Extension of as-if-Markov modeling to scaled payments

Landmark estimators In multi-state life insurance, as-if-Markov modeling has recently been suggested as an alternative to Markov modeling in case of deterministic sojourn and transition payments. Incidental policyholder behavior, on the other hand, gives rise to duration-dependent payments in the form of so-called scaled payments. The goal of this paper is to establish as-if-Markov modeling also for scaled payments. To this end, we employ change of measure techniques to transfer the added complexity from the payments to an auxiliary probabilistic model. Based hereon, we show how to compute the accumulated cash ﬂow by solving a system of equations comparable to Kolmogorov’s forward equations for Markov chains, but with the transition rates replaced by certain forward transition rates related to the auxiliary probabilistic model. Finally, we provide feasible landmark estimators for these auxiliary forward transition rates subject to entirely random right-censoring. © 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
According to Gatzert (2009), implicit options and the rights to modify contractual conditions are an integral part of product design in life insurance, why proper market consistent valuation and risk management should take implicit options into account.Examples include the free policy (or paid-up) option, which when exercised ceases premiums payments at the expense of down-scaled benefits.In this paper, we extend the as-if-Markov approach of Christiansen (2021) for non-Markov models to allow for so-called scaled payments, which arise naturally in the context of policyholder behavior related to free policy conversion and stochastic retirement.
In the last decade, there has been a significant interest in valuation of life insurance payments in the presence of free policy behavior and stochastic retirement, see Henriksen et al. (2014); Buchardt and Møller (2015); Buchardt et al. (2015); Gad and Nielsen (2016).A common feature of these investigations is that the jump process governing the state of the insured is assumed to be Markovian or semi-Markovian.Actually, Markov chain modeling and semi-Markov modeling constitute the predominant approaches in multi-state life insurance, especially due to the lack of feasible alternatives in regards to both estimation and computation, cf.Chapter 1 in Furrer (2020); recent applications of Markov chain and semi-Markov models, besides some of the aforementioned references, include Guibert and Planchet (2018); Furrer (2019); de Mol van Otterloo and Alonso-García (2022).But as also explained in Section 1 of Christiansen (2021), forcing Markov assumptions in a situation where the Markov assumption is violated in general gives rise to additional systematic model risk.As an alternative, so-called as-if-Markov modeling has recently been proposed in Christiansen (2021).Under as-if-Markov modeling, life insurance liabilities are evaluated conditional only on the present state of the insured -but without actually imposing the Markov assumption.In Christiansen (2021), it is shown that valuation may then essentially proceed using classic formulas for Markov chain models, but with the transition rates replaced by certain so-called forward transition rates.Furthermore, these forward transition rates may be consistently estimated via landmarking, namely by the so-called landmark Nelson-Aalen estimators introduced in Putter and Spitoni (2018).
For multi-state models, the concept of forward transition rates originally emerged from the desire to calculate market values for life insurance liabilities in doubly stochastic Markov models, cf.Norberg (2010); Buchardt et al. (2019), where various proposals are studied and compared.Predating this work are the so-called forward mortalities for survival models introduced by Miltersen and Persson.In Miltersen and Persson, forward mortalities are defined implicitly such that classic formulas that apply for deterministic mortalities produce correct market values even for stochastic mortalities -if the stochastic mortality is replaced by its forward mortality.The shift of perspective from doubly stochastic Markov models to non-Markov models is due to Christiansen (2021).
Landmark estimation is essentially a stratification method, so compared to Markov modeling, as-if-Markov modeling trades systematic model risk for unsystematic estimation risk.Depending on the trade-off in practice, this may make as-if-Markov modeling the preferred choice.
A key requirement in Christiansen (2021) is that the sojourn and transition payments have to be deterministic, which is not the case for scaled payments, since the scaling factor typically depends on for instance the duration since exercise of the underlying policyholder option.In Furrer (2022), it has recently been shown how change of measure techniques may be used to transfer this added complexity in the payments to the probabilistic model.The main contribution of the present paper is a merger of the methods and results of Christiansen (2021) and Furrer (2022) to establish as-if-Markov modeling also for scaled payments.As a secondary contribution, we also unfold and clarify the concise consistency proofs of Christiansen (2021).
The paper is structured as follows.In Section 2, we present the mathematical setup.Section 3 contains a review of the contributions of Christiansen (2021); Furrer (2022) within the setup of the present paper, while Section 4 contains the main results.In Subsection 4.1, we show how the accumulated cash flow and prospective reserve of interest may be found by solving a system of equations comparable to Kolmogorov's forward equations for Markov chains, but involving certain auxiliary forward transition rates.In Subsection 4.2, we introduce and establish consistency of landmark estimators for the auxiliary forward transition rates and the corresponding accumulated cash flow.Finally, Section 5 discusses implementation and connotations for practice, while Section 6 concludes.

Setup
In this section, we describe the mathematical setup by specifying the core components of a multi-state model in life insurance, namely the probabilistic model for the state of the insured, the payments between the insured and insurer, and the (usage of) available information.Furthermore, we introduce the associated expected accumulated cash flows; they constitute our main objects of interest in regards to valuation.
The state of the insured is modeled by a non-explosive jump process Z = (Z t ) t 0 on a finite state space Z.Let ( , F , P ) be the underlying probability space.For simplicity, we suppose that Z 0 ≡ z 0 ∈ Z.The information generated by Z is denoted F Z = (F Z t ) t 0 .We associate with the jump process Z a multivariate counting process N with components N jk = (N jk (t)) t 0 given by (1) This common assumption in particular implies the previously stated non-explosion of Z .
We consider basic payments B = (B(t)) t 0 , describing benefits less premiums of some insurance contract.The basic payments are required to satisfy B(0 for right-continuous real functions of finite variation B j , j ∈ Z, and measurable real functions of finite variation b jk , j, k ∈ Z, j = k. Following Definition 5.1 in Christiansen (2021), the basic payments B thus admit a deterministic cash flow representation.
The inclusion of policyholder behavior such as free policy behavior and stochastic retirement leads to scaled payments which depend on the duration since free policy conversion or retirement and thus do not admit a deterministic cash flow representation, see for instance Furrer (2022).The purpose of this paper is to adapt the main program of Christiansen (2021) for basic payments to scaled payments.To this end, we suppose that Z = Z 0 ∪ Z 1 , that z 0 ∈ Z 0 , and that (2) so that the states Z 1 are absorbing for Z .Furthermore, we introduce the stopping time τ as the first hitting time for Z of Z 1 , that is so that τ may be interpreted as the exercise time of some policyholder option, and so that Z 0 and Z 1 would constitute the states preand post-exercise, respectively.The situation without policyholder behavior is recovered by setting Z 1 = ∅ in which case τ = ∞.The extension to multiple options is discussed in Section 6. Solely for simplicity, we assume that for all j ∈ Z 0 ; in other words, lump sum payments are assumed not to occur simultaneously with the exercise of the policyholder option.This assumption may be relaxed at the cost of an increase in notational burden.The scaled payments of interest B ρ = (B ρ (t)) t 0 are now given by B ρ (0) = B(0) and for some strictly positive F Z -predictable processes ρ(•, j, k), j ∈ Z 0 , k ∈ Z 1 , assumed bounded by one.Typically, the scaling factor ρ is determined as to maintain actuarial equivalence on some technical basis, see e.g.Buchardt and Møller (2015); Buchardt et al. (2015); Christiansen and Djehiche (2020); in practice, the scaling factor ρ is prevalently deterministic.The extension to scaling factors that are not bounded by one is discussed in Section 6.
As previously mentioned, we aim to adapt the as-if-Markov approach of Christiansen (2021) to the present setting.Consequently, we do not impose any conditions on the intertemporal dependence structure of Z .Instead, we restrict the information used for valuation, which we denote by G = (G t ) t 0 , to be of the form There are multiple reasons for considering reduced, typically non-monotone, information, including the desire to reduce the numerical complexity of computations, an absence of or an insufficient quality of data, and regulatory requirements, cf. the discussions in Section 1 and Section 3 of Christiansen and Furrer (2021) and Section 3 of Christiansen (2021).
If Z is actually Markovian and the payments admit a deterministic cash flow representation, then the restriction to information G in place of F Z is inconsequential.But in many applications, there is a lack of evidence supporting such an assumption on the intertemporal dependence structure of Z .In other words, and following the discussion in Section 9 of Christiansen (2021), the usage of Markov chain modeling is subject to added systematic risk, model risk, which does not vanish for large sample sizes.By restricting the information used for valuation, rather than imposing assumptions on the intertemporal dependence structure of Z , we avoid this model risk at the cost of additional unsystematic risk, estimation risk.
An alternative to the 'Markovian information' described by G could for instance be H = (H t ) t 0 given by which additionally keeps track of the scaling factor.This alternative is discussed in Section 6.Note that G and H agree pre-exercise, that is they agree up until the stopping time τ .
In the context of valuation of future liabilities in life insurance, so-called expected accumulated cash flows are essential.In accordance with Definition 2.2 of Buchardt et al. (2019), given information G, the expected accumulated cash flows A ρ associated with the payments B ρ are defined by In case of a suitably regular deterministic savings account κ and a finite maximal contract time 0 < T < ∞, the prospective reserve V ρ = (V ρ (t)) 0 t<T may then be calculated according to
(4) Especially in the case of 'Markovian information', it is natural to work with state-wise quantities.The state-wise expected accumulated cash flows A ρ i , i ∈ Z, are given by where the conditional expectation is to be read as the factorized conditional expectation Christiansen and Furrer (2021).It holds that (5) In the following, we focus on valuation at some fixed point in time t 0 0. To be specific, we develop probabilistic and statistical tools to estimate and compute the state-wise expected accumulated cash flows valued at this single point in time, namely

Background
In this section, we review and provide alternative perspectives on the recent contributions of Christiansen (2021); Furrer (2022) and related literature in the context of the setup of this paper.Merging the methods and results of Christiansen (2021); Furrer (2022) constitutes the starting point of our primary investigation, which is postponed to Section 4.

As-if-Markov approach
In this subsection we disregard policyholder behavior by letting Z 1 = ∅ leading to τ = ∞.In this case B ρ = B, which entails that the payments B ρ in particular admit a deterministic cash flow representation and which encourages us to suppress ρ in the notation.Let denote the transition probabilities, and introduce also the quantities Following Christiansen (2021), we define so-called (cumulative) forward transition rates i, jk We assume that In the spirit of Example 6.7 and Example 6.8 in Christiansen (2021), one may show that for fixed i ∈ Z, and also that Thus to compute the state-wise expected accumulated cash flows it essentially suffices to solve (6), which is a system of equations comparable to Kolmogorov's forward equations for Markov models.If Z is a Markov process, then the cumulative rates i, jk and if the transition rates are also continuous, then (6) reduces to the usual Kolmogorov's forward differential equations given by d ds If Z is a Markov process that does not admit transition rates on [t 0 , ∞), say transitions occur only at some given finite time points Both the former system of differential equations and this recursion are easy to implement and widely used by practitioners.The system of equations ( 6) is of a similar form and therefore equally easy to implement.
In Markov models, one may estimate the (cumulative) transition rates and the transition probabilities from empirical data by using the Nelson-Aalen and corresponding Aalen-Johansen estimator; these estimators in particular allow for left-truncation and right-censoring.For non-Markov model, it is proposed in Christiansen (2021) and elsewhere to instead use so-called landmark Nelson-Aalen and corresponding landmark Aalen-Johansen estimators, that retain many of the desirable properties of their regular counterparts, to estimate the (cumulative) forward transition rates and transition probabilities.Following Christiansen (2021), suppose that we observe a portfolio of n ∈ N insured: where R m , m = 1, . . ., n, are non-negative random variables that describe right-censoring, and where Z m , m = 1, . . ., n are replicated of Z .We suppose that the pairs (Z 1 , R 1 ), . . ., (Z n , R n ) are mutually independent and identically distributed and that R 1 is independent of Z 1 , that is to say right-censoring is entirely random.The latter assumption, as well as the inclusion of left-truncation, is discussed in more detail in Section 6. Denote by N m , m = 1, . . ., n, the associated multivariate counting processes.For fixed i ∈ Z let for j, k ∈ Z, j = k, be the counting processes of the sub-portfolio of insured sojourning in state i at time t 0 , and let for j ∈ Z be the corresponding at-risk processes.The landmark Nelson-Aalen estimator for i, jk (t 0 , •) is then defined via for j, k ∈ Z, j = k, and the corresponding landmark Aalen-Johansen estimator for the transition probabilities is defined as the solution to (6) with the (cumulative) forward transition rates replaced by their landmark Nelson-Aalen estimator, that is and initial values q p i, j (t 0 , t 0 ) = 1 { j=i} for j ∈ Z, see also Putter and Spitoni (2018); here and in the following we use the convention 0/0 := 0. In Christiansen (2021), it is shown that these estimators are consistent in the sense of uniform L 1 -convergence (under certain technical regularity conditions), cf.Theorem 7.3 and Theorem 7.4 of Christiansen (2021).Note that this in particular implies weak consistency of the estimators.Similarly, a consistent estimator of the state-wise expected accumulated cash flow A i (t 0 , •) can be obtained by replacing the (cumulative) forward transition rates and transition probabilities in ( 7) by their landmark Nelson-Aalen and Aalen-Johansen estimator.The proofs in Christiansen ( 2021) are based on Overgaard (2019b), which in particular serves to address some shortcomings in the technical arguments of Datta and Satten (2001); Putter and Spitoni (2018), cf.Section 1 in Nießl et al. (2021).In this paper, we -among other things -further unfold and clarify the proofs of Christiansen (2021).
Landmarking is essentially a stratification method, so one disadvantage of landmarking is data reduction and thus loss of power.If the Markov assumption is actually satisfied, it is obviously preferable to use the regular Nelson-Aalen and Aalen-Johansen estimators, which take into account the whole portfolio and not only the sub-portfolio of insured sojourning in state i at time t 0 .In case the model is only partially non-Markovian, it has recently been suggested to employ a hybrid concept striking a compromise between regular estimation and landmark estimation, see Maltzhan et al. (2021).

Change of measure techniques
Since the scaled payments B ρ include the term H , they do not admit a deterministic cash flow representation.In Furrer (2022), change of measure techniques are in a canonical framework used to transfer the term H from the payments to the probabilistic model.In the following, we adapt the result of Furrer (2022) to the present setup and cast the relevant change of measure explicitly in terms of a transformation of the jump process.
Let U be a real-valued random variable, and suppose that U is independent of Z and uniformly distributed on the open unit interval (0, 1).Define Z = ( Z t ) t 0 according to where ∇ is an artificial cemetery state.Note that Z is a non-explosive jump process on the finite state space Z := Z ∪ {∇} with Z 0 ≡ z 0 and with both {∇} and Z 1 absorbing.The information generated by Z is denoted F Z = (F Z t ) t 0 .We associate with the jump process Z a multivariate counting process N with components N jk = ( N jk (t)) t 0 given by N jk (t) = #{s ∈ (0, t] : Z s− = j, Z s = k}, t 0, for j, k ∈ Z, j = k.Since (1) holds, we also obtain Finally, consider auxiliary payments B = ( B(t)) t 0 given by B(0) ≡ b 0 and which correspond to the basic payments B but with Z and N replaced by Z and N, respectively.In particular, the payments B admit a deterministic cash flow representation with respect to Z (rather than Z ).
In Furrer (2022), it is presumed that the insurer utilizes full information.The quantity of interest is thus the expected accumulated cash flow associated with the payments B ρ given the information generated by Z , corresponding to Since U is assumed to be independent of Z , it does not matter whether one does or does not condition on information generated by U .By closely inspecting the arguments leading to Theorem 3.6 in Furrer (2022), it is possible to show that This identity is sensible due to the local property of the conditional expectation, cf.Lemma 8.3 in Kallenberg (2021).Furthermore, since and since it follows directly from local property of the conditional expectation, cf.Lemma 8.3 in Kallenberg (2021), that Referring to the definition of H and the fact that U is independent of Z , one may then show that In combination, the identities ( 10) and ( 11) provide a partial link between the expected accumulated cash flow associated with the payments B given the information generated by Z and the expected accumulated cash flow associated with the payments B ρ given the information generated by Z .In the next section, we utilize this partial link and the fact that the payments B admit a deterministic cash flow representation with respect to Z to extend the as-if-Markov approach outlined in Subsection 3.1 to scaled payments, that is to the case Z 1 = ∅.

Main results
This section containts the main contributions of the paper.In Subsection 4.1, we derive a system of equations -comparable to Kolmogorov's forward equations for Markov models -which allows for efficient computation of the state-wise expected accumulated cash flows.The system relies on certain (cumulative) auxiliary forward transition rates.Subsection 4.2 deals with the estimation of the quantities involved, in particular the (cumulative) auxiliary forward transition rates.

Valuation
We introduce auxiliary state-wise expected accumulated cash flows A i (t 0 , •), i ∈ Z, according to The following result, which builds on the review given in Subsection 3.2, provides a direct link between the state-wise expected accumulated cash flows of interest and the auxiliary state-wise expected accumulated cash flows.
By the law of iterated expectations and (10), we find that In particular, as desired.Let now i ∈ Z 1 , and suppose that P (Z By similar arguments as in the first part of the proof, but referring to (11) instead of ( 10), it follows that .
Recall that U is independent of Z and uniformly distributed on the open unit interval, so that Repeated usage of the law of iterated expectations then yields In conclusion, which serves to complete the proof.
Theorem 1 shows that one may compute the state-wise expected accumulated cash flows of interest by instead computing certain auxiliary state-wise expected accumulated cash flows.The importance of this result lies in the fact that the auxiliary payments admit a deterministic cash flow representation, so that the methods and results of Christiansen (2021) reviewed in Subsection 3.1 may be applied more or less directly.To this end, for i ∈ Z let Based hereon, define auxiliary (cumulative) forward transition rates i, jk As before, one may for fixed i ∈ Z show that and also that Note that ( 14) and ( 15) hold even if the assumption prescribed by ( 13) is dropped, as long as the involved functions are redefined as measures on the real line.From ( 12) and ( 15) it follows that to compute the state-wise expected accumulated cash flows of interest, it essentially suffices to solve ( 14), which is a system of equations comparable to Kolmogorov's forward equations for Markov models.This requires the estimation of the auxiliary (cumulative) forward transition rates and the associated auxiliary probabilities.In the next subsection, we show how this may be done using landmark Nelson-Aalen and corresponding landmark Aalen-Johansen estimators.

Estimation
In this subsection, we show how to estimate the auxiliary (cumulative) forward transition rates i, jk (t 0 , •), i ∈ Z, j, k ∈ Z, j = k, and associated auxiliary probabilities and state-wise expected accumulated cash flows from empirical data.Similar to Christiansen (2021), the results presented here serve only as a proof of concept and far from cover the full range of possible model settings.
Similar to Subsection 3.1, suppose that we observe a portfolio of n ∈ N insured: where R m , m = 1, . . ., n, are non-negative random variables that describe right-censoring.Let U 1 , . . ., U n be mutually independent and identically distributed real-valued random variables.Further, suppose that (U m ) n m=1 is independent of (Z m , R m ) n m=1 , and suppose that U 1 is uniformly distributed on the open unit interval (0, 1).Define the auxiliary data where m = 1, . . ., n.In the following, we denote by N m the multivariate counting process associated with Z m .For fixed i ∈ Z let for j, k ∈ Z, j = k, be the auxiliary counting processes of the sub-portfolio of insured sojourning in state i at time t 0 , and let for j ∈ Z be the corresponding at-risk processes.In the following, the landmark i ∈ Z is held fixed.
Definition 1.The landmark Nelson-Aalen estimator p i, jk (t 0 , •), j, k ∈ Z, j = k, for i, jk (t 0 , •), j, k ∈ Z, j = k, is defined via Definition 2. The landmark Aalen-Johansen estimator p p i, j (t 0 , •), j ∈ Z, for p i, j (t 0 , •), j ∈ Z, is defined as the solution to ( 14) with the auxiliary (cumulative) forward transition rates replaced by their landmark Nelson-Aalen estimator, that is and initial values The estimator p p i, j (t 0 , t 0 ) for p i, j (t 0 , t 0 ) is consistent in the sense of L 1 -convergence (as the size of the portfolio increases) and bounded, which are crucial properties for the proof of Theorem 3 below.Alternatively, we may replace p p i, j (t 0 , t 0 ) by any other consistent and bounded estimator of p i, j (t 0 , t 0 ).
In the following, we fix i ∈ Z with P (Z t 0 = i) > 0. For a finite maximal contract time T > t 0 and p ∈ [1, ∞), we define the p-variation norm • [p] := • ∞ + • v p , where • ∞ is the supremum norm on [t 0 , T ] and • v p is the total p-variation on [t 0 , T ]; a compilation of fundamental properties for the p-variation norm may be found in Appendix A. The next theorem establishes consistency (as the size of the portfolio increases) of the landmark Nelson-Aalen estimators.The theorem is in the same vein as Theorem 7.3 in Christiansen ( 2021), but it also deals with the transformation from Z to Z .Recall that U 1 , . . ., U n are mutually independent and identically distributed real-valued random variables and that (U m ) n m=1 is independent of (Z m , R m ) n m=1 .
Theorem 2. Suppose that For an individual insured with state process Z , auxiliary state process Z , and right-censoring time R, we write for the censored landmark auxiliary counting process and for the censored landmark auxiliary state indicator process.Their expectations are denoted by Based hereon we define a (cumulative) forward transition rate i, jk (t 0 , •) via i, jk (t 0 , t 0 ) = 0 and i, jk (t 0 , dt) = q i, j (t 0 , t−) q i, jk (t 0 , dt), t 0 < t < ∞.
Note that i, jk (t 0 , t 0 ) = i, jk (t 0 , t 0 ) by definition.Let now t > t 0 .Under the assumption that U is independent of (Z , R) and that R and Z are independent, it follows from the law of iterated expectations that where the last line follows from Campbell's theorem.In similar fashion, In combination, we find that where the finiteness follows from ( 13).So it suffices to show that This is in particular true if for any M ∈ N and where p N i (t 0 , •) is given by We first prove (19).According to Markov's inequality, it holds that For m = 1, . . ., n we let as well as i, jk (t 0 , t) = 1 implies p I m i, j (t 0 , t−) = 1, by applying Jensen's inequality and by using the mutual independence of the random variables p Because of this inequality, the law of iterated expectations, Campbell's theorem, and (17), we obtain q i, j (t 0 , u) q i, jk (t 0 , du) Jensen's inequality and ( 21) imply that By arguing analogously to (22) but using the latter inequality and the Cauchy-Schwarz inequality for conditional expectations, we can show that 1 The random variable which appears in the last expectation, is bounded by the random variable i, jk (t 0 , du), which has expectation i, jk (t 0 , T ) < ∞, cf. ( 22).Markov's inequality for conditional expectations, the mutual independence of p N 1 i (t 0 , T ), . . ., p N n i (t 0 , T ), and the conditional dominated convergence theorem yield From ( 23) and ( 24) as well as the dominated convergence theorem, it thus holds that lim Finally, by applying the triangle inequality, (56), (25), and (20), we may conclude that lim for suitable real numbers c 1 , c 2 < ∞.The triangle inequality and (56) moreover yield So, in conjunction with Theorem 2, the dominated convergence theorem, and (56), we get Markov's inequality and ( 22) then yield which converges to zero for M → ∞, see also (13).Combining this fact with (39) establishes the desired result.
Let p A i (t 0 , •) be given by ( 15), but with the auxiliary probabilities p i, j (t 0 , •), j ∈ Z, and the auxiliary (cumulative) forward transition rates i, jk (t 0 , •), j, k ∈ Z, j = k, replaced by their landmark estimators p p i, j (t 0 , •), j ∈ Z, and p i, jk (t 0 , •), j, k ∈ Z, respectively.The next theorem shows that the consistency results for the landmark Nelson-Aalen estimator and the landmark Aalen-Johansen estimator carry over to the estimator p A i (t 0 , •) for the auxiliary state-wise expected accumulated cash flow A i (t 0 , •).
The consistency result for the state-wise expected accumulated cash flows p A i (t 0 , •) and A i (t 0 , •) carries over to the corresponding prospective reserves.Analogously to formula (4), let p V (t 0 ) and V (t 0 ) be the prospective reserves at time t 0 of p A i (t 0 , •) and A i (t 0 , •), respectively.The following corollary is then a consequence of Theorem 4 and (51).

Appendix A. p-variation
This appendix recollects the definition and fundamental properties of the p-variation norm for p ∈ [1, 2).We generally recommend to consult Dudley and Norvaiša (2011).For 0 t 0 < T < ∞ and d ∈ N we consider a matrix-valued function h : [t 0 , T ] → R d×d .Let • be any norm on R d×d .The supremum defines a semi-norm, so that the sum is a norm on the class of bounded multivariate functions with finite p-variation.Recall that we generally assume that p ∈ [1, 2).We have (50) see Lemma 3.45 in Dudley and Norvaiša (2011).Since any function of finite variation can be represented as the difference of two nondecreasing functions, we have Let g : [t 0 , T ] → R d×d be another matrix-valued function.There exists a finite constant k such that A B k A B for all A, B ∈ R d×d .