bond associated to v ε asB ε (t, x) = exp − x 0 v ε (t, y) dy − t 0 v ε (s, 0) ds .