HAL Id: hal-01238187
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Preprint submitted on 8 Dec 2015
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Partial regularity of viscosity solutions for a class of Kolmogorov equations arising from mathematical finance
M Rosestolato, Andrzej Swiech
To cite this version:
M Rosestolato, Andrzej Swiech. Partial regularity of viscosity solutions for a class of Kolmogorov equations arising from mathematical finance. 2015. �hal-01238187v2�
Partial regularity of viscosity solutions for a class of Kolmogorov equations arising from
mathematical finance
M. Rosestolato
∗A. ´ Swie
,ch
†Abstract
We study value functions which are viscosity solutions of certain Kolmogorov equations. Using PDE techniques we prove that they are C1,α regular on special finite dimensional subspaces. The problem has origins in pricing and hedging of derivatives of risky assets in mathematical finance.
Key words: viscosity solution, Kolmogorov equation, stochastic differential equation, delay problem, pricing and hedging problem.
AMS 2010 subject classification: 35R15, 49L25, 60H15, 91G80.
1 Introduction
In this paper we study partial regularity of viscosity solutions for a class of Kolmogorov equations. Our motivation comes from mathematical finance, more precisely from pric- ing and hedging of a derivative of a risky asset whose volatility as well as the claim price may depend on the past history of the asset. Our Kolmogorov equations are thus associated to stochastic delay problems. They are linear second order partial differen- tial equations in an infinite dimensional Hilbert space with a drift term which contains an unbounded operator and a second order term which only depends on a finite dimen- sional component of the Hilbert space. Such equations are typically investigated using the notion of the so-calledB-continuous viscosity solutions (see [10,12,20]). We impose conditions under which our Kolmogorov equations have unique B-continuous viscosity solutions. However general Hamilton-Jacobi-Bellman equations associated to stochas- tic delay optimal control problems which are rewritten as optimal control problems for stochastic differential equations (SDE) in an infinite dimensional Hilbert space are diffi- cult, not well studied yet, and few results are available in the literature.
We work directly with the value function here since its partial regularity is of inter- est in the pricing/hedging problem and it is well known that under our assumptions the
∗LUISS Guido Carli, Rome, e-mail: [email protected]. This research has been partially sup- ported by the INdAM-GNAMPA project “Equazioni stocastiche con memoria e applicazioni” (2014).
†School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA, e-mail:
value function is the uniqueB-continuous viscosity solution of the Kolmogorov equation (see e.g. [10,12]). We thus never use the theory ofB-continuous viscosity solutions. In- stead our strategy for proving partial regularity of the value function is the following. We consider SDEs with smoothed out coefficients and the unbounded operator replaced by its Yosida approximations and study the corresponding value functions with smoothed out payoff function. The new value functions are Gâteaux differentiable and converge on compact sets to the original value functions. They also satisfy their associated Kol- mogorov equations. We then prove that their finite dimensional sections are viscosity solutions of certain linear finite dimensional parabolic equations for which we establish C1,α estimates. Passing to the limit with the approximations, these estimates are pre- served givingC1,α partial regularity for finite dimensional sections of the original value function.
Partial regularity results for first order unbounded HJB equations in Hilbert spaces associated to certain deterministic optimal control problems with delays have been ob- tained in [11]. The technique of [11] relied on arguments using concavity of the data and strict convexity of the Hamiltonian and provided C1regularity on one-dimensional sec- tions corresponding to the so-called “present” variable. Here the equations are of second order, we rely on approximations and parabolic regularity estimates, and we obtain regu- larity onm-dimensional sections. The reader can also consult [16] for various global and partial regularity results for bounded HJB equations in Hilbert spaces (see also [19]).
We refer the reader to [16, 17, 10] for the theory of viscosity solutions for bounded second order HJB equations in Hilbert spaces and to [10, 12,20] for the theory of the so-called B-continuous viscosity solutions for unbounded second order HJB equations in Hilbert spaces. A fully nonlinear equation with a similar separated structure to our Kolmogorov equation (3.14) but with a nonlinear unbounded operator A was studied in [13]. For classical results about Kolmogorov equation in Hilbert spaces we refer the reader to [7].
The plan of the paper is the following. In the rest of the Introduction we explain the financial motivation of our problem. Section2contains notation and various results about mild solutions of the SDE, their extensions to a bigger space with a weaker topol- ogy related to the original unbounded operatorA, and various approximation results. In Section3 we study viscosity solutions of the approximating equations, investigate finite dimensional sections of viscosity solutions, and prove their regularity.
1.1 Motivation from finance
One motivation for the present study comes form a classical problem in financial mathe- matics, that is the pricing/hedging problem of a derivative of some risky assets.
For the sake of simplicity, we consider a financial market composed of two assets: a risk free assetB (a bond price), and a risky asset R (a stock price). We assume that B follows the deterministic dynamics dBs=rBsds, where r is the short rate of interest, and thatR follows the dynamics
(dRs=rRsds+ν(s,Rs)dWs s∈(t,T],
Rt=x0, (1.1)
where (Ω,F={Ft}t∈[0,T],P) is a filtered probability space,T>0 is the maturity, andx0∈R
is the initial datum at time t∈[0,T). Let us assume thatνsatisfies the usual Lipschitz assumptions, and denote byRt,x0 the unique strong solution of SDE (1.1).
Given a function ϕ: R → R, the pricing problem consists in finding a “fair price”
process {πs}s∈[t,T] of the European simple T-claim ϕ(Rt,xT 0). The hedging problem con- sists in finding a replicating self-financing portfolio strategy for the derivative ϕ(RTt,x0), that is a couple of real-valued processes {(hBs,hRs)}s∈[t,T] such that the portfolio Vs = hBsBs+hRsRs, composed of hBs shares of B and of hRs shares of R, follows the dynam- ics dVs=hBsdBs+hRsdRsand has its final value given byVT=ϕ(RTt,x0).
Under suitable conditions onνandϕ, a classical argument based on the assumption that the market is free of arbitrage possibilities (see [1]) shows that the priceπis given by the formula
πs=e−r(T−s)Eh
ϕ(Rt,xT 0)|Fs
i . By the Markov property,πcan be represented in the form
πs=u(s,Rst,x0), where u(t,x0)=e−r(T−t)Eh
ϕ(RTt,x0)i
∀(t,x0)∈[0,T]×R,∀s∈[t,T]. (1.2) If u is regular enough to apply Itô’s formula, (1.2) can be used to show that u solves to the following Kolmogorov-type partial differential equation
(ut+rxDxu+12ν2(t,x)D2xu−ru=0 (t,x)∈(0,T]×R,
u(T,x)=ϕ(x) x∈R. (1.3)
By using (1.3), Itô’s formula entails the following representation formula, for s∈[t,T], u(s,Rst,x0)=u(t,x0)+
Z s
t
ru(w,Rt,xw0)dw+ Z s
t
Dx0u(w,Rwt,x0)ν(w,Rwt,x0)dWw, (1.4) which provides the following solutions for the hedging problem
hBs = u(s,Rt,xs 0)−Dx0u(s,Rst,x0)Rt,xs 0
Bs and hRs =Dx0u(s,Rt,xs 0) ∀s∈[t,T). (1.5) Observe that the assumption that the market is free of arbitrage possibilities reduces the pricing problem to the hedging problem (see [1]). Observe also that there are three essential features of the model that allow to implement the program above:
(1) The Markov property ofR, which makes (1.2) possible.
(2) The existence ofDx0u, which lets the portfolio strategy be defined by (1.5)
(3) The availability of Itô’s formula and the fact thatusolves to (1.3), in order to derive (1.4).
Let us now consider a slightly more general risky asset R, in which the volatility depends not only on the valueRsofR at times, but also on the entire past values ofR.
That is, the dynamics ofR has the following form
dRs=rRsds+ν(t,Rs, {Rs0+s}s0∈(−∞,0))dWs s∈(t,T]
Rt=x0
Rt0=x1(t0) t0∈(−∞,t),
(1.6)
where x0∈Rand x1: (−∞, 0)→Ris a given deterministic funtion belonging toL2(R−,R), expressing the past history of the stock price R up to time t. We also would like to face the case in which the European claim depends itself on the history of R, that is it has the formϕ(RT, {Rt}t∈(−∞,T)).
A natural question is if we can implement the standard arguments above in or- der to solve the pricing and the hedging problems, as done in the case in which R is given by (1.1). More precisely, we would like to write the price process πt of the claim ϕ(RT, {Rt}t∈(−∞,T)) through a functionuwhich can be characterized as a solution (in some sense) of a PDE analogous to (1.3), and then show that such a solution has enough regu- larity in order to define the hedging strategy (assuming that such a strategy is somehow analogous to (1.5)). Actually, we will see that it is sufficient foruto be differentiable with respect to a special direction.
If Rt,(x0,x1) solves (1.6), then in general it is not Markovian. Moreover, since both the claimϕand the functionu, now defined by
u(t,x0,x1) :=e−(T−t)Eh ϕ¡
RTt,x0,x1,©
Rt,x0,x1
t0
ª
t0∈(−∞,T)
¢i
, ∀(t,x0,x1)∈[0,T]×R×L2(R−,R), are path-dependent, PDE (1.3) would now be path-dependent, and it would be necessary to employ a stochastic calculus for path-dependent Itô processes in order to relate u with the PDE, as done for the non-path-dependent case. A classical workaround tool to regain Markovianity and to avoid the complications of a path-dependent stochastic calculus consists in rephrasing the model in a functional space setting. Of course we lose something by doing so, that is the dynamics evolves in an infinite dimensional space (see [2]). We briefly recall how the rephrasing works.
Let us introduce the Hilbert space H:=R×L2(R−,R), and the functions F: [0,T]×H−→H, (x0,x1)7−→(rx0, 0)
Σ: [0,T]×H−→H, (x0,x1)7−→(ν(t,x0,x1), 0). (1.7) OnH, consider the strongly continuous semigroup of translations,i.e.the family { ˆSt}t∈[0,T]
of linear continuous operators defined by
Sˆt:H→H, (x0,x1)7→(x0,x1(t+ ·)χ(−∞,−t)+x0χ[−t,0]).
The infinitesimal generator ˆAof ˆSis given by
Aˆ:D( ˆA)−→H, (x0,x1)7−→(0,x10), where
D( ˆA)=©
(x0,x1)∈H: x1∈W1,2(R−) andx0=x1(0)ª . Now consider the H-valued dynamics
(dXˆs=¡AˆXˆs+F¡ t, ˆXs¢¢
ds+Σ¡t, ˆXs
¢dWs s∈(t,T],
Xˆt=(x0,x1), (1.8)
where (x0,x1)∈H. Under usual Lipschitz assumptions onν, it can be shown that (1.8) has a unique mild solution ˆXt,(x0,x1). The link between (1.6) and (1.8) is given by the following equation:
Xˆst,(x0,x1)=¡
Rst,(x0,x1),©
Rt,(x0,x1)
s0+s
ª
s0∈(−∞,0)
¢, (1.9)
whereRt,(x0,x1)denotes the unique strong solution of (1.6). Observe that ˆX is Markovian and no path-dependency appears in the coefficientsF,Σ. This is the natural rephrasing of the dynamics of R to get a Markovian setting for which the basic tools of stochastic calculus (such as Itô’s formula) are available.
We need a further step to let the model studied in the paper apply to the financial problem we are considering. We rephrase (1.8) as an SDE in the same Hilbert space H, but with a maximal dissipative unbounded operator. To this aim, we observe that A:=Aˆ−12I is a maximal dissipative operator generating the semigroup of contractions S :={St := e−t/2Sˆt}t∈R+. Let us define G(t,x) :=F(t,x)+2x, (t,x)∈[0,T]×H. Denote by Xt,(x0,x1)the unique mild solution of the SDE
(d Xs=(A Xs+G(t,Xs))ds+Σ(t,Xs)dWs s∈(0,T],
Xt=(x0,x1). (1.10)
It is not difficult to see that ˆXt,(x0,x1)=Xt,(x0,x1). Indeed, if { ˆAλ}λ>1/2 denote the Yosida approximations of ˆA, then the strong solution of
(d Xλ,s=¡AˆλXλ,s+F¡
t,Xλ,s¢¢
ds+Σ¡t,Xλ,s
¢dWs s∈(t,T],
Xλ,t=(x0,x1), (1.11)
coincides with the strong solution Xλt,(x0,x1) of
Xλ,s=
µµ Aˆλ−1
2
¶
Xλ,s+G¡
t,Xλ,s¢
¶
ds+Σ¡t,Xλ,s
¢dWs s∈(t,T], Xλ,t=(x0,x1),
(1.12) by the very definition and by uniqueness of strong solutions. Recalling that strong and mild solutions coincide when the linear term appearing in the drift is bounded (see [6]), Xλt,(x0,x1) solves (1.12) in the mild sense. Now observe that ˆAλ−12I generates the semigroup ˆSλ :={ ˆSλ,t := e−t/2eAˆλt}t∈R+. Since eAˆλt →Sˆt strongly as λ→ +∞, we have also ˆSλ,t →St strongly. Then the mild solution Xλt,(x0,x1) converges to the mild solution Xt,(x0,x1) asλ→ +∞(see e.g. the argument used to show Proposition 2.9-(ii)). Similarly, Xλt,(x0,x1) solves (1.11) in the mild sense and then Xλt,(x0,x1)→ Xˆt,(x0,x1) as λ→ +∞. We thus conclude that ˆXt,(x0,x1)=Xt,(x0,x1) in the suitable space of processes where the well- posedness of the SDEs and the convergences above are considered.
It follows that equivalence (1.9) can be rewritten as Xst,(x0,x1)=¡
Rst,(x0,x1),©
Rt,(xs0 0,x1)
+s
ª
s0∈(−∞,0)
¢. (1.13)
Having (1.13), the hedging problem (and then the pricing problem), is reduced to proving that a representation analogous to (1.4) is possible for the process©
u(s,Xst,(x0,x1))ª
s∈[t,T], where
u(t, (x0,x1))=e−r(T−t)Eh ϕ³
Xt,(x0,x1)
T
´i
∀(t, (x0,x1))∈[0,T]×H. (1.14) More precisely, thanks to the special structure ofΣin SDE (1.10), ifuhas enough regu- larity to perform the computations, it turns out that, fors∈[t,T],
u³
s,Xst,(x0,x1)´
=u(t, (x0,x1))+ Z s
t
ru³
w,Xwt,(x0,x1)´ dw +
Z s t
Dx0u³
w,Xwt,(x0,x1)´ ν³
w,Xwt,(x0,x1)´ dWw,
(1.15)
and the only derivative of u appearing in the above formula is the directional deriva- tive Dx0u with respect to the variable x0, representing the “present”, according to the rephrasingR X.
The aim of this paper is to show the regularity of the function u, defined by (1.14), with respect to the component x0, when all the data are assumed to be Lipschitz with respect to a particular norm associated to the operator A.
2 Preliminaries
2.1 Notation
Let (Ω,P,F) be a complete probability space, let T >0, and let F={Ft}t∈[0,T] be a fil- tration on (Ω,F,P) satisfying the usual conditions. Define ΩT =Ω×[0,T]. Denote by P the σ-algebra in ΩT generated by the sets As×(s,t], where As∈Fs, 0≤s<t≤T, and A0×{0}, where A0∈F0. An element of P is called a predictable set. We denote R−=(−∞,0],R+=[0,+∞).
Let (F,| · |F)1be a real separable Banach space. We define the following spaces:
(i) Forp≥1,LPp (F) :=LpP(ΩT,F) is the Banach space ofF-valued predictable processes X such that
|X|Lp
P(F):= µ
E
·Z T
0 |Xt|Fpdt
¸¶1/p
< +∞. (ii) HPp(F) is the subspace of elementsX ofLPp (F) such that
|X|Hp
P(F):= sup
t∈[0,T]
¡E£
|Xt|Fp¤¢1/p
< +∞. HPp(F), when endowed with the norm| · |Hp
P(F), is a Banach space. We will consider HPp(F) also with other norms. Forγ>0, define
|X|Hp
P(F),γ:= sup
t∈[0,T]
³ e−γt¡
E£
|Xt|Fp¤¢1/p´ . The norms| · |Hp
P(E) and| · |Hp
P(E),γare equivalent.
Letn≥0, k≥0,T>0, and letE,F be real separable Banach spaces.
(iii) Gs1(E,F) denotes the space of continuous functionsf:E→F such that the Gâteaux derivative∇f(x) exists for every x∈E, the function
∇f:E→L(E,F) is strongly continuous and
sup
x∈E
|∇f(x)|L(E,F)< +∞.
When E is a Hilbert space and F =R, we will identify ∇f with an element of E through the Riesz representation E∗=E.
1We use the same symbol| · |to denote the norm of a normed space. The context should let avoid any confusion. If not, we will clarify the space of reference with a subscript.
(iv) Gs0,1([0,T]×E,F) denotes the space of continuous functions f: [0,T]×E→F, that the Gâteaux derivative in thexvariable∇xf(t,x) exists for everyx∈E, the function
∇xf: [0,T]×E→L(E,F) is strongly continuous and
sup
(t,x)∈[0,T]×E
|∇xf(t,x)|L(E,F)< +∞.
(v) C1
b(E,F) denotes the space of continuous functions f:E→F, continuously Fréchet differentiable, and such that
sup
x∈E
|D f(x)|L(E,F)< +∞, where D f denotes the Fréchet derivative of f.
(vi) C0,1([0,T]×E,F) denotes the space of continuous functions f: [0,T]×E→F, con- tinuously Fréchet differentiable with respect to the second variable.
(vii) C0,1b ([0,T]×E,F) denotes the space of continuous functions f: [0,T]×E→F, con- tinuously Fréchet differentiable with respect to the second variable, and such that
sup
(t,x)∈[0,T]×E
|Dxf(t,x)|L(E,F)< +∞,
where Dxf denotes the Fréchet derivative of f with respect to x.
When F=R, we dropR and simply write LpP, HPp,Gs1(E), Gs0,1(E),C0,1b ([0,T]×E), and C0,1
b ([0,T]×E).
Though the notation could appear to be misleading, observe that if f ∈C0,1
b ([0,T]× E,F) or f∈C1b(E,F), then f is not supposed to be bounded.
Let m>0 be a positive integer, and letU be an open subset ofRm. Leta,b be real numbers such thata<b. DefineQ:=[a,b)×Uand∂PQ:=∂U×[a,b]∪{b}×U.
(viii) Forα∈(0, 1),C1+α(Q) denotes the space of continuous functions f:Q→Rsuch that Dxf(t,x) exists classically for every (t,x)∈Q, and such that
|f|C1+α(Q):= |f|∞+ |Dxf|∞+ sup
(t,x),(s,y)∈Q (t,x)6=(s,y)
|u(s,y)−u(t,x)− 〈Dxf(t,x),y−x〉m|
¡|t−s| + |x−y|2m
¢(1+α)/2 < +∞, where| · |∞ is the supremum norm, and| · |m and〈·,·〉m are the Euclidean norm and scalar product inRm respectively.
(ix) Forα∈(0, 1),C1loc+α((0,T)×Rm) denotes the space of continuous functions f: (0,T)× Rm→R such that, for every point (t,x)∈(0,T)×Rm, there exists ε>0 and a, b∈ (0,T), witha<b, such that f ∈C1+α([a,b)×B(x,ε))2.
(x) For p≥1,W1,2,p(Q) denotes the usual Sobolev space of functions f ∈Lp(Q), whose weak partial derivatives ut, fxi and fxixj belong to Lp(Q). W1,2,p(Q) is equipped with the norm
|f|W1,2,p(Q):=
³
|f|Lpp(Q)+ |ft|Lpp(Q)+ |Dxf|Lpp(Q)+ |D2xf|Lpp(Q)
´1/p
.
2B(x,ε) denotes the open ball centered atxof radiusε.
2.2 H
B-extensions of mild solutions of SDEs
Letm≥1, and letH1be a real separable Hilbert space with scalar product〈·,·〉H1. Define H:=Rm×H1. Whenever xis a point of H, we will denote byx0the component of xlying inRm, and by x1the component ofxin H1. We endowHwith the natural scalar product
〈(x0,x1), (y0,y1)〉:= 〈x0,y0〉m+ 〈x1,y1〉H1 ∀(x0,x1), (y0,y1)∈H.
Let G: [0,T]×H→H andσ: [0,T]×H→L(Rm). We will consider the following as- sumptions on them.
Assumption 2.1. The functionsG andσare continuous, and there existsM>0such that
|G(t,x)−G(t,y)|H+ |σ(t,x)−σ(t,y)|L(Rm)≤M|x−y|H ∀(t,x), (t,y)∈[0,T]×H.
We associate toσthe following function:
Σ: [0,T]×H→L(Rm,H), defined by
Σ(t,x)y=(σ(t,x)y, 01) (2.1) for (t,x)∈[0,T]×H, y∈Rm, and where 01 denotes the origin inH1.
The following assumption will be standing for the remaining part of the work.
Assumption 2.2. S is a strongly continuous semigroup of contractions, with A as its infinitesimal generator.
LetW be a standardm-dimensional Brownian motion with respect to the filtrationF. Fort∈[0,T) and x∈H, consider the SDE
(d Xs=(A Xs+G(s,Xs))ds+Σ(s,Xs)dWs s∈(t,T]
Xt=x. (2.2)
It is well known (see [6, Ch. 7]) that, under Assumption 2.1, for p≥2, there exists a unique mild solution inHPp(H) to (2.2), that is a unique processXt,x∈HPp(H) such that
Xst,x=
x s∈[0,t]
Ss−tx+ Z s
t
Ss−wG(w,Xwt,x)dw+ Z s
t
Ss−wΣ(w,Xwt,x)dWw s∈(t,T] and, for every t∈[0,T], the map
H→HPp(H), x7→Xt,x (2.3)
is continuous and Lipschitz.
For future reference, we state existence and uniqueness of mild solution in the fol- lowing proposition, where we also show continuity int, and we introduce tools useful for later proofs.
Proposition 2.3. For any p≥2, under Assumption2.1, there exists a unique mild solu- tion Xt,x∈HPp(H)to SDE(2.2), and the map
[0,T]×(H,| · |)→HPp(H), (t,x)7→Xt,x (2.4) is continuous in(t,x), and Lipschitz inx, uniformly in t.
Proof. Since the arguments are standard, we just give a sketch of proof. Let t∈[0,T].
Define the map
Φ(t;·,·) : H×HPp(H)→HPp(H) by
Φ(t;x,Z)s:=
x s∈[0,t)
Ss−tx+ Z s
t
Ss−wG(w,Zw)dw+ Z s
t
Ss−wΣ(w,Zw)dWw s∈[t,T].
Letγ>0. By Assumption2.1, we have sup
t∈[0,T]
e−γptE£
|G(t,Z)−G(t,Z0)|p¤
≤Mp|Z−Z0|Hp p
P(H),γ ∀Z,Z0∈HPp(H) (2.5) sup
t∈[0,T]
e−γptEh
|σ(t,Z)−σ(t,Z0)|L(pRm)
i
≤Mp|Z−Z0|Hp p
P(H),γ ∀Z,Z0∈HPp(H). (2.6) It is easy to see that by (2.5), (2.6), the linearity ofΦ(t;x,Z) inx, and [7, Ch. 7, Proposition 7.3.1], there existsγ>0, depending only on p,T,M, such that
sup
(t,x)∈[0,T]×H
|Φ(t;x,Z)−Φ(t;x,Z0)|Hp
P(H),γ≤1
2|Z−Z0|Hp
P(H),γ ∀Z,Z0∈HPp(H). (2.7) This shows that, for every (t,x)∈[0,T]×H, there exists a unique fixed pointXt,x∈HPp(H) ofΦ(t;x,·). Such a fixed point is the mild solution of (2.2).
The continuity of (2.4) is also standard. We sketch a slightly different argument. Let {tn}n∈N be a sequence converging to t in [0,T]. By standard estimates on the integrals defining Φ (for the stochastic integral using Burkholder-Davis-Gundy’s inequality), by sublinear growth ofG(t,x) andσ(t,x) in x uniformly in t, and by Lebesgue’s dominated convergence theorem, we have
n→+∞lim Φ(tn;x,Z)=Φ(t;x,Z) inHPp(H), ∀(x,Z)∈H×HPp(H). (2.8) Then, by (2.8), by (2.7), and by [3, Lemma 3.5-(i)], we have
nlim→+∞Xtn,x=Xt,x inHPp(H), ∀x∈H. (2.9) This shows the continuity intof Xt,x. We notice that
sup
(t,Z)∈[0,T]×HPp(H)
|Φ(t;x,Z)−Φ(t;x0,Z)|Hp
P(H),γ≤ |x−x0|H ∀x,x0∈H. (2.10) By applying [3, Lemma 3.5-(ii)], we obtain
sup
t∈[0,T]
|Xt,x−Xt,x0|Hp
P(H),γ≤2|x−x0|H ∀x,x0∈H. (2.11)
By (2.9) and by (2.11) we conclude that the map
[0,T]×H→HPp(H), (t,x)7→Xt,x is continuous and Lipschitz continuous in xuniformly int.
We are going to endow H with a weaker norm, and give conditions such that the above continuity in (t,x) of Xt,xextends to the new norm. We will also make assumptions which will guarantee the Gâteaux differentiability of the mild solution with respect to the initial datum x in the space with the weaker norm and the strong continuity of the the Gâteaux derivative.
Let R:D(R)→ H be a densely defined linear operator such that R: D(R)→H has inverse R−1∈L(H). Then B=(R∗)−1R−1∈L(H) is selfadjoint and positive. For x∈H, define
|x|2B= 〈Bx,x〉 =¯
¯R−1x¯
¯
2 H
The spaceHendowed with the norm|·|Bis pre-Hilbert, since|·|Bis inherited by the scalar product〈x,y〉B= 〈B1/2x,B1/2y〉, where B1/2 is the unique positive self-adjoint continuous linear operator such that B=B1/2B1/2. Denote by HB the completion of the pre-Hilbert space (H,| · |B). With some abuse of notation, we also denote by| · |B the extension of| · |B
toHB. Now suppose that
StR⊂RSt ∀t∈R+. (2.12)
Observe that (2.12) implies
AR⊂R A. (2.13)
SinceR−1St=StR−1, we have
|Stx|B= |R−1Stx|H= |StR−1x|H≤ |R−1x|H= |x|B ∀t∈R+,x∈H.
We can then extend each St to an operator St∈L(HB) with the operator norm less than or equal to 1. By density of Hin HB, it is clear that the family {St}t∈R+ is a semigroup of contractions. Moreover, for x∈H,
lim
t→0+|Stx−x|B= lim
t→0+|R−1(Stx−x)|H= lim
t→0+|StR−1x−R−1x|H=0.
The above observations imply that the family {St}t∈R+is uniformly bounded and strongly continuous on a dense subspace of HB. Thus S is a strongly continuous semigroup on HB (see [9]).
By definition of| · |B,R: (D(R),| · |H)→(H,| · |B) is a full-range isometry. This implies the following facts:
(i) There exists a unique extensionRe: H→HB. (ii) Re andRe−1are isometries.
(iii) Re−1=Rg−1, whereRg−1:HB→His the unique continuous extension ofR−1.
Denote byR the operatorRe considered as an operatorHB⊃H=D(R)→HB. The above facts imply that R is a densely-defined full-range closed linear operator in HB, and that D(R) is a core for R.
We claim that
StR⊂R St ∀t∈R+. (2.14)
To prove it, let (x,R x)∈Γ(R), whereΓ(R) is the graph ofR. Since D(R) is a core for R, we can choose a sequence {(xn,R xn)}n∈N∈Γ(R) such that (xn,R xn)→(x,R x) inHB×HB. Then, recalling (2.12), we can write
StR x= lim
n→+∞StR xn= lim
n→+∞StR xn= lim
n→+∞RStxn= lim
n→+∞R Stxn,
where all the limits are considered inHB. This means that {R Stxn}n∈N is convergent in HB. We recall thatRis closed inHBand we observe thatStxn→StxinHBby continuity.
Thus we conclude thatR Stxn→R Stxin HB. This proves (2.14).
Now let A be the generator of the semigroup {St: HB→HB}t∈R+. Obviously A is an extension of A, i.e. Ax=Ax for x∈D(A). We will show that D(A)=R(D(A)). Using (2.14) we have forx∈HB,
t→0lim+ St−I
t x= lim
t→0+
St−I
t R R−1x=(by (2.14))= lim
t→0+RSt−I
t R−1x= lim
t→0+RSt−I t R−1x.
The last limit exists inHB if and only if the limit
tlim→0+
St−I t R−1x exists inH. Therefore we conclude that
D(A)=R(D(A)) and Ax=R AR−1x ∀x∈D(A). (2.15) Let us now in addition assume that
((a) AR=R A
(b) D(A)⊂D(R). (2.16)
Let {An}n≥1 be the Yosida approximations ofA. We claim that
(n−A)−1R⊂R(n−A)−1 ∀n≥1. (2.17) By (2.16), it follows that
D((n−A)−1R)=D(R)⊂H=D(R(n−A)−1).
Forx∈D(R),
(n−A)−1R x=(n−A)−1R(n−A)(n−A)−1x=(n−A)−1(n−A)R(n−A)−1x=R(n−A)−1x, where we use the fact that (n−A)−1x∈D(R A), because, by (2.16)(b),
A(n−A)−1x=n(n−A)−1x−x⊂D(A)+D(R)⊂D(R), (2.18)
and then we can invoke (2.16)(a). Then, forx∈D(R),
AnR x=n2(n−A)−1R x−nR x=n2R(n−A)−1x−nR x=R Anx, (2.19) that is
AnR⊂R An. (2.20)
We now claim that etAnR⊂R etAn. Letx∈D(R), whereetAn is the semigroup generated by An. By (2.18) and by (2.20),
AknR x=R Aknx ∀k∈N. (2.21) Fort∈R+, define
ym:=
m
X
k=0
tk k!Aknx.
By (2.18), ym∈D(R). Moreover, limm→+∞ym=etAnx, and (by (2.21))
m→+∞lim R ym= lim
m→+∞
m
X
k=0
tk
k!AknR x=etAnR x.
SinceR is closed, it follows that etAnx∈D(R), and R etAnx=etAnR x. Since this holds for everyx∈D(R), we conclude etAnR⊂R etAn.
Arguing as it was done for S, we obtain that every Sn can be uniquely extended to the semigroup etAn onHB generated by An, and (similarly to (2.15))
D(An)=R(D(An)) and Anx=R AnR−1x ∀x∈D(An). (2.22) We observe thatR(D(An))=R(H)=HB. If x∈H, by (2.15), (2.16), (2.17), and (2.22), we have
Anx=R AnR−1x=RnA(n−A)−1R−1x=RnA(n−A)−1R−1x
=n(R AR−1)(R(n−A)−1R−1)x=n(R AR−1)(n−A)−1x=nA(n−A)−1x, which can be written as
Anx=nA(n−A)−1x=Anx, ∀x∈H,
whereAnis the Yosida approximation ofA. Finally, since bothAnand Anare continuous onHB, and sinceH is dense inHB, we obtain
An=An,
and thenetAn=etAn, whereetAn is the semigroup generated by An. We collect the results above in the following proposition.
Proposition 2.4. Let R:D(R)⊂H→ H be a densely defined linear operator such that R−1∈L(H). Let HB be the Hilbert space obtained by completing H with respect to the norm induced by the scalar product defined by
〈x,y〉B:= 〈(R∗)−1R−1x,y〉 ∀x,y∈H.
Suppose that
StR⊂RSt ∀t∈R+. (2.23)
Then, for every t∈R+, there exists a unique continuous extension St of St to HB and the familyS:={St}t∈R+ is a strongly continuous semigroup of contractions on HB.
If we also suppose that
AR=R A and D(A)⊂D(R), (2.24)
then the Yosida approximations{An}n≥1of the infinitesimal generatorA of S are given by the unique continuous extensions toHB of the Yosida approximations {An}n≥1 of A, that is
An=An n≥1.
In the remaining of this section we will assume that (2.23) and (2.24) hold true.
Assumption 2.5. The functionsG andΣare Lipschitz with respect to the norm|·|B, with respect to the second variable and uniformly in the first one, that is there exists M>0 such that
|G(t,x)−G(t,y)|B+ |Σ(t,x)−Σ(t,y)|L(Rm,HB)≤M|x−y|B (2.25) for all t∈[0,T], x,y∈H. Denote by G (resp. Σ) the unique extension of G (resp. Σ) to a function from[0,T]×HBintoHB (resp. from[0,T]×HB intoL(Rm,HB)).
Under Assumption2.5we can consider the following SDE onHB (d Xs=¡
A Xs+G¡ s,Xs¢¢
ds+Σ¡s,Xs
¢dWs, s∈(t,T],
Xt=x, (2.26)
where x∈HB. By changing the reference Hilbert space from H to HB, we can apply Proposition2.3and say that SDE (2.26) has a unique mild solutionXt,xinHPp(HB), and [0,T]×HB →HPp(HB), (t,x)7→ Xt,x is continuous and | · |B-Lipschitz with respect to x, uniformly int.
Proposition 2.6. For any p≥2, under Assumptions 2.1 and 2.5, there exists a unique mild solution Xt,x∈HPp(HB)of SDE(2.26), and the map
[0,T]×HB→HPp(HB), (t,x)7→Xt,x (2.27) is continuous in (t,x), and Lipschitz in x, uniformly in t. If x∈H, Xt,x ∈HPp(H) and Xt,x=Xt,x, whereXt,x∈HPp(H)is the unique mild solution of SDE(2.2).
Proof. The first part follows from Proposition2.3. It remains to comment on the fact that Xt,x=Xt,x ifx∈H. The spaceHPp(H) is continuously embedded in HPp(HB). Thus, ifG andΣ satisfy Assumptions2.1 and2.5, and if the initial value x belongs to H, the mild solutionXt,x of (2.2) is also a mild solution of (2.26), and then, by the uniqueness of mild solutions,Xt,x=Xt,x inHPp(HB).
In order to obtain an a-priori estimate giving the regularity in which we are inter- ested, we will need to approximate mild solutions with other mild solutions of SDEs with smoother coefficients.
Proposition 2.7. Let G and σ satisfy Assumptions 2.1 and 2.5. There exist sequences {Gn}n∈N⊂C0,1
b ([0,T]×H,H),{Σn}n∈N⊂C0,1
b ([0,T]×H,L(Rm,H)), withΣn(t,x)y=(σn(t,x)y, 01) for someσn∈C0,1
b ([0,T]×H,L(Rm)), satisfying:
(i) For every n∈N, Gn and Σn have extensions Gn ∈C0,1
b ([0,T]×HB,HB) and Σn ∈ C0,1
b ([0,T]×HB,L(Rm,HB)).
(ii) For all(t,x), (t,y)∈[0,T]×HB, sup
n∈N|Gn(t,x)−Gn(t,y)|B≤M|x−y|B (2.28) sup
n∈N|Σn(t,x)−Σn(t,y)|L(Rm,HB)≤M|x−y|B. (2.29) (iii) For every compact setK⊂HB,
n→∞lim sup
(t,x)∈[0,T]×K
|G(t,x)−Gn(t,x)|B=0 (2.30)
nlim→∞ sup
(t,x)∈[0,T]×K
|Σ(t,x)−Σn(t,x)|L(Rm,HB)=0. (2.31)
Remark 2.8. We remark that, due to the fact that the range of Σ is finite-dimensional (see (2.1)), once the above continuity/differentiability/approximation conditions for Σn
are satisfied with respect toHB, they automatically hold forΣnwith respect toH.
Proof. Let {en}n∈Nbe an orthonormal basis of HBcontained inH. Forn∈N, let us define the functions
In:Rn→HB, y7→
n
X
k=1
ykek and
Pn:HB→Rn, x7→(〈x,e1〉B, . . . ,〈x,en〉B).
It is clear that|In|L(Rn,HB)=1 and |InPn|L(HB)=1. We observe also that, for everyn∈N, the linear operator
HB→H, x7→InPnx=
n
X
k=1
〈x,ek〉Bek is well defined and continuous. Denotecn:= |InPn|L(HB,H).
Let
ϕ(r) := (
e−
1
1−r2 ifr∈(−1, 1) 0 otherwise, and, for everyn∈N,
Cn:= µZ
Rnϕ(n|y|n)d y
¶−1
,
where| · |n denotes the Euclidean norm inRn. Define gn: [0,T]×Rn→H by standard mollification
gn(t,y) :=Cn¡
G(t,In·)∗ϕ(n| · |n)¢
(y)=Cn Z
RnG Ã
t,
n−1
X
k=0
zkek
!
ϕ(n|y−z|n)d z,
for all (t,y)∈[0,T]×Rn. We observe that gn is well-defined, becauseG is H-valued and continuous, andϕhas compact support. By Lebesgue’s dominated convergence theorem, gnis continuous.
Since the map Rn→R, z7→ ϕ(n|z|) is continuously differentiable and has compact support and since G is continuous, by a standard argument we can differentiate under the integral sign to obtain gnis differentiable with respect to yand
Dygn(t,y)v=nCn Z
RnG(t,Inz)ϕ0(n|y−z|n)〈y−z,v〉n
|y−z|n
d z.
By Lebesgue’s dominated convergence theorem, the map
[0,T]×Rn×Rn→H, (t,y,v)7→Dygn(t,y)v is continuous. Thus gn∈C0,1([0,T]×Rn,H). Define
Gn: [0,T]×HB→HB by
Gn(t,x) :=gn(t,Pnx)=Cn Z
RnG(t,InPnx−Inz)ϕ(n|z|n)d z ∀(t,x)∈[0,T]×HB. Since Gn([0,T]×HB)⊂H, we can also define Gn: [0,T]×H→ H by Gn(t,x) :=Gn(t,x) for every (t,x)∈[0,T]×H. Then Gn∈C0,1([0,T]×H,H) andGn∈C0,1([0,T]×HB,HB).
Moreover, by Assumption2.1,
|Gn(t,x)−Gn(t,x0)|H= |gn(t,Pnx)−gn(t,Pnx0)|H
≤Cn Z
Rn
¯¯G(t,InPnx−Inz)−G¡
t,InPnx0−Inz¢¯
¯Hϕ(n|z|n)d z
≤M¯
¯InPnx−InPnx0¯
¯H≤M cn|x−x0|B≤M cn|R−1|L(H)|x−x0|H,
(2.32)
for everyt∈[0,T] and x,x0∈H. Similarly, by Assumption2.5,
|Gn(t,x)−Gn(t,x0)|B= |gn(t,Pnx)−gn(t,Pnx0)|B
≤M¯
¯InPnx−InPnx0¯
¯B≤M¯
¯x−x0¯
¯B, (2.33)
for every t∈[0,T] and x,x0 ∈HB. Thus Gn ∈C0,1b ([0,T]×H,H) and Gn∈C0,1b ([0,T]× HB,HB).
To prove (2.30) for every compactK⊂HB, we first notice that sup
x∈K
|InPnx−x|B=εn→0 asn→ +∞.
