In the first part of this paper, we follow [9] and revise the Itô functional calculus by means of stochastic calculus via regularization. We conclude the first part of the paper by showing how our functional Itô formula is strictly related to the Itô formula derived in the framework of stochastic Banach space-valued computation via regularization, for the case of window processes. A taste of the notion of a regular solution in the Banach space framework was presented in Chap.
We first consider regular solutions, which we call strict solutions, in the framework of the functional Itô calculus. Rather, it is not necessary that our definition of a strong viscosity solution of the path-dependent semilinear Kolmogorov equation be continuous, as in the spirit of viscosity solutions. Proposition 1 Suppose that limit(3) exists in the sense of ucp. 0Y d−X of Y with respect to X exists.
It is a well-known result in the theory of semimartingales, but it also extends to the framework of finite quadratic variational processes. Therefore, if U is continuous with respect to the topology of C([−T,0]), then it admits a unique continuous extensionu: C([−T,0])→R. This example will play an important role in Sect.3 to justify a weaker idea of solution for the path-dependent semilinear.
Functional Derivatives and Functional Itô’s Formula
We say that you admit these second-order vertical derivatives ηif the second-order partial derivatives at(η|[−T,0[ η(0))ofu˜ with respect to its second argument, denoted by ∂aa2 u(η˜ |[ −T ,0[ η(0)), exists and we set iii) DVu and DV Vue exist everywhere on C([−T,0]) and are continuous. Suppose there exists a unique extension u:C([−T,0])→RofU (e.g. ifU is continuous with respect to the topology of C([−T,0])). Then, denoting by ∂ε+φthe right partial derivative ofφwith respect toεand using formula (17), we find.
It follows from a standard result of the differential calculus (see, for example, Corollary 1.2, Chapter 2, in [32]) that φ is continuously differentiable on [0,∞[×R with respect to its first argument. Note that our definition of a horizontal drain differs from that presented in [17] as it is based on the boundary on the left, while the definition proposed in [17] would lead to the formula. This difference between (16) and (26) is crucial for the proof of the functional Itô formula.
Indeed, as can be seen from the proof of Theorem 2, to prove Itô's formula we need to consider the expression Accepting definition (26), it is convenient to write I1(ε,t) as the sum of the two integrals. To conclude the proof of Itô's formula in the same sense as in Theorem 2, we should prove.
As an example, (27) is satisfied if we assume the following condition onu: there exists a constant˜ C >0 such that, for every ε >0,. This last condition is verified if, for example, u˜ is uniformly Lipschitz continuous with respect to the norm L1([−T,0])-onC([−T,0[), that is: there exists a constant C >0 e such that We conclude this subsection by providing Itô's functional formula for a map U : [0,T]×C([−T,0])→R also depends on the time variable.
Then we can give the following definitions. ii) ∂tube is everywhere on I×C([−T,0]) and is continuous;. iv) DVu and DV Vu exist everywhere on I×C([−T,0]) and are continuous. We can now state the Itô functional formula, the proof of which is not reported, because it can be done in the same way as Theorem 2.
Comparison with Banach Space Valued Calculus via Regularization
More precisely, our goal is to identify the building blocks of our functional Itô formula (19) with the terms appearing in the Itô formula derived in Theorem 6.3 and Sect. Although it is expected that the vertical derivative DVU can be identified with the term Dd xδ0U of the Fréchet derivative, it is more difficult to guess which terms the horizontal derivative DHU corresponds to. We denote by BV([−T,0]) the set of càdlàg bounded variation functions on [−T,0], which is a Banach space when equipped with the norm.
We will denote by E⊗Ftensor the algebraic product of EandF, defined as the set of elements of the formv=n. i=1ei⊗fi, for some positive integers, where e∈ Eand f ∈ F. ii). We will refer to Eˆ⊗πF as the tensor product of the Banach spaces E and F. iv) If E and F are Hilbert spaces, we denote Eˆ⊗hF the Hilbert tensor product, which is still a Hilbert space taken as the complement of E⊗F for the scalar product e⊗ f,e⊗ f := e,eEf,fF, for any ,e∈ Eand f, f∈ F. v) The symbols Eˆ⊗2π and⊗2 denote, respectively, the Banach space Eˆ⊗πAnd element ⊗e of the algebraic tensor productE⊗E. Eˆ⊗πFis isomorphic to the space of continuous bilinear formsBi(E,F), endowed with the norm · E,F defined as.
So we have the following sequence of canonical identifications:(Eˆ⊗πF)∗∼=Bi(E,F)∼=L(E;F∗). i) DU, the first Fréchet derivative of U, belongs to C(E;E∗) and (ii) D2U, the second Fréchet derivative of U, belongs to C(E;L(E;E∗)). In the following we will refer to the term g4(x)δy(d x)⊗d ya as the diagonal component and tog4(x) as the diagonal element of μ. Taking into account that D·acu:C([−T,0])→ B V([−T,0]) is continuous and thus bounded on compact sets, it follows that f is bounded.
For our second representation result of DHU, we need the following generalization of the deterministic backward integral when the integrand is a measure. Theorem 8 If g is absolutely continuous and the density is cadlàg (still denoted by g), then Definition 18 is compatible with that in Definition 4. Let us also assume the following. i) D2x,Di agU(η), the diagonal element of the second-order derivative atη, has a series discontinuity that has zero measure with respect to [η] (especially if it is countable). ii).
By hypothesis, φη is a continuous map and is therefore uniformly continuous since [0,1]2 is a compact set. In conclusion, we have proved that all the integral terms on the right-hand side of (37), except I1(ε), admit a limit as ε goes to zero.
3 Strong-Viscosity Solutions to Path-Dependent PDEs
Strict Solutions
In the present subsection we give the definition of a strict solution to Eq. 43) and we state a result of existence and uniqueness. To do this we note that it follows from Proposition B.1 in [10] that there exists a positive constant c, which depends only on T and the constants Kandmap appearing in the statement of the current Theorem4, so that. We conclude this subsection with an existence result for the path-dependent heat equation, namely for Eq.
Then there exists a unique strict solutionU for the path-dependent heat Eq.(43), which is given by. Now, this follows from the integration by parts formula of Malliavin calculus, see, e.g. formula (1.42) in [31] (considering that Itô integrals are Skorohod integrals), that, for anyi =1,.
Towards a Weaker Notion of Solution: A Significant Hedging Example
However, it can be shown that this latter U is not regular enough to be a classical solution to Eq. 43), although it is "roughly" a solution to the path-dependent semilinear Kolmogorov equation (43). To characterize the map U, we note that it allows the probabilistic representation formula for all(t, η)∈ [0,T] ×C([−T,0]),. Recalling Remark3, it follows from the presence of sup−t≤x≤0η(x) among the arguments of f that U is not continuous with respect to the topology of C([−T,0]), therefore it cannot be a classical solution to Eq.
However, we note that sup−t≤x≤0η(x) is Lipschitz on (C([−T), so it will follow from Theorem 7 that U is a strongly viscous solution of the equation. Nevertheless, in this particular case, even if U is not a classical solution, we will prove that it is related to the classical solution of a certain finite-dimensional PDE. For this purpose, we begin to calculate an explicit form for f, for which it is useful to remember the following standard result.
Proof The regularity properties of f are derived from its explicit form, derived in Lemma 2, by simple calculations. As for Itô's formula (56), the proof can be carried out using the same principle as the standard Itô's formula. It is well known that if Q were an open set, Itô's formula would hold.
In particular, the basic tools for proving Itô's formula are the following Taylor expansions for the function f:. To prove the above Taylor formulas, note that they lie in the open set Q, using the regularity of f. Then, we can extend them to the closure of Q, since f and its derivatives are continuous in Q. The formula can be proved in the usual way. 43), the function f is a solution to a certain Cauchy problem, as stated in the following proposition.
Strong-Viscosity Solutions
This implies, using standard estimates for backward stochastic differential equations (see e.g. Proposition B.1 in [10]) and the polynomial growth condition for (Fn)n, that. Then we see that all the requirements in Theorem C.1 in [10] follow from assumptions and estimates (46), so the claim follows. We now prove an existence result for strong viscosity solutions to the path-dependent heat equation, namely to Eq.
Then there exists a subsequence (Un,kn,Hn,kn,Fn,kn)n that converges pointwise to (U,H,F) as n tends to infinity. Note that a slightly different definition of a highly viscous solution was used in [8], see Note 12(i); but, continuing in the same direction, we can prove the present result. Then there exists a unique strong viscosity solution U of the path-dependent heat equation (43) given by .
Note that (η−Λη)(−T)=(η−Λη)(0), therefore η−Λη can be periodically extended to the entire real line with periodT, so that we can extend it in Fourier series. Now let (hn,k)k∈N for every n ∈ N be a locally equicontinuous sequence of C2(Rn+2;R)functions, bounded uniformly in polynomial terms, such that hn,k converge pointwise to hn, tending to infinity. Open Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any non-commercial use, distribution, and reproduction in any medium, provided the original author(s) and source are acknowledged.
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