An infinite-dimensional approach to path-dependent Kolmogorov’s equations Franco Flandoli∗ Giovanni Zanco† Dipartimento di Matematica, Università di Pisa
arXiv:1312.6165v1 [math.PR] 20 Dec 2013
December 24, 2013
Abstract In this paper a Banach space framework is introduced in order to deal with finite-dimensional pathdependent stochastic differential equations. A version of Kolmogorov’s backward equation is formulated and solved both in the space of Lp paths and in the space of continuous paths using the associated stochastic differential equation, thus estabilishing a relation between path-dependent SDEs and PDEs in analogy with the classical case. Finally it is shown how to estabilish a connection between such Kolmogorov’s equation and the analogue finite-dimensional equation that can be formulated in terms of the path-dependent derivatives recently introduced by Dupire, Cont and Fournié.
1
Introduction
In the recent literature, a growing interest for path-dependent stochastic equations has arisen, due both to their mathematical interest and to their possible applications in finance. The path-dependent SDEs considered here will be of the form dX(t) = bt (Xt ) dt + σ dW (t) for t ∈ [t0 , T ], (1) Xt0 = γt0 where {W (t)}t≥0 is a Brownian motion in Rd , σ is a diagonalizable d × d matrix, the solution X(t) at time t takes values in Rd , the notation Xt stands for the path of the solution on the interval [0, t], bt is , for each t ∈ [0, T ], a map from a suitable space of paths to Rd , γt0 is a given path on [0, t0 ]. After the insightful ideas proposed by Dupire ([Dup09]) and Cont and Fournié ([CF13], [CF10a], [CF10b]), who introduced a new concept of derivative and developed a path-dependent It¯o formula which exhibits a first connection between SDEs and PDEs in the path-dependent situation, some effort was made into generalising some classical concept to this setting, like forward-backward systems and viscosity solutions (see [PW11], [TZ13], [EKTZar], [ETZ13a], [ETZ13b]). Also, depending on the approach, there are some similarities with investigations about delay equations, see for instance [FGG10], [GM06], [FMT10]. Some of these works formulate a path-dependent Kolmogorov equation associated to the path-dependent SDE (1). Several issues about such Kolmogorov’s equation are of interest. The purpose of our work is to prove existence of classical C 2 solutions and to develop a Banach space framework suitable for this problem. To this aim we follow the classical method based on the probabilistic representation formula in terms of solutions to the SDE, which however, as explained in detail below, requires a new nontrivial analysis in our framework. ∗
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1
1.1
Notation
We will use the following notations throughout the paper (in addition to those introduced above): T will stand for a fixed finite time-horizon; Xt (r) will stand again for the value of X at r, r < t. Stochastic processes will be denoted with upper-case letters, while greek lower-case letters will be used for determinitic paths, most of the times seen as points in some paths space. As long as no stochastics are involved, one can always think of a path γ as defined on the whole interval [0, T ] and read γt as its restriction to [0, t]. By C([a, b]; Rd ) and D([a, b]; Rd ) we will denote respectively the space of continuous and càdlàg functions from the real interval [a, b] into Rd ; D([a, b); Rd ) will denote the set of càdlàg functions that have finite left limit also for t → b.
1.2
Main Results
A path-dependent non-anticipative function is a family of functions b = {bt }t∈[0,T ] , each one being defined on D([0, t]; Rd ) with values in Rd and being measurable with respect to the canonical σ-field on D([0, t]; Rd ). Some possible examples of path-dependent functions are the following: (i) for g : [0, T ] × Rd → Rd smooth, consider the function Z t bt (γ(t)) = g(t, γ(s)) ds ; 0
(ii) for 0 ≤ t1 ≤ t2 ≤ · · · ≤ tn ≤ T fixed consider the function bt (γt ) = hi(t) γ(t1 ), . . . , γ ti(t)
where for each t ∈ [0, T ] the index i(t) ∈ {1, . . . , n} is such that ti(t) ≤ t < ti(t)+1 and, for each j ∈ {1, . . . , n}, hj : Rd×j → Rd is a given function with suitable properties; (iii) in dimension d = 1 consider the function bt (γt ) = sup γ(s) . s∈[0,t]
In order to formulate the path-dependent SDE (1) as an SDE in Banach spaces, we consider a as a couple (endpoint, path) in some infinite-dimensional space, as it is usually done for delay equations, and reformulate consequently equation (1) as the infinite-dimensional abstract SDE √ (2) dY (t) = AY (t) dt + B(t, Y (t)) dt + Σ dβ(t) for t ∈ [t0 , T ] Y (t0 ) = y. (understood in mild sense) where A is the derivative operator, B is a sufficiently smooth (in Fréchet sense) nonlinear operator with range in Rd × {0} and β is a finite-dimensional Brownian motion (section 2.1). We associate to it the backward Kolomogorov equation in integral form with terminal condition Φ Z u (t, y) − Φ (y) = t
T
1 hDu (s, y) , Ay + B (s, y)i ds + 2
Z t
T
d X
σj2 D2 u(s, y)(ej , ej ) ds
(3)
j=1
and the related concept of solution (section 3). Our main result, under suitable regularity assumptions on B and Φ, as explained in section 5 is the following: Theorem. The function u(s, y) = E [Φ (Y s,y (T ))] , where Y s,y (t) solves equation (2), is of class C 2,α and solves the backward Kolmogorov equation in the space of continuous functions. 2
Since under our assumptions all the integrands appearing in (3) are in L∞ , a posteriori the function u is Lipschitz in t and hence, by Rademacher’s theorem, differentiable almst everywhere with respect to t. Therefore for almost every t it satisfies Kolmogorov’s backward equation in its differential form d
∂u 1X 2 2 (t, y) + hDu(t, y), Ay + B(t, y)i + σ D u(t, y)(ej , ej ) , ∂t 2 j=1 j
u(T, ·) = Φ .
We moreover exhibit some links between our results and the path-dependent calculus developed by Cont and Fournié (section 6). In particular, thanks to the theorem stated above, we can prove the following result (again under some regularity assumptions compatible with those of the previous theorem): Theorem. The function νs (γs ) = E [f (X γt (T ))] , where X γs (t) is the solution to equation (1), solves the path-dependent backward Kolmogorov equation ( Pd Dt ν(γt ) + bt (γt ) · Dνt (γt ) + 21 j=1 σj2 Di2 νt (γt ) = 0 , (4) νT (γT ) = f (γT ) . in which the derivatives are understood as horizontal and vertical derivatives as definded in [CF13].
1.3
Some ideas about the proofs
We intend here to find regular solutions to the Kolmogorov equation, by analogy with the classical theory. To this aim the space of L2 paths would appear to be the easiest setting to work in; unfortunately there are no significant example of path-dependent functions, not even integral functions, that satisfy the natural condition of having uniformly continuous second Fréchet derivative in L2 . To include a wider class of functions one would want to formulate and solve equations (2) and (3) in the space of continuous paths, that in our framework is the space x C : = y = ( ϕx ) ∈ Rd × C([−T, 0); Rd ) s.t. x = lim ϕ(s) . s↑0
This leads to two issues: first, the operator B (our abstract realization of the functional b) takes values in a x
space larger than C , thus we have to consider paths with a single jump-discontinuity at the final time t = 0. But then the semigroup generated by A shifts such discontinuity so that we have to deal with paths with a single discontinuity at an arbitrary time t. The need to work with a linear space and possibly with a Banach space structure suggests the choice of D : = Rd × D [−T, 0); Rd with the uniform norm as the ambient space for our equations. The second issue comes along when we try to estabilish the link between the SDE and the PDE. As in the classical theory, we need to work with some It¯o-type formula. We decide not to use some version of the It¯o formula in Banach spaces due to the difficulties one encounters in defining a concept of quadratic variation there (see for example [DGRarb], [DGRara], [DGFRar], [RDG10]), although we intend to address this problem in our future works; we proceed therefore using a Taylor expansion, but we are not able to control the second order terms in spaces endowed with the uniform norm. Therefore we adopt the following strategy: we go back to an Lp setting with p ≥ 2 (recovering in this way at least examples like the integral functionals) and we develop rigorously the relation between the SDE and the PDE in this framework (section 4). We then introduce a keen approximation procedure to extend our results to the space of continuous paths (section 5). This step requires us to introduce an additional assumption that remarks again the deep effort that is needed in order to obtain a satisfactory general theory already in the easiest case of regular coefficients.
3
2 2.1
The stochastic equation Framework
From now onwards fix a time horizon 0 < T < ∞ and a filtered probability space Ω, F, {Ft }t∈[0,T ] , P . We introduce the following spaces: C : = Rd × ϕ ∈ Cb ([−T, 0); Rd ) : ∃ lim ϕ(s) , s↑0 x C : = y = ( ϕx ) ∈ C s.t. x = lim ϕ(s) , s↑0
d
d
D : = R × Db ([−T, 0); R ), Dt : = y = ( ϕx ) ∈ D s.t. ϕ is discontinuous at most in the only point t , Lp : = Rd × Lp ((−T, 0); Rd ), p ≥ 2. All of them apart from Lp are Banach spaces with respect to the norm k ( ϕx ) k2 = |x|2 + kϕk2∞ , while Lp is a Banach space with respect to the norm k ( ϕx ) k2 = |x|2 + kϕk2p ; the space D turns out to be not separable with respect to this norm but this will not undermine our method. With these norms we have the natural relations x
C ⊂ C ⊂ D ⊂ Lp x
x
with continuous embeddings. We remark that C , C and D are dense in Lp while neither C nor C are dense in D. The choice for the interval [−T, 0] is made in accordance with most of the classical literature on delay equations. x
Notice that the space C has not the structure of a product space; notice also that it is isomorphic to the space C([−T, 0]; Rd ). As said above, we consider a family b = {bt }t∈[0,T ] of functions bt : D([0, t]; Rd ) → Rd adapted to the canonical filtration and we formulate the path dependent stochastic differential equation dX(t) = bt (Xt ) dt + σ dW (t) for t ∈ [t0 , T ], (5) Xt0 = γt0 where σ is a diagonalizable d × d matrix and W is a d-dimensional Brownian motion. b can also be seen as an Rd -valued function on the space D = ∪t D([0, t]; Rd ). To reformulate the path-dependent SDE (5) in our framework we need to introduce two linear bounded operators: for every t ∈ [0, T ] define the restriction operator Mt : D([−T, 0); Rd ) −→ D([0, t); Rd ) Mt (ϕ)(s) = ϕ(s − t),
s ∈ [0, t)
(6)
and the backward extension operator Lt : D([0, t); Rd ) −→ D([−T, 0); Rd ) Lt (γ)(s) = γ(0)1[−T,−t) (s) + γ(t + s)1[−t,0) (s), s ∈ [−T, 0).
(7)
Since the extension in the definition of Lt is arbitrary, one has that Mt Lt γ = γ while in general Lt Mt ϕ 6= ϕ.
4
(8)
Note also that both Lt and Mt map continuous functions into continuous functions. Set moreover Mt ϕ(s) s ∈ [0, t) x f Mt ( ϕ ) (s) = x s = t. Now given a functional b on D as in (5) one can define a function ˆb on [0, T ] × D setting ˆb (t, ( ϕx )) = ˆb(t, x, ϕ) : = bt M ft ( ϕx ) ;
(9)
(10)
conversely if ˆb is given one can obtain a functional b on D setting bt (γ) : = ˆb(t, γ(t), Lt γ).
(11)
The idea is simply to traslate and extend (or restrict) the path in order to pass from one formulation to another. For instance the functional of example (i) in section 1 would define a function ˆb on [0, T ] × D given by (i0 ) ˆb (t, ( ϕx )) =
Z
t
g(ϕ(s − t)) ds. 0
We consider again the path-dependent SDE (5) with the initial condition given now by a path ϕ on [−T + t0 , t0 ] and its terminal value x = ψ(t0 ), for s ∈ [t0 , T ] dX(s) = bs (Xs ) ds + σ dW (s) X(t0 ) = x = ψ(t0 ) (12) X(s) = ψ(s) for s ∈ [−T + t0 , t0 ); Recall that by Xs we denote the path of X starting from 0 up to time s, not a portion of the path of X of lenght T , which would be anyway well defined in this setting. If X solves (12) (in some space), for t ∈ [t0 , T ] we set X(t) Y (t) = {X(t + s)}s∈[−T,0] and then differentiate with respect to t formally obtaining ! ! ˙ 0 X(t) ˙ (t) dY (t) bt (Xt ) σW n o n o = + = + . ˙ + s) ˙ + s) X(t 0 X(t 0 dt
(13)
s
s
It is therefore natural to define the operators A
x 0 : = , ϕ ϕ˙
ˆb (t, ( ϕx )) x B t, : = ϕ 0 and
√
x σx Σ : = ϕ 0
(14)
(15)
(16)
and to formulate the infinite dimensional SDE √ dY (t) = AY (t) dt + B(t, Y (t)) dt +
Σ dβ(t) , t ∈ [t0 , T ] ,
(130 )
where β is given by β(t) =
W (t) , 0 5
(17)
with some initial condition Y (t0 ) = y. Solutions of this SDE will always be understood to be mild solutions, that is, we want to solve Z t Z t √ Y (t) = e(t−t0 )A y + e(t−s)A B(s, Y (s)) ds + e(t−s)A Σ dβ(s). t0
(1300 )
t0
It is not difficult to show that if Y solves (130 ) then its first coordinate X(t) solves the original SDE (12).
2.2
Some properties of the convolution integrals
The operator A has different domains depending on the space we choose: Dom(A) = ( ϕx ) ∈ Lp : ϕ ∈ W 1,p −T, 0; Rd , ϕ (0) = x , x x 1 d Dom(Ax ) = ( ϕ ) ∈ C : ϕ ∈ C [−T, 0); R ; C
x
one can think to define A on Lp and then consider its restriction to D or to C , as the notation above emphasizes. It is well known (see theorem 4.4.2 in [BDPDM92]) that A is the infinitesimal generator of a strongly x
continuous semigroup both in Lp and in C ; it’s easy to check that it still generates a semigroup in D which is not uniformly continuous. Indeed we have that x x etA = (18) ϕ (ξ + t) 1[−T,−t] (ξ) + x1[−t,0] (ξ) ξ∈[−T,0] . ϕ This formula comes from the trivial delay equation dx (t) = 0, dt x (0) = x,
t≥0 x (ξ) = ϕ (ξ) for ξ ∈ [−T, 0] ;
its solution, for t ≥ 0, is simply x (t) = x. If we introduce the pair x (t) y (t) := x|[t−T,t] then y (t) = etA
x ϕ
.
However it still holds that ketA kL(D,D) ≤ C for t ∈ [0, T ]
(19) x
with C not depending on t. Moreover it is evident from (18) that etA maps Lp into Lp , D into D and C x
into C , but it maps C into D−t because an element of C is essentially a continuous function with a unique discontinuity at its endpoint, and the semigroup just shifts that discontinuity. In particular this happens for elements of Rd × {0}. Consider the stochastic convolution Z t Z t √ (s) Z t0 (t) := e(t−s)A Σ dβ (s) = e(t−s)A σ dW , t ≥ t0 . 0 t0
t0
6
It is not obvious to investigate Z t0 by infinite dimensional stochastic integration theory, due to the difficult nature of the Banach space D. However we may study its properties thatnks to the following explicit formulae. From now on we work in a set Ω0 ⊆ Ω of full probability on which W has continuous trajectories. For any ω ∈ Ω0 and x ∈ Rd we have √ σx x (t−s)A Σ = e σx1[−(t−s),0] (ξ) ξ∈[−T,0] 0 hence
!
Rt
σ dW (s) Rt 1 (·) σ dW (s) t0 [−(t−s),0]
t0
t0
Z (t) = because
t
Z
=
Z
(20)
t
1[−(t−s),0] (ξ) σ dW (s) = t0
σ (W (t) − W (t0 )) σ (W ((t + ·) ∨ t0 ) − W (t0 ))
1[0,t+ξ] (s) σ dW (s) . t0
x
From the previous formula we see that Z t0 (t) ∈ C , hence Z t0 (t) ∈ Lp . We have
t
Z 0 (t) x = 2 sup |σ (W ((t + ξ) ∨ t0 ) − W (t0 ))| C
ξ∈[−T,0]
hence (using the fact that r 7→ W (t0 + r) − W (t0 ) is a Brownian motion and applying Doob’s inequality) # " h h i i
4 4 2 4 E Z t0 (t) x ≤ 24 E (21) sup |σW (s)| ≤ C 0 E |W (t − t0 )| ≤ C 00 (t − t0 ) C
s∈[0,t−t0 ]
where C 0 and C 00 are suitable constants. Consequently the same property holds in Lp (possibly with a x
different constant) by continuity of the embedding C ⊂ Lp . Moreover from (20) we obtain that for ω fixed !
t
t
Z 0 (t) − Z 0 (s) x = C |W (t) − W (s)| + sup |W ((t + ξ) ∨ t0 ) − W ((s + ξ) ∨ t0 )| . C
ξ∈[−T,0]
Observe that (we suppose s < t for simplicity) for ξ ∈ [−T, 0] 0 W (t + ξ) − W (t0 ) for ξ ∈ [t0 − t, t0 − s] W ((t + ξ) ∨ t0 ) − W ((s + ξ) ∨ t0 ) = W (t + ξ) − W (s + ξ) for ξ ∈ [t0 − s, 0] and sup
|W (t − ξ) − W (t0 )| =
ξ∈[t0 −t,t0 −s]
|W (η)|,
sup η∈[0,t−s]
x
therefore Z t0 is a continuous process in C , since any fixed trajectory of W is uniformly continuous. The x
same property holds then in Lp again by continuity of the embedding C ⊂ Lp . We can argue in a similar way for F t0 : [t0 , T ] × L∞ ([t0 , T ]; D) → D, Z t t0 F (t, θ) = e(t−s)A B (s, θ (s)) ds . t0
From (15) using (18) one deduces that e(t−s)A B (s, θ(s)) =
and therefore Z
bs (θ(s)) bs (θ(s)) 1[−t+s] (ξ)
t
e(t−s)A B (s, θ(s)) ds =
t0
7
Rt
b (θ(s)) ds o t0 s nR t+ξ bs (θ(s)) ds t0 ξ
x
which shows that F t0 (t, θ) always belongs to C . Writing Y t0 ,y (t) = e(t−t0 )A y + F t0 (t, Y t0 ,y ) + Z t0 (t) x
x
we see immediately that, for any t ∈ [t0 , T ], Y t0 ,y (t) ∈ D if y ∈ D and Y t0 ,y (t) ∈ C if y ∈ C . This will be crucial in the sequel.
2.3
Existence, uniqueness and differentiability of solutions to the SDE
We state and prove here some abstract results about existence and differentiability of solutions to the stochastic equation √ dY (t) = AY (t) dt + B(t, Y (t)) dt + Σ dβ(t), Y (t0 ) = y, (130 ) with respect to the initial data. By abstract we mean that we consider a general B not necessarily defined through a given b as in previous sections. Also A can be thought here to be a generic infinitesimal generator of a semigroup which is strongly continuous in Lp and satisfies (19) in D. Although all these theorems are analogous to well known results for stochastic equations in Hilbert spaces (see for example [DPZ92]), we give here complete and exact proofs due to the lack of them in the literature for the case of time-dependent coefficients in Banach spaces, which is the one of interest here. We are interested in solving the SDE in Lp and in D; since almost all the proofs can be carried out in the same way for each of the spaces we consider and since we do not need any particular property of these spaces themselves, we state all our results in this section in a general Banach space E, stressing out possible distinctions that could arise from different choices of E. We will make the following assumption: Assumption 2.1. B ∈ L∞ 0, T ; Cb2,α (E, E) for some α ∈ (0, 1), where we have denoted by Cb2,α (E, E) the space of twice Fréchet differentiable functions ϕ from E to E, bounded with their first and second differentials, such that x 7→ D2 ϕ (x) is α-Hölder continuous from E to L (E, E; E) (the space of bilinear forms on E). The L∞ property in time means that the differentials are measurable in (t, x) and both the function, the two differentials and the Hölder parameters are bounded in time. Under these conditions, B, DB, D2 B are globally uniformly continuous on E (with values in E, L (E, E), L (E, E; E)) respectively and with a uniform in time continuity modulus. Theorem 2.2. Equation (130 ) can be solved in a mild sense path by path: for any y ∈ E, any t0 ∈ [0, T ] and every ω ∈ Ω0 there exists a unique function [t0 , T ] 3 t → Y t0 ,y (ω, t) ∈ E which satisfies identity (1300 ) Y t0 ,y (ω, t) = e(t−t0 )A y +
Z
t
e(t−s)A B(s, Y t0 ,y (ω, s)) ds +
t0
Z
t
√ e(t−s)A Σ dβ(ω, s).
(1300 )
t0
Such a function is continuous if E = Lp , it is only in L∞ if E = D. Proof. Thanks to the Lipschitz property of B the proof follows through a standard argument based on the contraction mapping principle. The lack of continuity in D is due to the fact that the semigroup etA is not strongly continuous in D. Theorem 2.3. For almost every ω ∈ Ω, for all t0 ∈ [0, T ] and t ∈ [t0 , T ] the map y 7→ Y t0 ,y (t, ω) is twice Fréchet differentiable and the map y 7→ D2 Y t0 ,y (t, ω) is α-Hölder continuous from E to L (E, E; E). Proof. Due to its lenght the proof is postponed to the appendix. Theorem 2.4. If the solution Y t0 ,y (t) is continous as a function of t with values in E then it has the Markov property. 8
Proof. This follows immediately from theorem 9.15 on [DPZ92]. Notice that there the authors require a different set of hypothesis which however are needed only for proving existence and uniqueness of solutions and not in the actual proof of the result. It therefore applies to our situation as well. In section 4 we will need the notion of continuity modulus for the second Fréchet derivative of a map from E into E, together with some of its properties; we summarize what we will need in the following general remark. Remark 2.5. Given a map R : E → L (E, E; R), we define its continuity modulus ω (R, r) =
sup ky−y 0 k
E ≤r
kR (y) − R (y 0 )kL(E,E;R) .
Let v : E → R be a function with two Fréchet derivatives at each point, uniformly continuous on bounded sets. Then there exists a function rv : E 2 → R such that 1 1 v (x) − v (x0 ) = hDv (x0 ) , x − x0 i + D2 v (x0 ) (x − x0 , x − x0 ) + rv (x, x0 ) 2 2 2 |rv (x, x0 )| ≤ ω D2 v, kx − x0 kE kx − x0 kE for every x, x0 ∈ E. Indeed, 1 v (x) − v (x0 ) = hDv (x0 ) , x − x0 i + D2 v (ξx,x0 ) (x − x0 , x − x0 ) 2 where ξv,x,x0 is an iternediate point between x0 and x, and thus |rv (x, x0 )| = D2 v (ξv,x,x0 ) − D2 v (x0 ) (x − x0 , x − x0 )
2 ≤ D2 v (ξv,x,x0 ) − D2 v (x0 ) L(E 2 ,R) kx − x0 kE 2 ≤ ω D2 v, kx − x0 kE kx − x0 kE . If D2 v is α-Holder continuous, namely
2
D v (y) − D2 v (y 0 ) 2 ≤ M ky − y 0 kα E L(E ,R) then
α ω D2 v, kx − x0 kE ≤ M kx − x0 kE
and thus
2+α
|rv (x, x0 )| ≤ M kx − x0 kE
3
.
The Kolmogorov equation
In this and the following section we introduce and solve the backward Kolmogorov equation in our infinitedimensional setting. The relation between the results we shall show and the finite-dimensional pathdependent SDE we started from will be investigated in section 6. Suppose for a moment we are working in a standard Hilbert-space setting, that is, in a space H = R × H where H is a Hilbert space. Then (see again [DPZ92]) the backward Kolmogorov equation, for the unknown u : [0, T ] × H → R, is ∂u 1 (t, y) + Tr ΣD2 u(t, y) + hDu(t, y), Ay + B (t, y)i = 0, ∂t 2
u(T, ·) = Φ,
(22)
where Φ is a given final condition and Du, D2 u represent the first and second Fréchet derivatives with respect to the variable y. Its solution, under suitable hypothesis on A, B, Σ and Φ, is given by u(t, y) = E Φ Y t,y (T ) (23) 9
where Y t,y (t) solves the associated SDE √ dY (s) = [AY (s) + B (s, Y (s))] ds +
s ∈ [t, T ], Y (t) = y
Σ dβ(s),
(130 bis)
in H. In our framework, where the spaces are only Banach spaces, we have to give a precise meaning to the Kolmogorov equation and prove its relation above with the SDE. x
As outlined in the introduction we would like to solve it on the space C , but since B(t, y) belongs to x
Rd ×{0} * C , in order to give meaning to the term hDu(t, y), B(t, y)i we need Du(t, y) to be a functional defined at least on C, which necessarily implies u to be defined on [0, T ] × C. Therefore we should solve (in mild sense) the SDE for y ∈ C and this implies that Y t,y (s) ∈ D−t+s for s 6= t; this in turn requires Φ to be defined at least on ∪s∈[t,T ] D−t+s in order for a function of the form (23) to be well defined. However the space ∪Ds is not a linear space, thus it turns out that it is more convenient, also for exploiting a Banach space structure, to formulate everything in D, that is u : [0, T ] × D → R. Therefore we interpret h·, ·i in this setting as the duality pairing between D0 and D. For the trace term, if we denote by e1 , . . . , ed an orthonormal basis of Rd where σ diagonalizes, i.e. σej = σj ej for some real σj (in any of the spaces considered up to now), we could complete it to an orthonormal system {en } in H obtaining that X 2 2 Tr ΣD2 u(t, y) = σj hD u(t, y)ej , ej i; j
hence, by analogy, also when working in D we interpret the trace term as d X σj2 D2 u(t, y)(ej , ej ). Tr ΣD2 u(t, y) =
(24)
j=1
Moreover we consider Kolmogorov equation in its integrated form with respect to time, that is, given a (sufficently regular; see below) real function Φ on D we seek for a solution of the PDE Z u (t, y) − Φ (y) =
T
hDu (s, y) , Ay + B (s, y)i ds + t
1 2
Z t
T
d X
σj2 D2 u(s, y)(ej , ej ) ds.
(25)
j=1
Here one can see one of the difficulties in working with Banach spaces: the second order term in the equation comes from the quadratic variation of the solution of the SDE, but in such spaces there is no general way of defining a quadratic variation (although, as mentioned at the beginning, some results in this direction are appearing in the literature recently). Although we will seek for such a u, when dealing with the equation we will always choose y to be in Dom(Ax ), to let all the terms appearing there be well defined. C All these observations lead to our definition of solution to (25); first we say that a functional u on [0, T ]×D belongs to L∞ 0, T ; Cb2,α (D, R) if it is twice Fréchet differentiable on D, u, Du and D2 u are bounded, the map x 7→ D2 u(x) is α-Hölder continuous from D to L (D, D; D) (the space of bilinear forms on D), the differentials are measurable in (t, x) and the function, the two differentials and the Hölder parameters are bounded in time Definition 3.1. Given Φ ∈ Cb2,α (D, R), we say that u : [0, T ] × D → R is a classical solution of the Kolmogorov equation with final condition Φ if 0 u ∈ L∞ 0, T ; Cb2,α (D, R) ∩ C ([0, T ] × D, R) for some α0 ∈ (0, 1), and satisfies identity (25) for every t ∈ [0, T ] and y ∈ Dom Ax , with the duality C terms understood with respect to the topology of D. 10
It will be clear in section 5 that the restriction y ∈ Dom Ax is necessary and that it would not be C possible to obtain the same result choosing y in some larger space. Our aim is to show that, in analogy with the classical case, the function u(t, y) = E Φ(Y t,y (T )) solves equation (25). However we are not able to prove this result directly, due essentially to the lack of an appropriate It¯o-type formula for our setting. Therefore we will proceed as follows: first we are going to show how to prove such a result in Lp , then we will show that if the problem is formulated in D it is possible to approximate it with a sequence of Lp problems; the solutions to such approximanting problems will be finally shown to converge to a function that solves Komogorov’s backward PDE in the sense of definition 3.1. All the above discussion about the meaning of Kolmogorov’s equation applies verbatim to the space Lp . A solution in Lp is defined in a straightforward way as follows: Definition 3.2. Given Φ ∈ Cb2,α (Lp , R), we say that u : [0, T ] × Lp → R is a solution of the Kolmogorov equation in Lp with final condition Φ if 0 u ∈ L∞ 0, T ; Cb2,α (Lp , R) ∩ C ([0, T ] × Lp , R) for some α0 ∈ (0, 1), and satisfies identity (25) for every t ∈ [0, T ] and y ∈ Dom (A), with the duality terms understood with respect to the topology of Lp .
4
Solution in Lp
The choice to work in a general Lp space instead of working with the Hilbert space L2 could seem unjustified at first sight. As long as solving Kolmogorov’s equation in Lp is only a step towards solving it in D through approximations it would be enough to develop the theory in L2 , where the results needed are well known. Nevertheless we give and prove here this more general statement for Lp spaces for some reasons. First, the proof shows a method to obtain this kind of result without actually using a It¯o-type formula, but only a Taylor expansion; the difference is tiny but it allows to work in spaces where there is no It¯o formula to apply. Second, the proof points out where a direct argument of this kind (which is essentially the classical scheme for these results) fails. Last, also the easiest examples do not behave well in L2 : recall for instance example (i) with its “delay”-version ˆb (t, ( ϕx )) x B t, = ϕ 0 where ˆb is given in (i0 ). The second Fréchet derivative of B with respect to y = ( ϕx ) is simply R t g 00 (ϕ(s − t))ψ(s − t)χ(s − t) ds x D2 B (t, y) ( ψ1 , ( xχ2 )) = 0 0 which is not uniformly continuous in ( ϕx ) in L2 , since for z = ( ϕx11 ) kD2 B(t, y) − D2 B(t, z)kL(L2 ,L2 ;L2 ) = Z t 00 g ϕ(s − t) − ϕ1 (s − t) · |ψ(s − t)| · |χ(s − t)| ds = sup χ,ψ∈L2 kχk,kψk≤1
≤ kg 000 k
0
Z χ,ψ∈L2 kχk,kψk≤1
t
|ϕ(s − t) − ϕ1 (s − t)| · |χ(s − t)| · |ψ(s − t)| ds
sup 0
11
can not be bounded in terms of kϕ − ϕ1 kL2 if ϕ, ϕ1 , χ and ψ are only in L2 , and uniform continuity is essential in our proof, as it is in all classical cases. D2 B is however uniformly continous in L4 . This shows that proving the result in Lp is already enough to deal with some examples, without the need to go further in the development of the theory. If B satisfies assumption 2.1 with E = Lp , theorems 2.2, 2.3 and 2.4 yield that the SDE √ dY (s) = [AY (s) + B (s, Y (s))] ds + Σ dβ(s) , s ∈ [t, T ] , Y (t) = y (130 bis) admits a unique mild solution Y t0 ,y (t) in Lp which is continuous in time, Cb2,α with respect to y and has the Markov property. Theorem 4.1. Let Φ : Lp → R be in Cb2,α and let assumption 2.1 hold in Lp . Then the function u (t, y) := E Φ Y t,y (T ) , (t, y) ∈ [0, T ] × Lp , is a solution of the Kolmogorov equation in Lp with final condition Φ. Proof. Throughout this proof k · k will denote the norm in Lp and h·, ·i will denote duality between Lp and 0 Lp , where p1 + p10 = 1. Since Φ ∈ Cb2,α (Lp , R), theorem 2.3 assures that the function u has the regularity properties required by the definition of solution. We have thus to show that it satisfies equation (25). Recall that we choose y in the domain of A. Step 1. Fix t0 ∈ [0, T ]. From Markov property, for any t1 > t0 in [0, T ], we have u (t0 , y) = E u t1 , Y t0 ,y (t1 ) because E Φ Y t0 ,y (T ) = E E Φ Y t0 ,y (T ) |Y t0 ,y (t1 ) h i = E E Φ Y t1 ,w (T ) w=Y t0 ,y (t1 ) = E u t1 , Y t0 ,y (t1 ) . From Taylor formula applied to the function y 7→ u (t, y) we have D E u t1 , Y t0 ,y (t1 ) − u t1 , e(t1 −t0 )A y = Du t1 , e(t1 −t0 )A y , Y t0 ,y (t1 ) − e(t1 −t0 )A y 1 + D2 u t1 , e(t1 −t0 )A y Y t0 ,y (t1 ) − e(t1 −t0 )A y, Y t0 ,y (t1 ) − e(t1 −t0 )A y 2 1 + ru(t1 ,·) Y t0 ,y (t1 ) , e(t1 −t0 )A y 2 where
2
ru(t1 ,·) Y t0 ,y (t1 ) , e(t1 −t0 )A y ≤ ω D2 u (t1 , ·) , Y t0 ,y (t1 ) − e(t1 −t0 )A y Y t0 ,y (t1 ) − e(t1 −t0 )A y 0
(for the definitions of r and ω see Remark 2.5 in the appendix). Due to the Cb2,α (Lp , R)-property, uniform in time, we have
2+α0
. ru(t1 ,·) Y t0 ,y (t1 ) , e(t1 −t0 )A y ≤ M Y t0 ,y (t1 ) − e(t1 −t0 )A y Recall that Y t0 ,y (t1 ) − e(t1 −t0 )A y = F t0 t1 , Y t0 ,y + Z t0 (t1 ) Z t1 F t0 t1 , Y t0 ,y = e(t1 −s)A B s, Y t0 ,y (s) ds t0
12
and E Z t0 (t1 ) = 0 h
4 i 2 E Z t0 (t1 ) ≤ CZ4 (t1 − t0 )
t
F 0 t1 , Y t0 ,y ≤ CA kBk ∞,Lp (t1 − t0 ) Hence, recalling u (t0 , y) = E [u (t1 , Y t0 ,y (t1 ))], u (t0 , y) − u t1 , e(t1 −t0 )A y = D E = Du t1 , e(t1 −t0 )A y , E F t0 t1 , Y t0 ,y i 1 h + E D2 u t1 , e(t1 −t0 )A y F t0 t1 , Y t0 ,y + Z t0 (t1 ) , F t0 t1 , Y t0 ,y + Z t0 (t1 ) 2 i 1 h + E ru(t1 ,·) Y t0 ,y (t1 ) , e(t1 −t0 )A y . 2 Step 2. Now let us explain the strategy. Given t ∈ [0, T ], taken a sequence of partitions πn of [t, T ], of the form t = tn1 ≤ ... ≤ tnkn +1 = T of [t, T ] with |πn | → 0, we take t0 = tni and t1 = tni+1 in the previous identity and sum over the partition πn to get u (t, y) − Φ (y) + In(1) = In(2) + In(3) + In(4) where In(1) :=
kn X
n n u tni+1 , y − u tni+1 , e(ti+1 −ti )A y
i=1
In(2) :=
kn D X
h n iE n n n Du tni+1 , e(ti+1 −ti )A y , E F ti tni+1 , Y ti ,y
i=1
In(3) :=
kn h n n 1X E D2 u tni+1 ,e(ti+1 −ti )A y 2 i=1 n i n n n n n F ti tni+1 , Y ti ,y + Z ti tni+1 , F ti tni+1 , Y ti ,y + Z ti tni+1 k
In(4) :=
n h n i n n 1X E ru(tn ,·) Y ti ,y tni+1 , e(ti+1 −ti )A y . i+1 2 i=1
We want to show that T
Z
(1)
(I) lim In = −
hDu (s, y) , Ayi ds if y ∈ Dom (A),
n→∞
(II) lim
n→∞
(III) lim
n→∞
t (2) In
Z
T
hDu (s, y) , B (s, y)i ds,
= t
(3) In
1 = 2
Z t
T
d X
σj2 D2 u (s, y) (ej , ej ) ds,
j=1
(4)
(IV ) lim In = 0. n→∞
13
Step 3. We have, for y ∈ Dom (A) (in this case kn X
u
tni+1 , y
−u
n n tni+1 , e(ti+1 −ti )A y
d tA dt e y
=−
kn Z X
i
i
=−
k n Z tn X i+1 i
Z =− t
tn i
= AetA y) n tn i+1 −ti
Du tni+1 , esA y , AesA y ds
0
D E n n Du tni+1 , e(s−ti )A y , Ae(s−ti )A y ds
kn D T X
E n n Du tni+1 , e(s−ti )A y , Ae(s−ti )A y 1[tni ,tni+1 ] (s) ds
i
tA
The semigroup e is strongly continuous in Lp therefore it converges to the identity as t goes to 0; hence, since y is fixed, taking the limit in n yields (I) applying the dominated convergence theorem. Step 4. The function By standard properties of the Bochner integral we have * + Z tni+1 kn X n n n n (tn −t )A (t t −s)A ,y Du ti+1 , e i+1 i y , E e i+1 B s, Y i (s) ds = tn i
i=1
=
kn X
Z E
i=1
Z =E t
tn i+1
tn i
D
E n n n n Du tni+1 , e(ti+1 −ti )A y , e(ti+1 −s)A B s, Y ti ,y (s, ω) ds
kn D T X
E n n n n Du tni+1 , e(ti+1 −ti )A y , e(ti+1 −s)A B s, Y ti ,y (s, ω) 1[tni ,tni+1 ] (s) ds;
i=1
now arguing as in the previous step it’s easy to prove that this quantity converges to Z T hDu(s, y), B(s, y)i ds. t (3)
Step 5. First split each of the addends appearing in In as follows: n n n n n n n n D2 u tni+1 , e(ti+1 −ti )A y F ti tni+1 , Y ti ,y + Z ti tni+1 , F ti tni+1 , Y ti ,y + Z ti tni+1 = n n n n n n = D2 u tni+1 , e(ti+1 −ti )A y F ti tni+1 , Y ti ,y , F ti tni+1 , Y ti ,y + n n n n n + D2 u tni+1 , e(ti+1 −ti )A y F ti tni+1 , Y ti ,y , Z ti tni+1 + n n n n n + D2 u tni+1 , e(ti+1 −ti )A y Z ti tni+1 , F ti tni+1 , Y ti ,y + n n n n + D2 u tni+1 , e(ti+1 −ti )A y Z ti tni+1 , Z ti tni+1 . Let us give the main estimates. We have h h
i 2 i E D2 u t, e(t−t0 )A y F t0 t, Y t0 ,y , F t0 t, Y t0 ,y ≤ D2 u ∞,Lp E F t0 t, Y t0 ,y
2 2 2 ≤ D2 u ∞,Lp CA kBk∞,Lp (t − t0 ) and h h
i 2 i1/2 h t0 2 i1/2 E Z (t) E D2 u t, e(t−t0 )A y F t0 t, Y t0 ,y , Z t0 (t) ≤ D2 u ∞,Lp E F t0 t, Y t0 ,y
2 3/2 ≤ D u ∞,Lp CA CZ kBk∞,Lp (t − t0 ) hence the first three terms give no contribution when summing up over i, because they are estimated by a power of ti+1 − ti greater than 1. Therefore it remains to show that the term kn X
h n i n n n E D2 u tni+1 , e(ti+1 −ti )A y Z ti tni+1 , Z ti tni+1
i=1
14
(26)
converges to
Rt t0
σ 2 D2 u(s, y)(e, e) ds. To this aim we recall that Z tni+1 n n (r) Z ti tni+1 = e(ti+1 −r)A σ dW 0 tn
i
σ W tni+1 − W (tni ) = σ W tni+1 + · ∨ tni − W (tni ) i Z0 . =: σ Z1i We split again (26) into kn h i i X n n Z0 , Z0 E D2 u tni+1 , e(ti+1 −ti )A y + 0 0 i=1
i n n 0 Z0 , + D2 u tni+1 , e(ti+1 −ti )A y + Z1i 0 n n 0 Z0i i + D2 u tni+1 , e(ti+1 −ti )A y , + Z1 0 i n n 0 0 i i + D2 u tn , e(ti+1 −ti )A y , . i+1
Z1
Z1
For the first term we have, using It¯o isometry, that kn X
h i i i n n Z0 , Z0 E D2 u tni+1 , e(ti+1 −ti )A y = 0 0
i=1
=
d X
σj2
j=1
kn X
n n D2 u tni+1 , e(ti+1 −ti )A y (ej , ej ) (tni+1 − tni )
i=1
Rt Pd and the right-hand side in this equation converges to j=1 σj2 t0 D2 (s, y)(ej , ej ) ds thanks to the strong continuity of etA . For the second term we can write n n 0 Z0i , i E D2 u tni+1 , e(ti+1 −ti )A y (27) Z 0 1
i h
≤ D2 u ∞,Lp E W tni+1 − W (tni ) W tni+1 + · ∨ tni − W (tni ) Lp ! p1 Z tni+1 −tni
p ≤ D2 u ∞,Lp E W tni+1 − W (tni ) |W (r)| dr 0
Z 21
2 2 E ≤ D u ∞,Lp E W tni+1 − W (tni )
! p2 12
n tn i+1 −ti
p
|W (r)| dr
0
1 1
2 ≤ D u ∞,Lp tni+1 − tni 2 tni+1 − tni p E
1+ p1 ≤ D2 u ∞,Lp tni+1 − tni ,
! p2 12 sup n [0,tn i+1 −ti ]
p
(|W (r)| )
(28)
using It¯o isometry and Burkholder-Davis-Gundy inequality, thus it converges to zero when summing over i and letting n go to ∞. The third term can be shown to go to zero in the exact same way and by the same estimates as above one obtains that 2 n n 0 0 ≤ tni+1 − tni 1+ p , E D2 u tni+1 , e(ti+1 −ti )A y Z1i , Z1i 15
hence it follows that also this term gives no contribution when passing to the limit. Step 6. Since
2+α0
ru(t,·) Y t0 ,y (t) , e(t−t0 )A y ≤ M Y t0 ,y (t) − e(t−t0 )A y we have that
2+α0 h i
t0 ,y (t−t0 )A t0 ,y (t−t0 )AE (t) − e y y ≤ M E Y (t) , e E ru(t,·) Y ≤C
0 0 h h
4 i 2+α 4 i 2+α 4 4 E F t0 t, Y t0 ,y + E Z t0 (t)
!
0
1+ α2
≤ C (t − t0 ) (4)
and from this one proves that limn→∞ In = 0. Remark 4.2. The point in which the above argument fails when working directly in D is item (III) of step 2. Indeed step 5, which is the proof of the convergence in (III), can not be carried out when working with the sup-norm: if we start again from (27) using the norm of D we would end up with the estimate 2 2 0 tn −tn A n ( ) Z0i , ≤ D u i+1 i tni+1 − tni (280 ) y E D u ti+1 , e Z1i ∞,Lp 0 which is not enought to obtain the convergence to 0 that we need.
5
x
Solution in C
We now show how to use Lp approximations in order to obtain classical solutions of Kolmogorov’s equations in the sense of definition 3.1. As in the previous we will assume that B satisfied assumption 2.1 for E = D, that is B ∈ L∞ 0, T ; Cb2,α (D, D) for some α ∈ (0, 1). Suppose we have a sequence {Jn } of linear continuous operators from Lp (−T, 0; Rd ) n→∞ into C([−T, 0]; Rd ) such that Jn ϕ −→ ϕ uniformly for any ϕ ∈ C([−T, 0]; Rd ). By Banach-Steinhaus theorem we have that sup kJn kL(C;C) < ∞; however we need a slightly stronger property, namely that n
kJn f k∞ < CJ kf k∞ for all f with at most one jump, uniformly in n. Then we can define the sequence of operators Bn : [0, T ] × Lp → Rd × {0} Bn (t, y) = Bn (t, ( ϕx )) = Bn (t, x, ϕ) : = B (t, x, Jn ϕ) .
(29)
We will often write Jn ( ϕx ) for ( Jnxϕ ) in the sequel. It can be easily proved that if B satisfies assumption 2.1 in D then for every n the operator Bn satisfies assumption 2.1 both in D and in Lp . Thus if we consider the approximated SDE √ ˜ dYn (t) = AYn (t) dt + Bn (t, Yn (t)) dt + Σ dβ(t), Yn (s) = y ∈ Lp (30) by theorem 2.2 it admits a unique path by path mild solution Yns,y such that, thanks to theorem 2.3, the map t 7→ Yns,y (t) si in Cb2,α . Suppose also we are given a final condition Φ : D → R for the backward Kolmogorov equation (25) associated to the original problem with B; approximations Φn can be defined in the exact same way. We have then a sequence of approximated backward Kolmogorov’s equations in Lp , namely Z un (t, y) − Φ (y) =
T
hDun (s, y) , Ay + Bn (s, y)i ds + t
16
1 2
Z t
T
d X j=1
σj2 D2 un (s, y) (ej , ej ) ds (31)
with final condition un (T, ·) = Φn . Theorem 4.1 yields in fact that for each n the function un (s, y) = E [Φn (Yns,y (T ))]
(32) x
x
is a solution to equation (31) in LP . If we choose the initial condition y in the space C then Yns,y (t) ∈ C as well for every n and every t ∈ [s, T ] and therefore we have uniform convergence of Jn Yns,y to Yns,y . An example of a sequence {Jn } satisfying the required properties can be constructed as follows: for any ε ∈ 0, T2 define a function aε : [−T, 0] → [−T, 0] as −T + ε aε (x) = x −ε
if x ∈ [−T, −T + ε] if x ∈ [−T + ε, −ε] if x ∈ [−ε, 0].
Then choose any C ∞ (R; R) function ρ such that kρk1 = 1, 0 ≤ ρ ≤ 1 and supp(ρ) ⊆ [−1, 1] and define a sequence {ρn } of mollifiers by ρn (x) := nρ(nx). Finally set, for any ϕ ∈ L1 (−T, 0; Rd ) Z
0
Jn ϕ(x) := −T
ρn a n1 (x) − y ϕ(y) dy.
(33)
We will need one further assumption, together with the required properties for Jn that we write again for future reference. Assumption 5.1. There exists a sequence of linear continuous operators Jn : Lp (−T, 0; Rd ) → C([−T, 0]; Rd ) n→∞ such that Jn ϕ −→ ϕ uniformly for any ϕ ∈ C([−T, 0]; Rd ) and supn kJn ϕk∞ < CJ kϕk for every ϕ that has at most one jump and is continuous elsewhere. x
The drift B and the final condition Φ are such that for any s ∈ [−T, 0], any r ≥ s, any y ∈ C and for almost every a ∈ [−T, 0] the following hold: 1 1 DB (r, y) Jn 1[a,0] −→ DB (r, y) 1[a,0] ; 1 1 hDΦ(y), Jn 1[a,0] i −→ hDΦ(y), 1[a,0] i; 1 1 1 , 1[a,0] −→ 0; − 1[a,0] D2 Φ(y) Jn 1[a,0] 1 1 1 D2 Φ(y) 1[a,0] , Jn 1[a,0] − 1[a,0] −→ 0; 1 1 1 1 , Jn 1[a,0] D2 Φ(y) Jn 1[a,0] − 1[a,0] − 1[a,0] −→ 0. 1 Remark 5.2. Assumption 5.1 implies that the same set of properties holds if we substitute 1[a,0] with ψ(0) any element q = ∈ D−a , that is, it has at most one jump and no other discontinuities; this happens ψ because any such ψ is the sum of a continuous function and an indicator function, and all the derivatives appearing in the above assumptions are linear. Remark 5.3. The infinite dimensional operators associated to examples (i) and (ii) in section 1 through (15) satisfy this assumption if we choose Jn as in (33). Example (ii) does not satisfy assumption 5.1 for every a but only for a 6= tj , n = 1, . . . , n. We state and prove now the main result in this work. Theorem 5.4. Let Φ ∈ C 2,α (D, R) be given and let assumption 2.1 hold for E = D. Under assumption 5.1 the function u : [0, T ] × D → R given by u(t, y) = E Φ Y t,y (T ) , (34)
17
where Y t,y is the solution to equation (130 bis) in D, is asolution of the Kolmogorov equation with final
condition Φ, that is, for every (t, y) ∈ [0, T ] × Dom Ax it satisfies identity C
Z
T
u (t, y) − Φ (y) = t
1 hDu (s, y) , Ay + B (s, y)i ds + 2
Z t
T
d X
σj2 D2 u(s, y)(ej , ej ) ds.
(25)
j=1
Proof. Let Bn , Φn , Yn and un be as above. The proof will be divided into some steps that will prove the following: for y ∈ Dom Ax C
x
Yns,y (t) → Y s,y (t) in C for every t uniformly in ω; un (s, y) → u(s, y) = E [Φ (Y s,y (T ))] for every s pointwise in y; equation (31) converges to equation (25) for any t ∈ [0, T ]. Step 1 We first need to compute
s,y
Yn (t) − Y s,y (t) x C
Z t
Z t
(t−r)A s,y (t−r)A s,y
= e B (r, Y (r)) dr − e B (r, Y (r)) dr n n
x s s
Z t
C Z t
≤ e(t−r)A Bn (r, Y s,y (r)) dr − e(t−r)A B (r, Y s,y (r)) dr
x + s s
C
Z t Z t
e(t−r)A Bn (r, Yns,y (r)) dr − e(t−r)A Bn (r, Y s,y (r)) dr
x .
s
s
(35) (36)
C
For the term (35) recall that e(t−r)A Bn (r, Y s,y (r)) = e(t−r)A B (r, Jn Y s,y (r)) and that, thanks to the properties of Jn , n→∞
Jn Y s,y (r) −→ Y s,y (r) x
in C , hence by continuity of B B (r, Jn Y s,y (r)) −→ B (r, Y s,y (r))
(37)
pointwise as functions of r. Since B is uniformly bounded in r ∈ [s, t], by the dominated convergence theorem Z t Z t lim e(t−r)A Bn (r, Y s,y (r)) dr = e(t−r)A B (r, Y s,y (r)) dr; n→∞
s
s
Z t
Z t
(t−r)A s,y (t−r)A s,y
e Bn (r, Y (r)) dr − e B (r, Y (r)) dr
x < ε s
s
C
for n big enough. Consider now (36) ; Z t
Z t
e(t−r)A Bn (r, Yns,y (r)) dr − e(t−r)A Bn (r, Y s,y (r)) dr x
C s s Z t ≤C kB (r, Jn Yns,y (r)) − B (r, Jn Y s,y (r))k dr s Z t ≤C KB kYns,y (r) − Y s,y (r)k dr. s
18
(38)
because, for any ψ ∈ C, kJn ψk∞ ≤ CJ kψk∞ and therefore kJn yk ≤ CJ kyk. Hence this and (38) yield, by Gronwall’s lemma, kYns,y (t) − Y s,y (t)kx ≤ εeT CKB C
for any ε > 0 and n big enough. This implies that
Yns,y (t)
x
converges to Y s,y (t) in C for any t. x
Step 2 It is now easy to deduce that un (s, y) converges to u(s, y) for any s, y ∈ C . In fact un (s, y) − u(s, y) ≤ ≤ E |Φn (Yns,y (T )) − Φn (Y s,y (T ))| + E |Φn (Y s,y (T )) − Φ (Y s,y (T ))| and for almost any ω |Φn (Yns,y (T )) − Φn (Y s,y (T ))| ≤ KΦ kYns,y (T ) − Y s,y (Y )k and |Φn (Y s,y (T )) − Φ (Y s,y (T ))| ≤ KΦ kJn Y s,y (T ) − Y s,y (T )k , both of which are arbitrarily small for n large enough; now since B is bounded and we assumed that 4 E kZk is finite, we can apply again the dominated convergence theorem (integrating in the variable ω) to conclude this argument. Step 3 We now approach the convergence of the term hDun (s, y), Bn (s, y)i; it is enough to consider ageneric sequence g˜n → g˜ in R, to which we associate the corresponding sequence gn = g˜0n → g = g0˜ in C ⊂ D. We remark here that the duality D0 hDun (s, y), gn iD is well defined and equals Lp0 hDun (s, y), gn iLp ; a simple proof of this fact is the following: un is Fréchet differentiable both on D and on Lp and its Gâteaux derivatives along the directions in D are of course the same in D and in Lp , therefore also the Fréchet derivatives must be equal. Now | D0 hDun , gn iD −
D 0 hDu, giD |
= |hDun , gn − gi + hDun − Du, gi| ≤ |hDun − Du, gi| + |hDun , gn − gi| = |A| + |B| .
We show that for s, y fixed the set {Dun (s, y)}n is bounded in D0 . From the definition of u we have that for h ∈ D hDun (s, y), hi = hDΦn (Yns,y (T )) , DYns,y (T )hi and it is easily shown that DΦn (ˆ y ) = DΦ(Jn yˆ)Jn for any yˆ ∈ D. DΦ is boundend by assumption, whereas by the required properties of Jn kJn DYns,y (T )hkD ≤ CJ kDYns,y (T )hkD ; since the kDYn k are uniformly bounded by a constant depending only on etA and on DB, we have that the Dun ’s are uniformly bounded as well and therefore B → 0 as gn → g. The term A requires some work: first write (suppressing indexes s, y and T ) A = hDΦn (Yn ), DYn gi − hDΦ(Y ), DY gi = hDΦn (Yn ), (DYn − DY )gi + hDΦn (Yn ) − DΦ(Y ), DY gi = A1 + A2 , A2 = hDΦn (Yn ) − DΦn (Y ), DY gi + hDΦn (Y ) − DΦ(Y ), DY gi = A21 + A22 . x
Since the Lipschitz constants of DΦn are uniformly bounded in C we have that |A21 | ≤ kDΦn (Yn ) − DΦn (Y )kD0 kDY gkD ≤ C kYn − Y k kDY gk 19
and the last line goes to zero as n goes to infinity. For A22 write |A22 | = |hDΦ(Jn Y )Jn , DY gi − hDΦ(Y ), DY gi| ≤ |hDΦ(Jn Y )Jn , DY gi − hDΦ(Y )Jn , DY gi| |hDΦ(Y )Jn , DY gi − hDΦ(Y ), DY gi| ≤ KDΦ kJn Y − Y k kDY gk + |hDΦ(Y )Jn , DY gi − hDΦ(Y ), DY gi| ; the first term goes to zero by properties of Jn , the second one thanks to assumption 5.1: this is beacause from the defining equation for DY one easily sees that for any ( g0 ) ∈ C the second component of DY g has a unique discontinuity point, and our assumption is made exactly in order to be able to control the convergence of these terms. Now we consider A1 : DYns,y (T )g−DY s,y (T )g = Z T = e(T −r)A DBn (r, Yns,y (r)) [DYns,y (r) − DY s,y (r)] g dr s T
Z
e(T −r)A [DBn (r, Yns,y (r)) − DB (r, Y s,y (r))] DY s,y (r)g dr
+
(39)
s
= A11 + A12 and A12 can be written as Z T A12 = e(T −r)A [DBn (r, Yns,y (r)) − DBn (r, Y s,y (r))] DY s,y (r)g dr s T
Z
e(T −r)A [DBn (r, Y s,y (r)) − DB (r, Y s,y (r))] DY s,y (r)g dr
+ s
= A121 + A122 whence Z
T
kDY s,y (r)gk kDB (r, Jn Yns,y (r)) − DB (r, Jn Y s,y (r))k dr
kA121 k ≤ C s
Z
T
kDY s,y (r)gk kDB(r, ·)k kJn Yns,y (r) − Jn Y s,y (r)k
≤C s
Z ≤C
T
kDY s,y (r)gk kDB(r, ·)k kYns,y (r) − Y s,y (r)k dr
s
that goes to zero; for A122 k[DBn (r, Y s,y (r)) −DB (r, Y s,y (r))]DY gk ≤ kDB (r, Jn Y s,y (r)) − DB (r, Y s,y (r)) kkJn DY s,y (r)gk+ + kDB (r, Y s,y (r)) [Jn DY s,y (r)g − DY s,y (r)g] k ≤ KDB kJn Y s,y (r) − Y s,y (r)k kDY s,y (r)gk+ + kDB (r, Y s,y (r)) [Jn DY s,y (r)g − DY s,y (r)g] k where the last line goes to zero thanks to assumption 5.1 again, and therefore A122 goes to zero by the dominated convergence theorem. From (39) and this last argument it follows that for any fixed ε > 0 Z kDYn (T )g − DY (T )gk ≤ C
T
kDBn kkDYns,y (r)g − DY s,y (r)gk dr + ε
(40)
s
for n large enough. Since kDBn k is bounded uniformly in n and in r we can use Gronwall’s lemma to prove that kDYns,y (T )g − DY s,y (T )gk → 0, and since kDΦn k are uniformly bounded as well we can 20
conclude that also A1 → 0 as n → ∞. Putting together all the pieces we just examined we obtain the desired convergence of hDun , Bn i to hDu, Bi. Step 4 All the procedures used in the previous steps apply again to treat the convergence of the term hDun (s, y), Ayi, no further passages are needed; therefore we omit the computations and go on to the term involving the second derivatives. Step 5 We will study only the convergence of D2 un (s, y)(e1 , e1 ) since the σj ’s are constants and the passage from one to d dimensions is trivial. We will drop the subscript 1 in the computations to simplify notations. We can proceed as follows (suppressing again s, y and T ): |D2 un (s, y)(e, e)−D2 u(s, y)(e, e)| ≤ ≤ D2 Φn (Yn ) (DYn e, DYn e) − D2 Φ(Y ) (DY e, DY e) + hDΦn (Yn ), D2 Yn (e, e)i − hDΦ(Y ), D2 Y (e, e)i = |C| + |D|. The kind of computations needed are similar to those for the terms involving the first derivative. We first write C as C = D2 Φn (Yn ) (DYn e, DYn e) − D2 Φn (Yn ) (DY e, DY e) + + D2 Φn (Yn ) (DY e, DY e) − D2 Φ(Y ) (DY e, DY e) = C1 + C2 . For C1 just write |C1 | ≤ D2 Φn (Yn ) (DYn e − DY e, DYn e − DY e) + + D2 Φn (Yn ) (DY e, DYn e − DY e) + D2 Φn (Yn ) (DYn e − DY e, DY e) ≤ kD2 Φn (Yn )k kDYn e − DY ek2 + 2kDY ek kDYn e − DY ek and the last line goes to zero by the same reasoning as in A1 and the boundedness of kD2 Φn (Yn )k (uniformly in n). For C2 C2 = D2 Φ(Jn Y ) − D2 Φ(Y ) (Jn DY e, Jn DY e) + D2 Φ(Y ) (Jn DY e, Jn DY e) + − D2 Φ(Y ) (DY e, DY e) 2 = D Φ(Jn Y ) − D2 Φ(Y ) (Jn DY e, Jn DY e) + D2 Φ(Y ) (Jn DY e, Jn DY e − DY e) + + D2 Φ(Y ) (Jn DY e − DY e, DY e) 2 = D Φ(Jn Y ) − D2 Φ(Y ) (Jn DY e, Jn DY e) + D2 Φ(Y ) (Jn DY e − DY e, Jn DY e − DY e) + + D2 Φ(Y ) (DY e, Jn DY e − DY e) + D2 Φ(Y ) (Jn DY e − DY e, Jn DY e − Dye) . Last three terms go to zero by assumption 5.1, while the first one is bounded in norm by CJ kD2 Φkα kJn Y − Y kα kDY ek2 which goes to zero since kJn Y − Y k → 0. We now go on with D: D = hDΦn (Yn ), D2 Yn (e, e) − D2 Y (e, e)i + hDΦn (Yn ) − DΦ(Y ), D2 Y (e, e)i = D1 + D2 and D2 is easy to handle since |D2 | ≤ |hDΦ(Yn ) − DΦn (Y ), D2 Y (e, e)i| + |hDΦn (Y ) − DΦ(Y ), D2 Y (e, e)i| 21
where the first term is bounded by kDΦn k kYn − Y k kD2 Y (e, e)k x
and therefore goes to zero as for A1 , and the second goes to zero since D2 Y (e, e) is in C and DΦn (y) x
converge to DΦ(y) for any y as functionals on C . Let’s now rewrite the right-hand term in the bracket defining D1 as D2 Yns,y (T )(e, e) − D2 Y s,y (T )(e, e) = (41) Z T = e(T −r)A D2 Bn (r, Yns,y (r)) (DYns,y (r)e, DYns,y (r)e) + s − D2 B (r, Y s,y (r)) (DY s,y (r)e, DY s,y (r)e) dr + Z +
T
e
(T −r)A
DBn (r, Yns,y (r)) D2 Yns,y (r)(e, e)+
s
− DB (r, Y
s,y
2
(r)) D Y
s,y
(r)(e, e) dr
= D11 + D12 . Proceding in a way similar to before we write the integrand in D11 as a sum D2 Bn (Yn )(DYe , DYn e)−D2 Bn (Yn )(DY e, DY e)+ D2 Bn (Yn ) − D2 B(Y ) (DY e, DY e) = D111 +D112 and notice that kD111 k = kD2 Bn (Yn )(DYn e − DY e, DYn e − DY e) + D2 Bn (Yn )(DYn e − DY e, DY e)+ + D2 Bn (Yn )(DY e, DYn e − DY e)k ≤ kD2 Bn (Yn )k kDYn e − DY ek2 + 2kDY ek kDYn e − DY ek which can be treated as in A1 since the norms kD2 Bn (Yns,y (r)) k are bounded uniformly in n and r, and that D112 can be shown to go to zero pointwise in r thanks to assumption 5.1 and to the α-hölderianity of D2 Bn in the same way as for C2 . By dominated convergence D11 is thus shown to converge to 0. To finish studying D1 (hence D) we need to rewrite the integrand in D12 as DBn (Yn )D2 Yn (e, e) − DB(Y )D2 Y (e, e) = = DBn (Yn ) D2 Yn − D2 Y (e, e)+ + [DBn (Yn ) − DBn (Y )] D2 Y (e, e) + [DBn (Y ) − DB(Y )] D2 Y (e, e) = DBn (Yn ) D2 Yn − D2 Y (e, e) + [DBn (Yn ) − DBn (Y )] D2 Y (e, e)+ + DB(Jn Y ) Jn D2 Y (e, e) − D2 Y (e, e) + [DB(Jn Y ) − DB(Y )] D2 Y (e, e). The second term in last sum is bounded in norm by kDBn (r, ·)k kYn − Y k kD2 Y (e, e)k which goes to zero since Yn → Y and kDBn k are uniformly bounded (as already noticed before); the norm of the third term goes to zero because it is bounded by kDB(Jn Y )k kJn D2 Y (e, e) − D2 Y (e, e)k; the norm of last term goes to zero as well by the Lipschitz property of DB. Taking into account all these observations and the fact that D11 has already been shown to converge to zero, we can use Gronwall’s lemma in (41) to obtain that D2 Yns,y (T )(e, e) − D2 Y s,y (T )(e, e) → 0. 22
This together with the uniform boundedness of DΦn (Yn ) finally yields the convergence to zero of D. At last an application of the dominated convergence theorem with respect to the variable s in all integral terms appearing in the Klomogorov equation concludes the proof. Remark 5.5. Since u is given as an integral of functions which are L∞ in the variable t, it is a Lipschitz function, hence differentiable almost everywhere thanks to a classic result by Rademacher. Therefore a posteriori it satisfies the differential form of Komogorov’s equation d
∂u 1X 2 2 (t, y) + hDu(t, y), Ay + B(t, y)i + σ D u(t, y)(ej , ej ) , u(T, ·) = Φ . ∂t 2 j=1 j for almost every t ∈ [0, T ].
6
Comparison with path-dependent calculus
We conclude this work estabilishing some connections between our results and objects and those defined by Dupire and succesively developed by Cont and Fournié. We recall here the definitions of the pathwise derivatives given in [CF13]. For a function ν = {νt }t , νt : D [0, T ]; Rd → Rd the i-th vertical derivative at γt (i = 1, . . . , d) is defined as νt γthei − νt (γt ) (42) Di νt (γt ) = lim h→0 h where γthei (s) = γt (s) + hei 1{t} (s); we denote the vertical gradient at γt by Dνt (γt ) = D1 νt (γt ), . . . , Dd νt (γt ) ; higher order vertical derivatives are defined in a straightforward way. The horizontal derivative at γt is defined as νt+h (γt,h ) − νt (γt ) Dt ν (γt ) = lim (43) h h→0+ where γt,h (s) = γt (s)1[0,t] (s) + γt (t)1[t,t+h] (s) ∈ D [0, t + h]; Rd . The connection between a functional b of paths and the operator B was essentially a matter of definition, as carried out in (2.1) - (11). To estabilish some relations between Fréchet derivatives of B and horizontal and vertical derivatives of b is much less obvious; some results are given by the following theorem. Theorem 6.1. Suppose u : [0, T ] × D → R is given and define, for each t ∈ [0, T ], νt : D([0, t]; Rd ) → R as νt (γ) : = u(t, γ(t), Lt γ), in the same way as in (11). Then the vertical derivatives of νt coincide with the partial derivatives of u with respect to the second variable (i.e. the present state), that is, Di νt (γ) =
∂ u(t, x, Lt γ), i = 1, . . . , d ∂x
(44)
The same result holds true also if u is given from ν as in (10). Furthermore let γt ∈ Cb1 ([0, t]) and let again u be given and define ν as above. Then Dt ν(γt ) =
∂u (t, γ(t), Lt γt ) + hDu(t, γ(t), Lt γt ), (Lt γt )0+ i ∂t
where h·, ·i is the duality between D and D0 , Du is the Fréchet derivative of u with respect to ϕ and the lower script + denotes right derivative.
23
Proof. Both claims in the theorem are proved through explicit calculations starting from the definition of derivatives. From the definition of vertical derivative one gets 1 νt γ h − νt (γ) h→0 h 1 u t, γ h (t), Lt γ h − u(t, γ(t), Lt γ) = lim h→0 h 1 u t, γ(t) + h, Lt γ h − u(t, γ(t), Lt γ) = lim h→0 h ∂ = u(t, x, Lt γ) ∂xi
Di νt (γ) = lim
This proves the first part of the theorem. For the second part suppose first that there is no explicite dependence on t in u. Then 1 [u (γt,h (t), Lt+h γt,h ) − u (t, γt (t), Lt γt )] h→0 h 1 = lim [u (γt (t), Lt+h γt,h ) − u (t, γt (t), Lt γt )] h→0 h n 1h γ (t + s) [−t − h, 0) = lim u γt (t), γt,h (0) [−T, −t − h) t,h h→0 h n i γ (t + s) [−t, 0) − u γt (t), γt (0) [−T, −t) t γt (t) [−h, 0) 1 γt (t + s + h) [−t, −h) = lim u γt (t), γt (t + s + h) [−t − h, −t) h→0 h γt (0) [−T, −t − h) γt (t + s) [−h, 0) γ (t + s) [−t, −h) t . − u γt (t), [−t − h, −t) γt (0)
Dt b(γt ) = lim
[−T, −t − h)
γt (0)
Last line can be written as 1 u (γt (t), Lt γt + Nt,h γt ) − u (γt (t), Lt γt ) h→0 h
(45)
0 [−T, −t − h) γt (t + h + s) − γt (0) [−t − h, −t) . Nt,h γt (s) = γt (t + h + s) − γt (t + s) [−t, −h) γt (t) − γ(t + s) [−h, 0)
(46)
lim
where
Nt,h γt is a continuous function that goes to 0 as h → 0; moreover, recalling that in the definition of horizontal derivative h is greater than zero, we see that ¯ s.t. s < −t − h, ¯ hence Nt,h γ(s) = 0 ∀h < h ¯ and limh→0+ 1 Nt,h γ(s) = 0 = (i) for s ∈ [−T, −t) ∃h h 0 (Lt γ) (s); + (ii) for s = −t, since Nt,h γ(−t) = γ(h) − γ(0) we have h1 Nt,h γt (−t) → dds Lt γt (−t) = 0 (Lt γt )0+ (−t) = γ+ (0);
¯ s.t. s < −h ¯ < 0, hence (iii) for s ∈ (−t, 0) ∃h 0 γ+ (t + s) = γ 0 (t + s) = (Lt γt )0 (s); Therefore and, since γ ∈ Cb1 ,
1 h Nt,h γt (s)
=
1 h→0+ Nt,h γt (s) −−−−→ (Lt γt )0+ (s) h (Lt γt )0+ (s) = (Lt γt )0 (s) ∀ s 6= −t. 24
1 h
[γt (t + s + h) − γt (t + s)] →
(47)
Again since γt ∈ C 1 with bounded derivative h1 Nt,h γt converges to (Lt γt )0+ also uniformly. Keeping into account (45) and the definition of Fréchet derivative, one gets 1 [u(γt (t), Lt γt + Nt,h γt ) − u(γt (t), Lt γt )] h 1 = lim [Dϕ u(γt (t), Lt γt ) · Nt,h γt + ξ(h)] h→0 h
Dt b(γt ) = lim
h→0
where ξ is infinitesimal with respect to kNt,h γt k as h → 0, 1 h→0 h
Z
0
kNt,h γt k ξ(h) h→0 h kNt,h γt k
∇ϕ u(γt (t), Lt γt )(s)Nt,h γt (s) ds + lim
= lim
−T
= h∇ϕ u(γt (t), Lt γt ), (Lt γt )0+ i by the dominated convergence theorem. If now u depends explicitely on t just write 1 1 [νt+h (γt,h ) − νt (γ)] = [u (t + h, γ(t), Lt+h γt,h ) − u (t, γ(t), Lt γ)] h h 1 = [u (t + h, γ(t), Lt+h γt,h ) − u (t, γ(t), Lt+h γt,h )] + h 1 + [u (t, γ(t), Lt+h γt,h ) − u (t, γ(t), Lt γ)] ; h the first term in the last line converges to the time derivative of u while the second can be treated exactly as above. Thanks to this result we can reinterpret the infinite dimensional Kolmogorov equation (25) in terms of the horizontal and vertical derivatives introduced in the previous section. Consider the Kolmogorov equation with horizontal and vertical derivatives, namely ( Pd Dt ν(γt ) + bt (γt ) · Dνt (γt ) + 12 j=1 σj2 Dj2 νt (γt ) = 0 , (48) νT (γT ) = f (γT ) . Theorem 6.2. Let X γt be the solution to equation dX(t) = bt (Xt ) dt + σ dW (t) Xt0 = γt0
for t ∈ [t0 , T ],
(5)
Associate to bt and f the operators B and Φ as in (15); if such B and Φ satisfy the assumptions of theorem 5.4 then, for almost every t, the function νt (γt ) = E [f (X γt (T ))]
(49)
is a solution of the path dependent Kolmogorov equation (48) for all γ ∈ Cb1 such that γ 0 (0) = 0. Proof. Lift equation (5) to the infinite dimensional SDE (130 ) defining the operators A, B and Σ as in the previous sections; associate then to this last equation the PDE (25) with final condition given by f ( ϕx ) . Φ (( ϕx )) = f M x
Fix t: with our choice of γ the element y = (γ(t), Lt γt ) is in C therefore, if B and Φ satisfy assumptions 2.1 and 5.1, theorem 5.4 guarantees that u(s, y) = E [Φ (Y s,y (T ))] is a solution to the Kolmogorov equation. Notice that solving this equation for s ≥ t involves only a piece (possibly all) of the path γt , so that our “artificial” lenghtening by means of Lt is used only for defining all objects in the right way but does not come into the solution of the equation. Of course in principle one can solve the infinite dimensional 25
PDE for any s ∈ [0, T ], anyway we are interested in solving it at time t: indeed if we now define ν through u by means of (11) we have that νt (γt ) = u(t, γ(t), Lt γt ) h i f Y t,y (T ) =E f M = E [f (X γt (T ))] . 0
Recalling remark 5.5 and noticing that (Lt γt )+ = A (Lt γt ) thanks to the assumption that γ 0 (0) = 0, we can apply for almost every t theorem 6.1 obtaining that equations (25) and (48) coincide. Remark 6.3. If in the above proof one can show that the function u which solves (25) is in fact differentiable with respect to t for every t ∈ [0, T ], then theorem 6.2 hold everywhere, i.e. the function ν defined by (49) solves equation (48) for every t ∈ [0, T ].
Appendix: proof of theorem 2.3 We start from a simple estimate; for y, k ∈ E we have kY t0 ,y+k (t) − Y t0 ,y kE =
Z t h
(t−t )A i (t−s)A t0 ,y+k t0 ,y 0
(k) + e B s, Y (s) − B s, Y (s) ds = e
t0
E
Z
t
≤ CkkkE + CkDBk∞
kY t0 ,y+k (s) − Y t0 ,y (s)kE ds
t0
hence, by Gronwall’s lemma, sup kY t0 ,y+k (t) − Y t0 ,y (t)kE ≤ C˜Y kkkE .
(A1)
t
First derivative We introduce the following equation for the unknown ξ t0 ,y (t) taking values in the space of linear bounded operators L(E, E) ξ
t0 ,y
(t) = e
(t−t0 )A
Z
t
+
e(t−s)A DB s, Y t0 ,y (s) ξ t0 ,y (s) ds.
t0
Existence and uniqueness of a solution in L∞ 0, T ; L(E, E) follow again easily from the contraction mapping principle, since
Z t
Z t
(t−s)A t0 ,y
e DB s, Y (s) [ξ (s) − ξ (s)] ds ≤ CkDBk kξ1 (s) − ξ2 (s)kL(E,E) ds; 1 2 ∞
t0
L(E,E)
26
t0
Moreover, by Gronwall’s lemma, kξ t0 ,y (t)kL(E,E) ≤ Cξ uniformly in t. Now for k ∈ E we compute rt0 ,y,k (t) : = Y t0 ,y+k (t) − Y t0 ,y (t) − ξ t0 ,y (t)k Z t e(t−s)A B s, Y t0 ,y+k (s) − B s, Y t0 ,y (s) ds− = t0
Z
t
e(t−s)A DB s, Y t0 ,y (s) ξ t0 ,y (s)k ds
t0
Z
t
e(t−s)A
=
DB s, αY t0 ,y+k (s) + (1 − α)Y t0 ,y (s) Y t0 ,y+k (s) − Y t0 ,y (s) dα 0 t0 ,y t0 ,y − DB s, Y (s) ξ (s)k ds
t0
Z
1
Z
t
e(t−s)A DB s, Y t0 ,y (s) rt0 ,y,k (s) ds
= t0
Z
t
t0
Z
1
DB s, αY t0 ,y+k (s) + (1 − α)Y t0 ,y (s) dα 0 − DB s, Y t0 ,y (s) Y t0 ,y+k (s) − Y t0 ,y (s) ds.
e(t−s)A
+
Recalling (A1) we get kr
t0 ,y,k
Z
t
(t)kE ≤ CkDBk∞
krt0 ,y,k (s)kE ds +
t0
Z t Z
+ CkkkE
t0
1
DB s, αY
t0 ,y+k
(s) + (1 − α)Y
t0 ,y
(s) dα − DB s, Y
0
t0 ,y
(s)
ds
L(E,E)
which yields, by Gronwall’s lemma, krt0 ,y,k (t)k ≤ Ckkk2 . Therefore ξ t0 ,y (t)k = DY t0 ,y (t)k We proceed with an estimate about the continuity of ξ k∈E
t0 ,y
∀k ∈ E.
(t) with respect to the initial condition y. For h,
kξ t0 ,y+k (t)h − ξ t0 ,y (t)hkE =
Z t
t0 ,y+k t0 ,y (t−s)A t0 ,y+k t0 ,y
= e DB s, Y (s) ξ (s)h − DB s, Y (s) ξ h ds
t0 E
Z t
(t−s)A t0 ,y+k t0 ,y+k t0 ,y+k t0 ,y
≤ e DB s, Y (s) ξ (s)h − DB s, Y (s) ξ (s)h ds
t0
Z t
E
+ e(t−s)A DB s, Y t0 ,y+k (s) ξ t0 ,y (s)h − DB s, Y t0 ,y (s) ξ t0 ,y (s)h ds
t0 E Z t
t ,y+k
ξ 0 ≤ CkDBk∞ (s)h − ξ t0 ,y (s)h E ds t0
Z
t
+C t0
t ,y
DB s, Y t0 ,y+k (s) − DB s, Y t0 ,y (s)
ξ 0 (s)h ds L(E,E) E
27
Z
t
≤ CkDBk∞ t0
t ,y+k
ξ 0 (s)h − ξ t0 ,y (s)h E ds
+ C · Cξ khkE kD2 Bk∞
Z
t
t0
Z
t ,y+k
Y 0 (s) − Y t0 ,y (s) E ds
t
≤ CkDBk∞ t0
t ,y+k
ξ 0 (s)h − ξ t0 ,y (s)h E ds + CCξ kD2 Bk∞ C˜Y (t − t0 )khkE kkkE .
Again by Gronwall’s lemma we get kξ t0 ,y+k (t)h − ξ t0 ,y (t)hkE ≤ C˜ξ khkE kkkE . Therefore ξ
t0 ,y
(A2)
(t) is uniformly continuous in y uniformly in t.
Second derivative Let us consider the operator U from C ([t0 , T ]; L(E, E; E)) in itself defined through the equation Z t U(Y )(t)(h, k) = e(t−s)A D2 B(s, Y t0 ,y (s))(ξ t0 ,y (s)h, ξ t0 ,y (s)k) ds+ t0
Z
t
+
e(t−s)A DB(s, Y t0 ,y (s))Y (s)(h, k) ds (A3)
t0
for h, k ∈ E (we identify L E, L(E, E) with L(E, E; E) in the usual way). Since sup kU(Y1 )(t)(h, k) − U(Y2 )(t)(h, k)kE ≤ CkDBk∞ T sup kY1 (t)(h, k) − Y2 (t)(h, k)kE
t,h,k
t,h,k
there exists a unique fixed point for U, which will be denoted by η t0 ,y (t)(h, k); furthermore simple calculations yield that kη t0 ,y (t)kL(E,E;E) ≤ Cη uniformly in t. We now compute: r˜t0 ,y,h,k (t) : =ξ t0 ,y+k (t)h − ξ t0 ,y (t)h − η t0 ,y (t)(h, k) Z t = e(t−s)A DB s, Y t0 ,y+k (s) ξ t0 ,y+k (s)h ds t0
Z
t
e(t−s)A DB s, Y t0 ,y (s) ξ t0 ,y (s)h ds
− t0 t
Z
e(t−s)A D2 B s, Y t0 ,y (s) ξ t0 ,y (s)h, ξ t0 ,y (s)k ds
− t0 t
Z
e(t−s)A DB s, Y t0 ,y (s) η t0 ,y (s)(h, k) ds
− t0
Z
t
=
e(t−s)A DB s, Y t0 ,y+k (s) ξ t0 ,y+k (s)h ds
t0
Z
t
−
e(t−s)A DB s, Y t0 ,y (s) ξ t0 ,y+k (s)h ds
t0 t
Z +
e(t−s)A DB s, Y t0 ,y (s) ξ t0 ,y+k (s)h ds
t0 t
Z −
e(t−s)A DB s, Y t0 ,y (s) ξ t0 ,y (s)h ds
t0 t
Z −
e(t−s)A D2 B s, Y t0 ,y (s) ξ t0 ,y (s)h, ξ t0 ,y (s)k ds
t0 t
Z −
e(t−s)A DB s, Y t0 ,y (s) η t0 ,y (s)(h, k) ds
t0
28
Z
t
e(t−s)A DB s, Y to ,y (s) r˜to ,y,h,k (s) ds
= t0
Z
t
e(t−s)A DB s, Y to ,y+k (s) − DB s, Y to ,y (s) ξ to ,y+k (s)h ds
+ t0 t
Z
e(t−s)A D2 B s, Y t0 ,y (s) ξ t0 ,y (s)h, ξ t0 ,y (s)k ds
− t0
Z
t
e(t−s)A DB s, Y to ,y (s) r˜to ,y,h,k (s) ds
= Z +
t0 t
e(t−s)A
Z
1
D2 B s, αY to ,y+k (s) + (1 − α)Y to ,y (s) dα ξ to ,y+k (s)h, Y to ,y+k (s) − Y to ,y (s) t0 0 − D2 B s, Y t0 ,y (s) ξ t0 ,y (s)h, ξ t0 ,y (s)k ds
Z
t
e(t−s)A DB s, Y to ,y (s) r˜to ,y,h,k (s) ds
= t0
Z
t
+
e
(t−s)A
t0
Z
1 2
D B s, αY
to ,y+k
(s) + (1 − α)Y
to ,y
2 t0 ,y (s) dα − D B s, Y (s) ·
0
· ξ to ,y+k (s)h, Y to ,y+k (s) − Y to ,y (s) ds Z t + e(t−s)A D2 B s, Y t0 ,s (s) ξ to ,y+k (s)h, Y t0 ,y+k (s) − Y t0 ,y (s) t0 − D2 B s, Y t0 ,y (s) ξ t0 ,y (s)h, ξ t0 ,y (s)k ds Z
t
e(t−s)A DB s, Y to ,y (s) r˜to ,y,h,k (s) ds
= t0
Z
t
e(t−s)A
+ t0
Z
1
D2 B s, αY to ,y+k (s) + (1 − α)Y to ,y (s) dα − D2 B s, Y t0 ,y (s) ·
0
· ξ to ,y+k (s)h, Y to ,y+k (s) − Y to ,y (s) ds Z t t0 ,y+k (t−s)A 2 t0 ,y + e D B s, Y (s) ξ (s)h, Y t0 ,y+k (s) − Y t0 ,y (s) − ξ t0 ,y+k (s)h, ξ t0 ,y (s)k t0 t0 ,y+k t0 ,y t0 ,y t0 ,y + ξ (s)h, ξ (s)k − ξ (s)h, ξ (s)k ds Z
t
=
e(t−s)A DB s, Y to ,y (s) r˜to ,y,h,k (s) ds
t0
Z
t
+ t0
e(t−s)A
Z
1
0 D2 B s, αY to ,y+k (s) + (1 − α)Y to y (s) dα − D2 B s, Y t0 ,y (s) ·
0
· ξ to ,y+k (s)h, Y to ,y+k (s) − Y to ,y (s) ds Z t + e(t−s)A D2 B s, Y t0 ,y (s) ξ t0 ,y+k (s)h, Y t0 ,y+k (s) − Y t0 ,y (s) − ξ t0 ,y (s)k ds t0 t
Z +
e(t−s)A D2 B s, Y t0 ,y (s) ξ t0 ,y+k (s)h − ξ t0 ,y (s)h, ξ t0 ,y (s)k ds
t0
29
These calculations imply that k˜ rt0 ,y,h,k (t)kE ≤ CA kDBk∞
Z
t
k˜ rt0 ,y,h,k (s)kE ds
t0
Z t Z
+ CA
1 2
D B s, αY
t0 ,y+k
(s) + (1 − α)Y
t0 ,y
2
dα − D B s, Y
t0 ,y
0
t0
(s)
·
L(E,E;E)
·kξ t0 ,y+k (s)hkE · kY t0 ,y+k (s) − Y t0 ,y (s)kE ds + CA kD2 Bk∞ + CA kD2 Bk∞
Z
t
kξ t0 ,y+k (s)hkE · kY t0 ,y+k (s) − Y t0 ,y (s) − ξ t0 ,y (s)kkE ds
t0 Z t
kξ t0 ,y+k (s)h − ξ t0 ,y (s)hkE · kξ t0 ,y (s)kkE ds
t0
Z
t
≤ CA kDBk∞
k˜ rt0 ,y,h,k (s)kE ds
t0
+ CA Cξ C˜Y khkE kkkE ·
Z t Z 1
2 t0 ,y+k t0 ,y 2 t0 ,y
ds D B s, αY (s) + (1 − α)Y dα − D B s, Y (s) ·
0 t0 L(E,E;E)
Z t Z 1
2 t0 ,α(y+k)+(1−α)y t0 ,y
+ CA Cξ kD Bk∞ khkE ξ (s)k dα − ξ (s)k
ds t0
0
E
+ CA Cξ C˜ξ T kD2 Bk∞ khkE kkk2E Z t ≤ CA kDBk∞ k˜ rt0 ,y,h,k (s)kE ds t0
+ C1 khkE kkkE ·
Z t Z 1
2 t0 ,y+k t0 ,y 2 t0 ,y
D B s, αY (s) + (1 − α)Y dα − D B s, Y (s) ds ·
0 t0 L(E,E;E)
Z t Z 1
t0 ,y+αk t0 ,y
dskkkE + C2 khkE ξ (s) dα − ξ (s)
t0
+
0
L(E,E)
C3 khkE kkk2E .
Finally by a application of Gronwall’s lemma k˜ rt0 ,y,h,k (t)kE ≤ C4 khkE · kkkE
Z t Z 1
2 t0 ,y+k t0 ,y 2 t0 ,y
· D B s, αY (s) + (1 − α)Y dα − D B s, Y (s)
t0
0
Z t Z
+
t0
ds
L(E,E;E)
1
ξ
t0 ,y+αk
(s) dα − ξ
t0 ,y
0
(s)
ds + kkkE
L(E,E)
and such quantity goes to 0 uniformly in khkE ≤ M ∀M > 0 when kkkE goes to 0 by Lebesgue’s dominated convergence theorem. Our next step is to study the continuity of the second derivative computed above. We have η t0 ,y (t)(h, k) − η t0 ,w (t)(h, k) = Z t e(t−s)A D2 B s, Y t0 ,y (s) ξ t0 ,y (s)h, ξ t0 ,y (s)k − D2 B s, Y t0 ,w (s) ξ t0 ,w (s)h, ξ t0 ,w (s)k ds t0
Z
t
+
e(t−s)A DB s, Y t0 ,y (s) η t0 ,y (s)(h, k) − DB s, Y t0 ,w (s) η t0 ,w (s)(h, k) ds
t0
= I1 + I2 ;
(A4) 30
then Z
t
Z
t
I1 = e D2 B s, Y t0 ,y (s) ξ t0 ,y (s)h, ξ t0 ,y (s)k − D2 B s, Y t0 ,w (s) ξ t0 ,y (s)h, ξ t0 ,y k t0 t0 ,y t0 ,w 2 t0 ,w t0 ,y 2 t0 ,w t0 ,w + D B s, Y (s) ξ (s)h, ξ k − D B s, Y (s) ξ (s)h, ξ (s)k ds (t−s)A
= Z
e(t−s)A D2 B s, Y t0 ,y (s) − D2 B s, Y t0 ,w (s) ξ t0 ,y (s)h, ξ t0 ,y (s)k ds
t0 t
+
e(t−s)A D2 B s, Y t0 ,w (s) ξ t0 ,y (s) − ξ t0 ,w h, ξ t0 ,y k ds
t0 t
Z +
e(t−s)A D2 B s, Y t0 ,w (s) ξ t0 ,w (s)h, ξ t0 ,y (s) − ξ t0 ,w (s) k ds
t0
and t
Z
e(t−s)A DB s, Y t0 ,y (s) η t0 ,y (s)(h, k) − η t0 ,w (s)(h, k) ds
I2 = Z
t0 t
e(t−s)A DB s, Y t0 ,y (s) − DB s, Y t0 ,w (s) η t0 ,w (s)(h, k) ds.
+ t0
Recalling all the previous estimates and the fact that both kY t0 ,y (t)kE and kξ t0 ,y (t)kL(E,E) are bounded uniformly in t, denoting with CH the Hölder constant of D2 B, we get
t ,y
η 0 (t)(h, k) − η t0 ,w (t)(h, k) E Z t
t ,y
Y 0 (s) − Y t0 ,w (s) α ξ t0 ,y (s)h ξ t0 ,y (s)k ds ≤ C · CH E E E t0
Z th
t ,y
t ,y
ξ 0 (s) − ξ t0 ,w (s) 1−α khkE ξ t0 ,y (s)k
ξ 0 (s) − ξ t0 ,w (s) α + CkD Bk∞ E L(E,E) L(E,E) t0 i
t ,w
t ,y
α 1−α + ξ 0 (s)h E ξ 0 (s) − ξ t0 ,w (s) L(E,E) ξ t0 ,y (s) − ξ t0 ,w (s) L(E,E) kkkE ds Z t
t ,y
η 0 (s)(h, k) − η t0 ,w (s)(h, k) ds + CkDBk∞ E 2
t0 t
Z + CkDBk∞
t0
t ,y
Y 0 (s) − Y t0 ,w (s) α Y t0 ,y (s) − Y t0 ,w (s) 1−α η t0 ,w (s)(h, k) ds E E E α
≤ C1 khkE kkkE ky − wkE + C2
Z
t
t0
t ,y
η 0 (s)(h, k) − η t0 ,w (s)(h, k) ds E
hence
t ,y
η 0 (t)(h, k) − η t0 ,w (t)(h, k) ≤ CkhkE kkkE ky − wkα E E which shows that the second Fréchet derivative of the map y 7→ Y t0 ,y (t) is α-Hölder.
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