In this paper, we then construct time-consistent dynamic risk measures on the setΩ of càdlàg paths to produce solutions to path-dependent semi-linear second-order PDEs. These dynamic risk measures are constructed using probability measures, solving a path-dependent martingale problem. In Sect.4 we construct a stable set of probability measures on the set Ω of càdlàg paths.
Then we prove that R(h) provides a viscosity supersolution and a viscosity subsolution for (2) on the set of continuous paths. In this section we introduce new notions for regular and viscosity solutions for second-order path-dependent PDEs on the set of càdlàg paths. In contrast to [20] and all papers that use the notions of continuity and derivative introduced by Dupire [12], we work with the Skorohod topology on the set of càdlàg paths.
The solution of the path-dependent PDE (1) is a progressive functionφ(t, ω), where belongs to IR+ and ω belongs to the set of paths càdlàg. Our solution construction is based on a martingale problem approach for path-dependent diffusions. 13, 14] introduced the notion of a viscosity solution of a path-dependent second-order PDE for a function v defined on a set of continuous paths.
A probability measure defined in (Ω, (Bt)) is a solution to the path-dependent martingale problem for La, starting from ω0 at timeref.
Multivalued Mapping and Continuous Selector
Stable Set of Probability Measures Associated to a Multivalued Mapping
For all r≥0 and allω∈Ω, there exists a unique solution to the martingale problem for La,aμ starting from ω at time r with μ(u, ω) = μ(u, ω, ω(u)). Moreover for all r < s, the map → Qas,ω,aμ is Bs measurable and is a regular probability distribution conditional on the given Qar,ω,aμBs. We first prove by induction that there exists a unique solutionQ to the martingale problem for La,aμ starting fromωin time and thatQ belongs to Q˜r,ω(Λ).
It follows from [22] that forQalmost allωinAk,j,Qsk,ω is a solution to the martingale problem forLa,aλk,j assuming ωat timesk. On the other hand, the constraint of QtotBski solves the martingale problem for La,aνwhereν ∈ ˜L(Λ) is associated with the subdivision (si)0≤i≤k−1andνcoincides with μonBsk. It follows from the induction hypothesis that the constraint of Q to Bsk is uniquely determined, coincides with Qar,ω,aν and belongs to Q˜r,ω(Λ).
On the other hand, it is easy to verify that the set {Qar,ω,aμ: μ∈ ˜L(Λ)} is stable.
5 Construction of Penalties
A progressively measurable process µ belongs to BMO(P) and has a BMO norm less than or equal to Cif for all stopping times τ,. We also recall from [15] that the stochastic exponent E(μ) of the BMO μ process is uniformly integrable and that the BMO norms with respect to P and P(E(μ).) are equivalent. We need to check that the penalties are well defined for all BMO processes and that they satisfy the local property and condition of the cocycle.
Assume that g satisfies (GC2), then the random variablesαs,t(Qar,ω,aμ) belong to L∞(Qar,ω) and are uniformly bounded for||μ||BMO(Qar,ω)≤C and r ≤s≤t ≤ T. It follows from the conditional Hölder inequality and the equivalence of the BMO norms with respect to Qar,ωandQar,ω,aμat. The first claim of 1 in the proposition then follows from Eq. of the proposition follows from Eq. 23) applied with Bsæk with the trivial sigma algebra.
6 Time Consistent Dynamic Risk Measures Associated to Path-dependent Martingale Problems
Normalized Time-Consistent Convex Dynamic Risk Measures
General Time-Consistent Convex Dynamic Risk Measures
The time consistency for stop times with a finite number of values follows from the stability property of the set of probability measures, as well as from the cocycle and local property of the penalties (see [5] in L∞case).
7 Strong Feller Property
Feller Property for Continuous Parameters
It follows from [3] that the set of probability measurements Q = {Qasnλ,ωn∗snxn, n ∈ IN} ∪ {Qas,ω∗λ sx} is weakly relatively compact and therefore tight. We now introduce a class of Bt-measurable functions that satisfy a continuity condition derived from the progressive continuity condition we introduced for progressive functions and from the continuity property proved in Theorem 1. Assume that the penalty is given by Eq. 20) for a given Caratheodory function g on IR+×Ω×IRns that satisfies the growth condition (GC1) or (GC2).
Feller Property for the Dynamic Risk Measure
For each function h∈Ct there exists a progressive mapping R(h) onto IR+×Ω, R(h)(t, ω)=h(ω), such that R(h) is lower semicontinuous on {(u , ω,x ), u≤t, ω=ω∗ux}and such that the following equation is satisfied. 41) (R(h) denotes the strict progressive mapping onto IR+×Ω×IR related to R(h) in the one-to-one correspondence presented in Section 2).
8 Existence of Viscosity Solutions for Path-dependent PDEs 8.1 Existence of Viscosity Supersolutions
Existence of Viscosity Subsolutions
Then for all(r, ω) and all processesμ Λvalued such thatμerP ×B(IRn) is measurable,μbelongs to BMO(Qar,ω)and|μ||BMO(Qar,ω) ≤C. It follows from [22] and from the uniqueness of the solution to the martingale problem for La,0 starting fromωat timeτ(ω)at. The function R(h) is lower semicontinuous in the viscosity sense, but it is not necessarily upper semicontinuous.
Therefore, we need to introduce the upper semicontinuous envelope R(h) in terms of viscosity according to Section 2.3. Assume that the function g is a Caratheodory function satisfying the growth condition (GC1) or (GC2) and that the Fenchel transform f of g is progressively continuous. The result then follows either from condition (GC1) or from (GC2) and from the unified BMO hypothesis with similar arguments as in the proof of Proposition 3.
Application of the time consistency of the risk measure (ρutn,ωv n) and of the following equations derived from Theorem4,. It follows from the definition of the risk measure(ρttnn,ωtnn+δn) that for all there is a processμninL˜(Λ) such. Given(tn, ωn), the probability measure Qtan,ωaμnni's solution to the martingale problemLa,aμnstarting fromωnat timetn.
Existence of Viscosity Solutions on the Set of Continuous Paths
9 Conclusion and Perspectives
The study of comparison theorems and continuity properties in this setting, as well as the study of solutions for path-dependent fully nonlinear PDEs, will be the subject of future work. Acknowledgments This work was partly carried out at the Center for Advanced Studies at the Norwegian Academy of Sciences and Letters (CAS, Oslo). It is a pleasure to thank Giulia Di Nunno and Fred Espen Benth for the invitation to contribute to the CAS program in Stochastics in Environmental and Financial Economics (SEFE) and to give a talk at the inaugural SEFE conference.
Appendix
Open Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. ii) α satisfies the cocycle condition: For allr≤s≤t, for allQinQ, αr,t(Q)=αr,s(Q)+EQ(αs,t(Q)|Fr) Recall the following result from [ 5]. Proposition 7 Given a stable set of probability measures of Qof and a penalty(αs,t) defined inQfulfilling the local property and the cocycle condition,.
