Abstract
We study path-dependent SDEs in Hilbert spaces. By using methods based on contractions in Banach spaces, we prove the Gâteaux differentiability of generic order n of mild solutions with respect to the starting point and the continuity of the Gâteaux derivatives with respect to all the data.
Keywords
- Stochastic functional differential equations in Hilbert spaces
- Gâteaux differentiability
- Contraction mapping theorem
AMS 2010 Subject Classification
This research has been partially supported by the ERC 321111 Rofirm.
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- 1.
\(B_b([0,T],H)\) denotes the space of bounded Borel functions \([0,T]\rightarrow H\).
- 2.
If \(x,x'\in X\), the segment \([x,x']\) is the set \(\{\zeta x+(1-\zeta )x'|\zeta \in [0,1]\}\).
- 3.
Recall notation at p. 5.
- 4.
We recall that \(B_b([0,T],H)\) is endowed with the norm \(|\cdot |_\infty \).
- 5.
The limits should be understood in the suitable spaces \(Y_k\). For instance, when computing \(\lim _{\varepsilon \rightarrow 0}\frac{\mathbf {I_1}}{\varepsilon }\), the object \(\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u+\varepsilon x_n)\) should be considered in the space \(Y_2\), which can be done thanks to the inductive hypothesis.
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4 Appendix
4 Appendix
Proof of Proposition 2.1 Suppose that the derivatives \(\partial ^j_{x_1\ldots x_j}f(u)\) exists for all \(u\in U\), \(x_1,\ldots ,x_j\in X_0\), \(j=1,\ldots ,n\), separately continuous in \(u,x_1,\ldots , x_j\). We want to show that \(f\in \mathscr {G}^{n}(U,Y;X_0)\).
We proceed by induction on n. Let \(n=1\). Since \(\partial _xf(u)\) is continuous in u, for all \(x\in X_0\), we have that \(X_0\rightarrow Y,\ x\mapsto \partial _xf(u)\) is linear ([11, Lemma 4.1.5]). By assumption, it is also continuous. Hence \(x \mapsto \partial _x f(u)\in L(X_0,Y)\) for all \(u\in U\). This shows the existence of \( \partial _{X_0}f\). The continuity of \(U\rightarrow L_s(X_0,Y),\ u\mapsto \partial _{X_0} f(u)\), comes from the separate continuity of (2.1) and from the definition of the locally convex topology on \(L_s(X_0,Y)\). This shows that \(f\in \mathscr {G}^{1}(U,Y;X_0)\).
Let now \(n>1\). By inductive hypothesis, we may assume that \(f\in \mathscr {G}^{n-1}(U,Y;X_0)\) and
Let \(x_n\in X_0\). The limit
exists in \(L_s^{(n-1)}(X_0^{n-1},Y)\) if and only if \(\Lambda \in L_s^{(n-1)}(X_0^{n-1},Y)\) and, for all \(x_1,\ldots ,x_{n-1}\in X_0\), the limit
holds in Y. By assumption, the limit (4.2) is equal to \( \partial ^n_{x_1\ldots x_{n-1}x_n}f(u)\), for all \(x_1,\ldots ,x_{n-1}\). Since, by assumption, \( \partial ^n_{x_1\ldots x_{n-1}x_n}f(u)\) is separately continuous in \(u,x_1,\ldots ,x_{n-1},x_n\), we have that the limit (4.1) exists in \(L_s^{(n-1)}(X_0^{n-1},Y)\) and is given by
Since u and \(x_n\) were arbitrary, we have proved that \( \partial _{x_n} \partial ^{n-1}_{X_0}f(u)\) exists for all u, \(x_n\). Moreover, for all \(x_1,\ldots ,x_n\in X_0\), the function
is continuous, by separate continuity of (2.1). Then \(\partial ^{n}_{x_1\ldots x_{n-1}x_n}f(u)\) is linear in \(x_n\). The continuity of
comes from the continuity of \(\partial ^{n}_{x_1\ldots x_{n-1}x}f(u)\) in each variable, separately. Hence (4.3) belongs to \(L_s(X_0,L_s^{n-1}(X_0^{n-1},Y))\) for all \(u\in U\). This shows that \( \partial _{X_0 }^{n-1}f\) is Gâteaux differentiable with respect to \(X_0\) and that
and shows also the continuity of
due to the continuity of the derivatives of f, separately in each direction. Then we have proved that \(f\in \mathscr {G}^{n}(U,Y;X_0)\) and that (2.2) holds.
Now suppose that \(f\in \mathscr {G}^{n}(U,Y;X_0)\). By the very definition of \( \partial _{X_0 }f\), \(\partial _xf(u)\) exists for all \(x\in X_0\) and \(u\in U\), it is separately continuous in u, x, and coincides with \( \partial _{X_0 }f(u).x\). By induction, assume that \(\partial ^{n-1}_{x_1\ldots x_{n-1}}f(u)\) exists and that
Since \( \partial _{X_0 }^{n-1}f(u)\) is Gâteaux differentiable, the directional derivative \(\partial _{x_n} \partial ^{n-1}_{X_0 }f(u)\) exists. Hence, by (4.4), the derivative \(\partial ^n_{x_1\ldots x_{n-1}x_n}f(u)\) exists for all \(x_1,\ldots ,x_{n-1},x_n\in X_0\). The continuity of \(\partial ^n_{x_1\ldots x_{n-1}x_n}f(u)\) with respect to u comes from the continuity of \( \partial _{X_0 }^nf\). The continuity of \(\partial ^n_{x_1\ldots x_{j}\ldots x_n}f(u)\) with respect to \(x_j\) comes from the fact that, for all \(x_{j+1},\ldots ,x_n\in X_0\), \(u\in U\),
belongs to \(L^{(j)}_s(X_0^j,Y)\). \(\blacksquare \)
Proof of Theorem 2.9 The proof is by induction on n. The case \(n=1\) is provided by Proposition 2.7.
Let \(n\ge 2\). Clearly, it is sufficient to prove that \(\varphi \in \mathscr {G}^{n}(U,Y_n)\) and that (2.11) holds true for \(j=n\). Since we are assuming that the theorem holds true for \(n-1\), we can apply it with the data
where \(\widetilde{h}_k:=h_{k+1}\), \(\widetilde{Y}_k:=Y_{k+1}\), for \(k=1,\ldots ,n-1\). According to the claim, the fixed-point function \(\widetilde{\varphi }\) of \(\widetilde{h}_1\) belongs to \(\mathscr {G}^{j}(U,\widetilde{Y}_{(n-1)-j+1})\), for \(j=1,\ldots ,n-1\), and formula (2.11) holds true for \(\widetilde{\varphi }\) and \(j=1,\ldots ,n-1\). Since \(\varphi (u)=(i_{2,1}\circ \widetilde{\varphi }) (u)\), for \(u\in U\), we have \(\varphi \in \mathscr {G}^{j}(U,\widetilde{Y}_{n-j})=\mathscr {G}^{j}(U,Y_{n-j+1})\), for \(j=1,\ldots ,n-1\), and
Then (2.11) holds true for \(\varphi \) up to order \(j=n-1\). In particular \(\varphi \in \mathscr {G}^{n-1}(U,Y_2)\), hence, for \(x_1,\ldots ,x_n\in X\), \(\epsilon >0\), we can write
where \(\mathscr {S}(\cdot )\) denotes the sum
for \(v\in U\). By recalling that \(\varphi \in \mathscr {G}^{j}(U,Y_{n-j+1})\), \(j=1,\ldots ,n-1\), hence by taking into account with respect to which space the derivatives of \(\varphi \) are continuous, we write
with (Footnote 5)
In a similar way,
Notice that
and
and
where
By collecting (4.7), (4.8), (4.9), (4.10), we obtain
Hence
and, by recalling (4.5), (4.6), we obtain
Finally, we can conclude the proof by recalling that \(I- \partial _{Y_1}h_1(u,\varphi (u))\) is invertible with strongly continuous inverse. \(\blacksquare \)
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Rosestolato, M. (2019). Path-Dependent SDEs in Hilbert Spaces. In: Cohen, S., Gyöngy, I., dos Reis, G., Siska, D., Szpruch, Ł. (eds) Frontiers in Stochastic Analysis–BSDEs, SPDEs and their Applications. BSDE-SPDE 2017. Springer Proceedings in Mathematics & Statistics, vol 289. Springer, Cham. https://doi.org/10.1007/978-3-030-22285-7_9
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