We will first consider the case where the coefficients Aj and B of the sto- chastic differential equation (2.37) are globally Lipschitz functions and have linear growth. Our aim is to show that the coordinates of the solution at each timet∈[0, T] belong to the spaceD1,∞=∩p≥1D1,p. To show this re- sult we will make use of an extension of the chain rule to Lipschitz functions established in Proposition 1.2.4.
We denote byDjt(F),t∈[0, T],j= 1, . . . , d, the derivative of a random variable F as an element of L2([0, T]×Ω;Rd) ≃ L2(Ω;H). Similarly we denote byDtj11,...,t,...,jNN(F) theNth derivative ofF.
Using Proposition 1.2.4, we can show the following result.
Theorem 2.2.1 Let X ={X(t), t∈[0, T]} be the solution to Eq. (2.37), where the coefficients are supposed to be globally Lipschitz functions with linear growth (hypotheses (h1) and (h2)). Then Xi(t)belongs to D1,∞ for any t∈[0, T]andi= 1, . . . , m. Moreover,
sup
0≤r≤t
E sup
r≤s≤T|DrjXi(s)|p
<∞,
and the derivative DjrXi(t) satisfies the following linear equation:
DjrX(t) = Aj(r, X(r)) + t
r
Ak,α(s)Djr(Xk(s))dWsα +
t r
Bk(s)DjrXk(s)ds (2.41) forr≤t a.e., and
DrjX(t) = 0
forr > ta.e., whereAk,α(s)andBk(s)are uniformly bounded and adapted m-dimensional processes.
Proof: Consider the Picard approximations given by X0(t) = x0,
Xn+1(t) = x0+ t
0
Aj(s, Xn(s))dWsj+ t
0
B(s, Xn(s))ds (2.42)
ifn≥0. We will prove the following property by induction onn:
(P) Xni(t)∈D1,∞ for all i= 1, . . . , m,n≥0, and t∈[0, T]; further- more, for allp >1 we have
ψn(t) := sup
0≤r≤t
E
sup
s∈[r,t]|DrXn(s)|p
<∞ (2.43) and
ψn+1(t)≤c1+c2
t 0
ψn(s)ds, (2.44)
for some constants c1, c2.
Clearly, (P) holds forn= 0. Suppose it is true forn. Applying Proposi- tion 1.2.4 to the random vectorXn(s) and to the functionsAij andBi, we deduce that the random variables Aij(s, Xn(s)) and Bi(s, Xn(s)) belong to D1,2 and that there exist m-dimensional adapted processes An,ij (s) = (An,ij,1(s), . . . , An,ij,m(s)) andBn,i(s) = (Bn,i1 (s), . . . , Bn,im(s)), uniformly boun- ded byK, such that
Dr[Aij(s, Xn(s))] = An,ij,k(s)Dr(Xnk(s))1{r≤s} (2.45) and
Dr[Bi(s, Xn(s))] = Bn,ik (s)Dr(Xnk(s))1{r≤s}. (2.46) In fact, these processes are obtained as the weak limit of the sequences {∂k[Aij∗αm](s, Xn(s)), m≥1}and{∂k[Bi∗αm](s, Xn(s)), m≥1}, where αm denotes an approximation of the identity, and it is easy to check the adaptability of the limit. From Proposition 1.5.5 we deduce that the random variablesAij(s, Xn(s)) andBi(s, Xn(s)) belong toD1,∞.
Thus the processes{Dlr[Aij(s, Xn(s))], s≥r}and{Dlr[Bi(s, Xn(s))], s≥ r} are square integrable and adapted, and from (2.45) and (2.46) we get
|Dr[Aij(s, Xn(s))]| ≤K|DrXn(s)|, |Dr[Bi(s, Xn(s))]| ≤K|DrXn(s)|. (2.47) Using Lemma 1.3.4 we deduce that the Itˆo integral t
0Aij(s, Xn(s))dWsj belongs to the spaceD1,2, and forr≤twe have
Drl[ t
0
Aij(s, Xn(s))dWsj] =Ail(r, Xn(r)) + t
r
Drl[Aij(s, Xn(s))]dWsj. (2.48) On the other hand,t
0Bi(s, Xn(s))ds∈D1,2, and forr≤twe have Drl[
t 0
Bi(s, Xn(s))ds] = t
r
Drl[Bi(s, Xn(s))]ds. (2.49)
From these equalities and Eq. (2.42) we see that Xn+1i (t) ∈ D1,∞ for all t∈[0, T], and we obtain
E
sup
r≤s≤t|DjrXn+1(s)|p
≤cp
&
γp+Tp−1Kp t
r
E
|DjrXn(s)|p ds
' , (2.50) where
γp= sup
n,j
E( sup
0≤t≤T|Aj(t, Xn(t))|p)<∞.
So (2.43) and (2.44) hold forn+ 1. From Lemma 2.2.1 we know that E
sup
s≤T|Xn(s)−X(s)|p
−→0
as n tends to infinity. By Gronwall’s lemma applied to (2.50) we deduce that derivatives of the sequenceXni(t) are bounded inLp(Ω;H) uniformly in n for all p≥ 2. Therefore, from Proposition 1.5.5 we deduce that the random variables Xi(t) belong toD1,∞. Finally, applying the operator D to Eq. (2.37) and using Proposition 1.2.4, we deduce the linear stochastic differential equation (2.41) for the derivative ofXi(t).
If the coefficients of Eq. (2.37) are continuously differentiable, then we can write
Aik,l(s) = (∂kAil)(s, X(s)) and
Bik(s) = (∂kBi)(s, X(s)).
In order to prove the existence of higher-order derivatives, we will need the following technical lemma.
Consider adapted and continuous processesα={α(r, t), t∈[r, T]} and V ={Vj(t),0≤t≤T, j= 0, . . . , d}such thatαism-dimensional andVjis uniformly bounded and takes values on the set of matrices of orderm×m.
Suppose that the random variables αi(r, t) and Vjkl(t) belong toD1,∞ for anyi, j, k, l, and satisfy the following estimates:
sup
0≤r≤T
E
sup
r≤t≤T|α(r, t)|p
< ∞, sup
0≤s≤T
E
sup
s≤t≤T|DsVj(t)|p
< ∞, sup
0≤s,r≤TE
sup
r∨s≤t≤T|Dsα(r, t)|p
< ∞, for anyp≥2 and anyj = 0, . . . , d.
Lemma 2.2.2 Let Y = {Y(t), r ≤ t ≤ T} be the solution of the linear stochastic differential equation
Y(t) =α(r, t) + t
r
Vj(s)Y(s)dWsj+ t
r
V0(s)Y(s)ds. (2.51) Then {Yi(t)} belongs to D1,∞ for any i = 1, . . . , m, and the derivative DsYi(t)verifies the following linear equation, fors≤t:
DjsY(t) = Dsjα(r, t) +Vj(s)Y(s)1{r≤s≤t}
+ t
r
[DjsVl(u)Y(u) +Vl(u)DsjY(u)]dWul +
t r
[DjsV0(u)Y(u) +V0(u)DjsY(u)]du. (2.52) Proof: The proof can be done using the same technique as the proof of Theorem 2.2.1, and so we will omit the details. The main idea is to observe that Eq. (2.51) is a particular case of (2.38) when the coefficients σj and b are linear. Consider the Picard approximations defined by the recursive equations (2.39). Then we can show by induction that the variablesYni(t) belong toD1,∞ and satisfy the equation
DjsYn+1(t) = Djsα(r, t) +Vj(s)Yn(s)1{r≤s≤t}
+ t
r
[DjsVl(u)Yn(u) +Vl(u)DsjYn(u)]dWul +
t r
[DjsV0(u)Yn(u) +V0(u)DjsYn(u)]du.
Finally, we conclude our proof as we did in the proof of Theorem 2.2.1.
Note that under the assumptions of Lemma 2.2.2 the solution Y of Eq.
(2.51) satisfies the estimates E
sup
0≤t≤T|Y(t)|p
< ∞, sup
0≤s≤tE
sup
r≤t≤T|DsY(t)|p
< ∞, for allp≥2.
Theorem 2.2.2 Let X be the solution of the stochastic differential equa- tion (2.37), and suppose that the coefficients Aij and Bi are infinitely dif- ferentiable functions inxwith bounded derivatives of all orders greater than or equal to one and that the functions Aij(t,0) and Bi(t,0) are bounded.
ThenXi(t)belongs toD∞ for allt∈[0, T], andi= 1, . . . , m.
Proof: We know from Theorem 2.2.1 that for any i = 1, . . . , m and any t ∈ [0, T], the random variable Xi(t) belongs to D1,p for all p ≥ 2.
Furthermore, the derivativeDjrXi(t) verifies the following linear stochastic differential equation:
DrjXi(t) =Aij(r, Xr) + t
r
(∂kAil)(s, X(s))DjrXk(s)dWsl +
t r
(∂kB)(s, X(s))DjrXk(s)ds. (2.53) Now we will recursively apply Lemma 2.2.2 to this linear equation. We will denote byDjr11,...,j,...,rNN(X(t)) the iterated derivative of orderN. We have to introduce some notation. For any subsetK={ǫ1<ã ã ã< ǫη}of{1, . . . , N}, we put j(K) =jǫ1, . . . , jǫη andr(K) =rǫ1, . . . , rǫη. Define
αil,j1,...,jN(s, r1, . . . , rN) =
(∂k1ã ã ã∂kνAil)(s, X(s))
ìDj(Ir(I11))[Xk1(s)]ã ã ãDj(Ir(Iνν))[Xkν(s)]
and
βij1,...,jN(s, r1, . . . , rN) =
(∂k1ã ã ã∂kνBi)(s, X(s))
ìDj(Ir(I11))[Xk1(s)]ã ã ãDr(Ij(Iνν))[Xkν(s)], where the sums are extended to the set of all partitions {1, . . . , N} = I1∪ ã ã ã ∪Iν. We also setαij(s) =Aij(s, X(s)). With these notations we will recursively show the following properties for any integerN ≥1:
(P1) For anyt∈[0, T],p≥2, andi= 1, . . . , m, Xi(t) belongs toDN,p, and
sup
r1,...,rN∈[0,T]
E
sup
r1∨ããã∨rN≤t≤T|Dr1,...,rN(X(t))|p
<∞. (P2) TheNth derivative satisfies the following linear equation:
Djr11,...,j,...,rNN(Xi(t)) = N ǫ=1
αijǫ,j1,...,jǫ−1,jǫ+1,...,jN(rǫ, r1, . . . , rǫ−1, rǫ+1, . . . , rN) +
t r1∨ããã∨rN
αil,j1,...,jN(s, r1, . . . , rN)dWsl
+βij1,...,jN(s, r1, . . . , rN)ds
(2.54) ift≥r1∨ ã ã ã ∨rN, and Djr11,...,j,...,rNN(X(t)) = 0 ift < r1∨ ã ã ã ∨rN.
We know that these properties hold forN = 1 because of Theorem 2.2.1.
Suppose that the above properties hold up to the indexN. Observe that αil,j1,...,jN(s, r1, . . . , rN) is equal to
(∂kAil)(s, X(s))Djr11,...,j,...,rNN(Xk(s))
(this term corresponds toν = 1) plus a polynomial function on the deriv- atives (∂k1ã ã ã∂kνAil)(s, X(s)) withν ≥2, and the processes Dj(I)r(I)(Xk(s)), with card(I) ≤ N −1. Therefore, we can apply Lemma 2.2.2 to r = r1∨ ã ã ã ∨rN, and the processes
Y(t) = Drj11,...,j,...,rNN(X(t)), t≥r,
Vjik(t) = (∂kAij)(s, X(s)), 1≤i, k≤m, j = 1, . . . , d, and α(r, t) is equal to the sum of the remaining terms in the right-hand side of Eq. (2.54).
Notice that with the above notations we have Drj!
αil,j1,...,jN(t, r1, . . . , rN)"
=αil,j1,...,jN,j(t, r1, . . . , rN, r) and
Djr!
βij1,...,jN(t, r1, . . . , rN)"
=βij1,...,jN,j(t, r1, . . . , rN, r).
Using these relations and computing the derivative of (2.54) by means of Lemma 2.2.2, we obtain
DjrDjr11,...,j,...,rNN(Xi(t))
= N ǫ=1
αijǫ,j1,...,jǫ−1,jǫ+1,...,jN,j(rǫ, r1, . . . , rǫ−1, rǫ+1, . . . , rN, r) +αij,j1,...,jN(r, r1, . . . , rN)
+ t
r1∨ããã∨rN
αil,j1,...,jN,j(s, r1, . . . , rN, r)dWsl
+βij1,...,jN,j(s, r1, . . . , rN, r)ds ,
which implies that property (P2) holds forN+1. The estimates of property (P1) are also easily derived. The proof of the theorem is now complete.
Exercises
2.2.1Letσandbbe continuously differentiable functions onRwith boun- ded derivatives. Consider the solutionX ={Xt, t∈[0, T]}of the stochastic differential equation
Xt=x0+ t
0
σ(Xs)dWs+ t
0
b(Xs)ds.
Show that fors≤twe have DsXt=σ(Xs) exp
t 0
σ′(Xs)dWs+ t
0
[b′−1
2(σ′)2](Xs)ds
.
2.2.2(Doss [84]) Suppose thatσis a function of classC2(R) with bounded first and second partial derivatives and thatbis Lipschitz continuous. Show that the one-dimensional stochastic differential equation
Xt=x0+ t
0
σ(Xs)dWs+ t
0
[b+1
2σσ′ ](Xs)ds (2.55) has a solution that can be written in the formXt=u(Wt, Yt), where
(i) u(x, y) is the solution of the ordinary differential equation
∂u
∂x =σ(u), u(0, y) =y;
(ii) for each ω ∈Ω,{Yt(ω), t≥0} is the solution of the ordinary differ- ential equation
Yt′(ω) =f(Wt(ω), Yt(ω)), Y0(ω) =x0, wheref(x, y) =b(u(x, y))
∂u
∂y
−1
=b(u(x, y)) exp(−x
0 σ′(u(z, y)dz).
Using the above representation of the solution to Eq. (2.55), show that Xtbelongs toD1,p for allp≥2 and compute the derivativeDsXt.