About classical solutions of the path-dependent heat equation

Cristina Di Girolami; Francesco Russo

doi:10.1515/rose-2020-2028

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About classical solutions of the path-dependent heat equation

Cristina Di Girolami und Francesco Russo

Veröffentlicht/Copyright: 5. Februar 2020

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Aus der Zeitschrift Random Operators and Stochastic Equations Band 28 Heft 1

Abstract

This paper investigates two existence theorems for the path-dependent heat equation, which is the Kolmogorov equation related to the window Brownian motion, considered as a C⁢([-T,0])-valued process. We concentrate on two general existence results of its classical solutions related to different classes of terminal conditions: the first one is given by a cylindrical not necessarily smooth random variable, the second one is a smooth generic functional.

Keywords: Infinite-dimensional analysis; Kolmogorov type equations; path-dependent heat equation; window Brownian motion

MSC 2010: 60H05; 60H30; 91G80

Communicated by Sergio Albeverio

Funding source: Istituto Nazionale di Alta Matematica ”Francesco Severi”

Award Identifier / Grant number: UUFMBAZ-2017-000859

Funding statement: The second named author would like to thank the first one for the kind invitation in the framework of GNAMPA project (n. prot. UUFMBAZ-2017-000859 31-07-2017) financially supported by Istituto di Alta Matematica Francesco Severi.

A Appendix: Malliavin and Fréchet derivatives

We need some technical results concerning the link between Fréchet and Malliavin derivatives in a separable Banach space that, for the moment, we set to be equal to ℝ. We need to apply Malliavin calculus related to the Brownian motion. Let T>0 and t∈[0,T] be fixed. We recall that

(A.1)W¯x:-Wt+x-Wt,x∈[0,T-t].

So the Wiener space will be C⁢([0,T-t]) with variable parameter in [0,T-t] and based on W¯. We consider the window Brownian element W¯T-t⁢(⋅) with values in C⁢([-(T-t),0]), defined as

W¯T-t⁢(x)=Wt+T-t⁢(x)-Wt=WT+x-Wt,x∈[-(T-t),0].

Lemma A.1.

Let G:C⁢([-(T-t),0])→R of class C1 be such that D⁢G has polynomial growth. Let Y∈D∞. Then G⁢(σ⁢W¯T-t⁢(⋅)) belongs to D1,2 and

(A.2)Drm⁢(G⁢(σ⁢W¯T-t⁢(⋅))⁢𝒴)=σ⁢∫]r-(T-t),0](Dd⁢y⁢G)⁢(σ⁢W¯T-t⁢(⋅))⁢𝒴+G⁢(σ⁢W¯T-t⁢(⋅))⁢Drm⁢𝒴,r∈[0,T-t]⁢a.e.

Proof.

The proof of this result needs some boring technicalities involving the approximation of a continuous function and its polygonal approximation. Formula (A.2) is stated in a particular case for instance in [13, Example 1.2.1]. ∎

A consequence of the previous lemma is the possibility of the differentiating

h=F⁢(YTt,η),F:C⁢([-T,0])→ℝ

of class C1 Fréchet. We remark that YTt,η=Gη⁢(σ⁢W¯T-t⁢(⋅)), where Gη:C⁢[-(T-t),0]→C⁢([-T,0]) is given by

Gη⁢(γ)={η⁢(x+T-t),x∈[-T,t-T[,η⁢(0)+γ⁢(T-t+x),x∈[t-T,0].

By Lemma A.1, if 𝒴∈𝔻∞,

(A.3)Dxm⁢(h⁢𝒴)=σ⁢∫]x-T+t,0]Dd⁢y⁢(F∘Gη)⁢(σ⁢W¯T-t⁢(⋅))⁢𝒴+F∘Gη⁢(σ⁢W¯T-t⁢(⋅))⁢Dxm⁢𝒴,x∈[0,T-t].

Remark A.2.

We remark that, for all γ∈C⁢([-T+t,0]), D⁢(F∘Gη)∈ℳ⁢([-T+t,0]).

We have, for ζ∈C⁢([-T+t,0]),

∫Dd⁢y⁢(F∘Gη)⁢(γ)⁢ζ⁢(y)=∫[t-T,0]Dd⁢y⁢F⁢(Gη⁢(γ))⁢ζ⁢(y).

So (A.3) gives, for x∈[0,T-t],

Dxm⁢(h⁢𝒴)=σ⁢∫]x-T+t,0](Dd⁢y⁢F)⁢(Gη⁢(σ⁢W¯T-t))⁢𝒴+(F∘Gη)⁢(σ⁢W¯T-t⁢(⋅))⁢Dxm⁢𝒴=σ⁢∫]x-T+t,0](Dd⁢y⁢F)⁢(YTt,η)⁢𝒴+F⁢(YTt,η)⁢Dxm⁢𝒴.

At this point, we have proved the following.

Proposition A.3.

Let H:C⁢([-T,0])→R be of class C1-Fréchet with polynomial growth. Let Y∈D∞. Then H⁢(YTt,η)⁢Y belongs to D1,2 and

Drm⁢(H⁢(YTt,η)⁢𝒴)=σ⁢∫]x-T+t,0](Dd⁢y⁢H)⁢(YTt,η)⁢𝒴+F⁢(YTt,η)⁢Drm⁢𝒴.

The previous proposition admits a generalization to the case when H:C⁢([-T,0])→ℝ is replaced by a functional

C⁢([-T,0])→(⊗^π⁡⊗^π⁡C⁢([-T,0])⋯C⁢([-T,0]))⏟n⁢times,n≥1.

Typically, an example will be Dn⁢H. We recall that

(⊗^π⁡⊗^π⁡C⁢([-T,0])⋯C⁢([-T,0])⏟n⁢times)*

can be isomorphically identified with the space of n-multilinear continuous functionals on C⁢([-T,0]). Proposition A.3 can be generalized as follows.

Proposition A.4.

Let H:C⁢([-T,0])→R of class Cn+1-Fréchet be such that Dn+1⁢F has polynomial growth. Let

𝒴∈𝔻∞⁢(⊗^π⁡⊗^π⁡C⁢([0,T-t])⋯C⁢([0,T-t])⏟n⁢𝑡𝑖𝑚𝑒𝑠).

Then 〈Dn⁢H⁢(YTt,η),Y〉 belongs to D1,2. Moreover, for a.e. r∈[0,T-t], we have

(A.4)Drm⁢(〈Dn⁢H⁢(YTt,η),𝒴〉)=σ⁢〈Dn+1⁢H⁢(YTt,η),1]r-T+t,0]⊗𝒴〉+〈Dn⁢H⁢(YTt,η),Drm⁢𝒴〉.

Remark A.5.

The function 1]r-T+t,0] can be considered as a test function ζ0. Indeed,

ζ0↦Dn+1⁢H⁢(YTt,η)⁢(ζ0⊗ζ1⊗⋯⁢ζn)

for fixed ζ1,…,ζn∈C⁢([-T,0]) is a measure.

Proof.

Avoiding to state too abstract results, the proof of Proposition A.4 is based on a generalization of Lemma A.1 replacing the value space ℝ with the separable Banach space B, setting

B=⊗^π⁡⊗^π⁡C⁢([-T,0])⋯C⁢([-T,0]).∎

Lemma A.6.

Let B be a separable Banach space. Let G:C⁢([-T+t,0])→B* be of class C1-Fréchet with polynomial growth. Let Y∈D∞⁢(B). Then G⁢(W¯T-t⁢(⋅))⁢Y∈D1,2⁢(B) and

Dr(〈G(W¯T-t(⋅)),𝒴〉BB*)=〈DG(W¯T-t(⋅)),1]r-T+t,0]⊗𝒴〉⊗^π⁡C⁢([-T,0])B(⊗^π⁡C⁢([-T,0])B)*+〈G(W¯T-t(⋅)),Drm𝒴〉

Remark A.7.

We remark the following.

D⁢G:C⁢([-T+t,0])→(⊗^π⁡C⁢([-T,0])B)*.
Proposition A.4 will be used for n=1,2,3.
Drm⁢𝒴∈B for almost all r.

Acknowledgements

The authors wish to thank the referee for the careful reading of the paper and the associated editor and the editor in chief as well.

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Received: 2018-12-09

Accepted: 2019-04-30

Published Online: 2020-02-05

Published in Print: 2020-03-01

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Artikel in diesem Heft

https://doi.org/10.1515/rose-2020-2028

Schlagwörter für diesen Artikel

Infinite-dimensional analysis; Kolmogorov type equations; path-dependent heat equation; window Brownian motion