Generalised Lyapunov Functions and Functionally Generated Trading Strategies
GGeneralised Lyapunov Functions and Functionally GeneratedTrading Strategies ∗ J OHANNES R UF K ANGJIANAN X IE Department of Mathematics, The London School of Economics and PoliticalScience, Houghton Street, London WC2A 2AE, United KingdomE-mail: [email protected] Department of Mathematics, University College LondonGower Street, London WC1E 6BT, United KingdomE-mail: [email protected]
January 25, 2018
Abstract.
This paper investigates the dependence of functional portfolio gener-ation, introduced by Fernholz (1999), on an extra finite variation process. Theframework of Karatzas and Ruf (2017) is used to formulate conditions on trad-ing strategies to be strong arbitrage relative to the market over sufficiently largetime horizons. A mollification argument and Koml´os theorem yield a generalclass of potential arbitrage strategies. These theoretical results are comple-mented by several empirical examples using data from the S&P 500 stocks.
Keywords : Additive generation; Lyapunov function; market diversity; multi-plicative generation; portfolio analysis; portfolio generating function; regularfunction; S&P 500; Stochastic Portfolio Theory
1. Introduction
E.R. Fernholz established Stochastic Portfolio Theory (SPT) to provide a theoreticaltool for applications in equity markets, and for analyzing portfolios with controlled be-havior; see Fernholz (1999) and Fernholz and Karatzas (2009), for example. SPT studiesso called functionally generated portfolios. The value of a functionally generated portfo-lio relative to the total market capitalization is merely a function, known as the so called master formula , of the market weights. This formula does not involve stochastic integra-tion or drifts, which makes the analysis very easy as the need for estimation is reduced.One very interesting topic following up this construction is the study of relative arbi-trage opportunities between functionally generated portfolios and the market portfolio.Fernholz (1999, 2001, 2002) states conditions for such relative arbitrage to exist over suf-ficiently large time horizons. To implement this relative arbitrage, trading strategies gen-erated by suitable portfolio generating functions are required. Karatzas and Ruf (2017)interpret portfolio generating functions as Lyapunov functions. More precisely, the super-martingale property of the corresponding wealth processes after an appropriate change ofmeasures is utilized to study the performance of functionally generated trading strate-gies. Relative arbitrage over arbitrary time horizons under appropriate conditions is alsostudied by Fernholz et al. (2017). ∗ We are grateful to Andrea Macrina and Daniel Schwarz for their detailed reading and helpful comments. a r X i v : . [ q -f i n . M F ] J a n ne offspring of portfolio generating functions is a generalized portfolio generatingfunction, which depends on an additional argument with continuous path and finite vari-ation. This is inspired by the fact that in practice, people tend to take historical data, suchas past performance of stocks, or statistical estimates, into consideration when construct-ing portfolios. Besides, this generalization provides additional flexibility in choosingportfolio generating functions. Section 3.2 of Fernholz (2002) formulates the concept oftime-dependent generating functions, and presents the master formula under this situa-tion. In the same framework, Strong (2014) shows an extension of the master formula toportfolios generated by functions that also depend on the current state of some continuouspath process of finite variation. Also based on Fernholz’s structure, Schied et al. (2016)provide a pathwise version of the relevant master formula. They also analyze exampleswhere the additional process is chosen to be the moving average of the market weights.All the above mentioned papers (Fernholz (2002), Strong (2014), and Schied et al.(2016)) make assumptions on the smoothness of the portfolio generating function withrespect to both the finite variation process and the market weights. In this paper, weweaken these assumptions such that the choice for the portfolio generating function isless restricted. To this end, we use a mollification argument and the Koml´os theorem.Then we study several examples empirically, using data from the S&P 500 index. An outline of the paper is as follows. Section 2 specifies the market model and recallsthe definitions of trading strategies and relative arbitrage. Section 3 first gives the defini-tions of regular functions and Lyapunov functions, and then presents sufficient conditionsfor a function to be regular and Lyapunov, respectively. The appendix presents the proofsof these results. Section 4 defines additive and multiplicative generation, and the cor-responding trading strategies and wealth processes. Section 4 also gives conditions forarbitrage relative to the market portfolio to exist. Section 5 describes the data involvedand the processing method to implement the empirical analysis. Section 6 contains sev-eral examples of portfolio generating functions and discusses empirical results. Section 7concludes.
2. Model setup
We fix a filtered probability space (cid:0) Ω , F ( ∞ ) , F ( · ) , P (cid:1) with F (0) = {∅ , Ω } and write ∆ d = (cid:40) ( x , · · · , x d ) (cid:48) ∈ [0 , d : d (cid:88) i =1 x i = 1 (cid:41) and ∆ d + = ∆ d ∩ (0 , d . We consider an equity market with d ≥ companies. We denote the market weightsprocess by µ ( · ) = (cid:0) µ ( · ) , · · · , µ d ( · ) (cid:1) (cid:48) . Here, µ i ( · ) is the market weight process of com-pany i computed by dividing the capitalization of company i by the total capitalizationof all d companies in the market, for all i ∈ { , · · · , d } . We assume that µ ( · ) is ∆ d -valued with µ (0) ∈ ∆ d + , and that µ i ( · ) is a continuous, non-negative semimartingale, forall i ∈ { , · · · , d } . As the constituent list of the stocks in the S&P 500 index changes over time, we avoid a survivorship bias by not restricting theanalysis to the current stocks in the S&P 500 index. Instead, we reconstruct the historical constituent list of the S&P 500 index andadjust the portfolios appropriately when the constituent list changes. o define a trading strategy for µ ( · ) , let us consider a process ϑ ( · ) = (cid:0) ϑ ( · ) , · · · , ϑ d ( · ) (cid:1) (cid:48) in R d , which is predictable and integrable with respect to µ ( · ) . We denote the collectionof all such processes by L ( µ ) .For such a process ϑ ( · ) ∈ L ( µ ) , we interpret ϑ i ( t ) as the number of shares in the stockof company i held at time t ≥ , for all i ∈ { , · · · , d } . Then V ϑ ( · ) = d (cid:88) i =1 ϑ i ( · ) µ i ( · ) can be interpreted as the wealth process corresponding to ϑ ( · ) . Definition 1. (Trading strategies). A process ϕ ( · ) ∈ L ( µ ) is called a trading strategy if V ϕ ( · ) − V ϕ (0) = (cid:90) · d (cid:88) i =1 ϕ i ( t )d µ i ( t ) . Remark . To convert a predictable process ϑ ( · ) ∈ L ( µ ) into a trading strategy ϕ ( · ) ,we adapt the measure of the “defect of self-financibility” of ϑ ( · ) , introduced in Proposi-tion 2.4 in Karatzas and Ruf (2017) and defined as Q ϑ ( · ) = V ϑ ( · ) − V ϑ (0) − (cid:90) · d (cid:88) i =1 ϑ i ( t )d µ i ( t ) . (1)As a result, the process ϕ ( · ) with components ϕ i ( · ) = ϑ i ( · ) − Q ϑ ( · ) + C, i ∈ { , · · · , d } , (2)where C can be any real constant, is a trading strategy for µ ( · ) .We are interested in the performances of different portfolios. Especially, we focus onstudying the conditions for the existence of so called relative arbitrage. Definition 2. (Arbitrage relative to the market). A trading strategy ϕ ( · ) is said to be relative arbitrage with respect to the market over a given time horizon [0 , T ] , for T ≥ ,if V ϕ ( · ) ≥ and V ϕ (0) = 1 , along with P (cid:2) V ϕ ( T ) ≥ (cid:3) = 1 and P (cid:2) V ϕ ( T ) > (cid:3) > . (3)If P (cid:2) V ϕ ( T ) > (cid:3) = 1 holds, we say that the relative arbitrage is strong over [0 , T ] . Remark . Definition 2 makes sense due to the fact that the wealth process of the marketportfolio at any time is given by V (1 , ··· , ( · ) = d (cid:88) i =1 µ i ( · ) = 1 . hen relative arbitrage exists over a given time horizon [0 , T ] when a non-negative wealthprocess V ϕ ( · ) has the same initial wealth as the market portfolio, the probability for V ϕ ( T ) to be greater than the market portfolio is strictly positive, and V ϕ ( T ) is not lowerthan the market portfolio.In the following sections, we study portfolio generating functions that depend on some R m -valued continuous process of finite variation on [0 , T ] , for T ≥ and some m ∈ N .We use Λ( · ) to denote such a process. This process allows for more flexibility in selectingportfolio generating functions. To this end, let W be some open subset of R m × R d suchthat P (cid:2)(cid:0) Λ( t ) , µ ( t ) (cid:1) ∈ W , ∀ t ≥ (cid:3) = 1 . (4)The following notations are introduced for the ranked market weights, which are stud-ied in Theorem 2 and Example 3. For a vector x = ( x , · · · , x d ) (cid:48) ∈ ∆ d , denote itscorresponding ranked vector as x = (cid:0) x (1) , · · · , x ( d ) (cid:1) (cid:48) , where max i ∈{ , ··· ,d } x i = x (1) ≥ x (2) ≥ · · · ≥ x ( d − ≥ x ( d ) = min i ∈{ , ··· ,d } x i are the components of x in descending order. Denote W d = (cid:110)(cid:0) x (1) , · · · , x ( d ) (cid:1) (cid:48) ∈ ∆ d : 1 ≥ x (1) ≥ x (2) ≥ · · · ≥ x ( d − ≥ x ( d ) ≥ (cid:111) ; then the rank operator R : ∆ d → W d is a mapping such that R ( x ) = x . Moreover,denote W d + = W d ∩ (0 , d .The ranked market weights process µ ( · ) is given by µ ( · ) = R (cid:0) µ ( · ) (cid:1) = (cid:0) µ (1) ( · ) , · · · , µ ( d ) ( · ) (cid:1) (cid:48) , (5)which is itself a continuous, ∆ d -valued semimartingale whenever µ ( · ) is a continuous, ∆ d -valued semimartingale (see Theorem 2.2 in Banner and Ghomrasni (2008)). At last,let W be some open subset of R m × R d such that P (cid:2)(cid:0) Λ( t ) , µ ( t ) (cid:1) ∈ W , ∀ t ≥ (cid:3) = 1 . (6)
3. Generalized regular and Lyapunov functions
In this section, we consider two classes of portfolio generating functions, regular andLyapunov functions, which are introduced in Karatzas and Ruf (2017). We generalizethese notions here to allow for the additional process Λ( · ) . Definition 3. (Regular function). A continuous function G : W → R is said to be regular for Λ( · ) and µ ( · ) if1. there exists a measurable function DG = ( D G, · · · , D d G ) (cid:48) : W → R d such thatthe process ϑ ( · ) = (cid:0) ϑ ( · ) , · · · , ϑ d ( · ) (cid:1) (cid:48) with components ϑ i ( · ) = D i G (cid:0) Λ( · ) , µ ( · ) (cid:1) , i ∈ { , · · · , d } , (7)is in L ( µ ) ; and . the continuous, adapted process Γ G ( · ) = G (cid:0) Λ(0) , µ (0) (cid:1) − G (cid:0) Λ( · ) , µ ( · ) (cid:1) + (cid:90) · d (cid:88) i =1 ϑ i ( t )d µ i ( t ) (8)is of finite variation on the interval [0 , T ] , for all T ≥ . Definition 4. (Lyapunov function). A regular function G : W → R is said to be a Lyapunov function for Λ( · ) and µ ( · ) if, for some function DG as in Definition 3, the finitevariation process Γ G ( · ) of (8) is non-decreasing.In the next example, we discuss sufficient conditions for a smooth function to beregular or Lyapunov. Example 1.
Consider a C , function G : W → R . Then Itˆo’s lemma yields G (cid:0) Λ( · ) , µ ( · ) (cid:1) = G (cid:0) Λ(0) , µ (0) (cid:1) + (cid:90) · m (cid:88) v =1 ∂G∂λ v (cid:0) Λ( t ) , µ ( t ) (cid:1) dΛ v ( t )+ (cid:90) · d (cid:88) i =1 ∂G∂x i (cid:0) Λ( t ) , µ ( t ) (cid:1) d µ i ( t )+ 12 d (cid:88) i,j =1 (cid:90) · ∂ G∂x i ∂x j (cid:0) Λ( t ) , µ ( t ) (cid:1) d (cid:10) µ i , µ j (cid:11) ( t ) . Set now ϑ i ( · ) = ∂G (Λ( · ) ,µ ( · )) ∂x i and Γ G ( · ) = − (cid:90) · m (cid:88) v =1 ∂G∂λ v (cid:0) Λ( t ) , µ ( t ) (cid:1) dΛ v ( t ) − d (cid:88) i,j =1 (cid:90) · ∂ G∂x i ∂x j (cid:0) Λ( t ) , µ ( t ) (cid:1) d (cid:10) µ i , µ j (cid:11) ( t ) . (9)Obviously, the process Γ G ( · ) has finite variation on [0 , T ] , for all T ≥ . Hence G is aregular function.Moreover, if the process Γ G ( · ) is non-decreasing, then G is not only a regular function,but also a Lyapunov function. For instance, this holds if G is non-decreasing in everydimension with respect to the first argument and Λ( · ) is decreasing in every dimension,and G is concave with respect to the second argument.Below we give sufficient conditions for a function G to be regular (Lyapunov). To thisend, recall the open set W from (4). Theorem 1.
For a continuous function G : W → R , consider the following conditions.(ai) On any compact set V ⊂ W , there exists a constant L = L ( V ) ≥ such that, forall ( λ , x ) , ( λ , x ) ∈ V , | G ( λ , x ) − G ( λ , x ) | ≤ L (cid:107) λ − λ (cid:107) . aii) G ( · , x ) is non-increasing, for fixed x , and Λ( · ) is non-decreasing in every dimen-sion.(bi) G is differentiable in the second argument. Moreover, on any compact set V ⊂ W ,there exists a constant L = L ( V ) ≥ such that, for all ( λ, x ) , ( λ, x ) ∈ V , (cid:13)(cid:13)(cid:13)(cid:13) ∂G∂x ( λ, x ) − ∂G∂x ( λ, x ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ L (cid:107) x − x (cid:107) . (bii) G ( λ, · ) is concave, for fixed λ .If one of the conditions (ai) or (aii) holds and one of the conditions (bi) or (bii) holds, G is a regular function for Λ( · ) and µ ( · ) . Moreover, in the case that (aii) and (bii) hold, G is Lyapunov. The proof of Theorem 1 is given in the appendix. A generalized version of It ˆo’s formulastudied in Krylov (2009) is related but can only be applied in a Markovian setting.
Remark . In contrast to Theorem . in Karatzas and Ruf (2017), even if G can beextended to a continuous function concave in the second argument, G may not be Lya-punov. A counterexample is given in Example 2. Therefore, for the generalized case,Theorem . in Karatzas and Ruf (2017) cannot be applied, and instead we have to usemodified conditions such as given by Theorem 1. Example 2.
Assume that µ ( · ) ∈ ∆ d with (cid:104) µ , µ (cid:105) ( t ) > , for all t > , and that Λ( · ) = γ d (cid:88) i =1 (cid:104) µ i , µ i (cid:105) ( · ) , where γ is a constant.Define the concave quadratic function G ( λ, x ) = λ − d (cid:88) i =1 x i , λ ∈ R , x ∈ ∆ d . Then from (9) we have Γ G ( · ) = − (cid:90) · dΛ( t ) + d (cid:88) i =1 (cid:90) · d (cid:104) µ i , µ i (cid:105) ( t ) = (1 − γ ) d (cid:88) i =1 (cid:104) µ i , µ i (cid:105) ( · ) . Observe that Γ G ( · ) is decreasing for γ > ; hence G is not a Lyapunov function for Λ( · ) and µ ( · ) , although it is concave in its second argument.Define now G ( λ, x ) = − G ( λ, x ) . Then we have Γ G ( · ) = − Γ G ( · ) . Therefore, if γ > holds, Γ G ( · ) is increasing; hence G is Lyapunov although convex in its second argument.Recall the ranked market weights process µ ( · ) defined in (5) and the open set W from(6). heorem 2. If a function G : W → R is regular for Λ( · ) and µ ( · ) = R (cid:0) µ ( · ) (cid:1) , then thecomposition G = G ◦ R is regular for Λ( · ) and µ ( · ) . We refer to the appendix for the proof of Theorem 2.The following example concerns a function G which is not in C , . Example 3.
Assume that µ ( · ) ∈ ∆ d + and consider the C , function G ( λ, x ) = − λ d (cid:88) l =1 x ( l ) log x ( l ) + 1 − d (cid:88) l = d +1 x l ) , λ ∈ R , x ∈ W d + , where d and d are positive integers with d < d ≤ d . According to Example 1, G isregular for Λ( · ) and µ ( · ) . In particular, the corresponding measurable function D G as inDefinition 3 can be chosen with components D l G ( λ, x ) = − λ log x ( l ) − λ, if l ∈ { , · · · , d }− x ( l ) , if l ∈ { d + 1 , · · · , d } , otherwise . (10)In this case, Itˆo’s lemma yields G (cid:0) Λ( · ) , µ ( · ) (cid:1) = G (cid:0) Λ(0) , µ (0) (cid:1) + (cid:90) · d (cid:88) l =1 D l G (cid:0) Λ( t ) , µ ( t ) (cid:1) d µ ( l ) ( t ) − Γ G ( · )+ (cid:90) · d (cid:88) l =1 µ ( l ) ( t ) log µ ( l ) ( t )dΛ( t ) (11)with D l G given in (10) and Γ G ( · ) = 12 (cid:90) · d (cid:88) l =1 Λ( t ) µ ( l ) ( t ) d (cid:10) µ ( l ) , µ ( l ) (cid:11) ( t ) + (cid:90) · d (cid:88) l = d +1 d (cid:10) µ ( l ) , µ ( l ) (cid:11) ( t ) . (12)Denote the number of components of x = ( x , · · · , x d ) (cid:48) ∈ ∆ d that coalesce at a givenrank l ∈ { , · · · , d } by N l ( x ) = d (cid:88) i =1 { x i = x ( l ) } . Then by Theorem 2.3 in Banner and Ghomrasni (2008), the ranked market weights pro-cess µ ( · ) has components µ ( l ) ( · ) = µ ( l ) (0) + (cid:90) · d (cid:88) i =1 { µ i ( t )= µ ( l ) ( t ) } N l (cid:0) µ ( t ) (cid:1) d µ i ( t ) + d (cid:88) k = l +1 (cid:90) · d Λ ( l,k ) ( t ) N l (cid:0) µ ( t ) (cid:1) − l − (cid:88) k =1 (cid:90) · d Λ ( k,l ) ( t ) N l (cid:0) µ ( t ) (cid:1) , l ∈ { , · · · , d } , (13)where Λ ( i,j ) ( · ) with ≤ i < j ≤ d is the local time process of the continuous semi-martingale µ ( i ) ( · ) − µ ( j ) ( · ) ≥ at the origin. y Theorem 2, the function G ( λ, x ) = G (cid:0) λ, R ( x ) (cid:1) = − λ d (cid:88) l =1 d (cid:88) i =1 { x i = x ( l ) } N l ( x ) x i log x i + 1 − d (cid:88) l = d +1 d (cid:88) i =1 { x i = x ( l ) } N l ( x ) x i is regular for Λ( · ) and µ ( · ) , since G is regular for Λ( · ) and µ ( · ) .Now, let us assume that Λ( · ) is of the form Λ( · ) = ξ ∧ (cid:0) ξ ∨ Λ (cid:48) ( · ) (cid:1) , where ξ and ξ are two positive constants with ξ < ξ , and the process Λ (cid:48) ( · ) is of finitevariation. Let G ( λ (cid:48) , x ) = G (cid:0) ξ ∧ (cid:0) ξ ∨ λ (cid:48) (cid:1) , x (cid:1) , for all λ (cid:48) ∈ R and x ∈ ∆ d + . Then with D l G and Γ G ( · ) given in (10) and (12), respectively, inserting (13) into (11) yields G (cid:0) Λ (cid:48) ( · ) , µ ( · ) (cid:1) = G (cid:0) Λ (cid:48) (0) , µ (0) (cid:1) + (cid:90) · d (cid:88) i =1 D i G (cid:0) Λ (cid:48) ( t ) , µ ( t ) (cid:1) d µ i ( t ) − Γ G ( · ) , where D i G ( λ (cid:48) , x ) = d (cid:88) l =1 { x i = x ( l ) } N l ( x ) D l G (cid:0) ξ ∧ (cid:0) ξ ∨ λ (cid:48) (cid:1) , R ( x ) (cid:1) , i ∈ { , · · · , d } , and Γ G ( · ) =Γ G ( · ) − (cid:90) · d (cid:88) l =1 µ ( l ) ( t ) log µ ( l ) ( t ) { ξ ≤ Λ (cid:48) ( t ) ≤ ξ } dΛ (cid:48) ( t ) − d − (cid:88) l =1 d (cid:88) k = l +1 (cid:90) · D l G (cid:0) Λ( t ) , R ( µ ( t )) (cid:1) N l (cid:0) µ ( t ) (cid:1) d Λ ( l,k ) ( t )+ d (cid:88) l =2 l − (cid:88) k =1 (cid:90) · D l G (cid:0) Λ( t ) , R ( µ ( t )) (cid:1) N l (cid:0) µ ( t ) (cid:1) d Λ ( k,l ) ( t ) . Observe that G is regular for Λ (cid:48) ( · ) and µ ( · ) , yet it is not in C , .
4. Functional generation and relative arbitrage
In Karatzas and Ruf (2017), two types of functional generation, additive and multiplica-tive generation, are constructed to study the properties of relative values of functionallygenerated portfolios. In this section, we first discuss the generalized versions of thesefunctional generations and corresponding properties. Then we consider sufficient condi-tions for strong arbitrage relative to the market to exist.
Recall the open set W from (4). efinition 5. (Additive generation). For a regular function G : W → R and the process ϑ ( · ) given in (7), the trading strategy ϕ ( · ) with components ϕ i ( · ) = ϑ i ( · ) − Q ϑ ( · ) + C, i ∈ { , · · · , d } , in the manner of (2) and (1), and with the real constant C = G (cid:0) Λ(0) , µ (0) (cid:1) − d (cid:88) j =1 µ j (0) D j G (cid:0) Λ(0) , µ (0) (cid:1) , (14)is said to be additively generated by the regular function G . Proposition 1.
The trading strategy ϕ ( · ) , generated additively by a regular function G : W → R , has components ϕ i ( · ) = D i G (cid:0) Λ( · ) , µ ( · ) (cid:1) + Γ G ( · ) + G (cid:0) Λ( · ) , µ ( · ) (cid:1) − d (cid:88) j =1 µ j ( · ) D j G (cid:0) Λ( · ) , µ ( · ) (cid:1) , (15) for all i ∈ { , · · · , d } . Moreover, the wealth process of ϕ ( · ) is given by V ϕ ( · ) = G (cid:0) Λ( · ) , µ ( · ) (cid:1) + Γ G ( · ) . (16) Proof.
The proposition is proven exactly like Proposition . in Karatzas and Ruf (2017). (Multiplicative generation). For a regular function G : W → (0 , ∞ ) , let theprocess ϑ ( · ) be given in (7) and assume that /G (cid:0) Λ( · ) , µ ( · ) (cid:1) is locally bounded. Considerthe process (cid:101) ϑ ( · ) ∈ L ( µ ) with components (cid:101) ϑ i ( · ) = ϑ i ( · ) × exp (cid:32)(cid:90) · dΓ G ( t ) G (cid:0) Λ( t ) , µ ( t ) (cid:1) (cid:33) , i ∈ { , · · · , d } . (17)Then the trading strategy ψ ( · ) with components ψ i ( · ) = (cid:101) ϑ i ( · ) − Q (cid:101) ϑ ( · ) + C, i ∈ { , · · · , d } , in the manner of (2) and (1), and with C given in (14), is said to be multiplicativelygenerated by the regular function G . Proposition 2.
The trading strategy ψ ( · ) , generated multiplicatively by a regular function G : W → (0 , ∞ ) with /G (cid:0) Λ( · ) , µ ( · ) (cid:1) locally bounded, has components ψ i ( · ) = V ψ ( · ) (cid:32) D i G (cid:0) Λ( · ) , µ ( · ) (cid:1) − (cid:80) dj =1 µ j ( · ) D j G (cid:0) Λ( · ) , µ ( · ) (cid:1) G (cid:0) Λ( · ) , µ ( · ) (cid:1) (cid:33) , (18) for all i ∈ { , · · · , d } , where the wealth process of ψ ( · ) is given by V ψ ( · ) = G (cid:0) Λ( · ) , µ ( · ) (cid:1) exp (cid:32)(cid:90) · dΓ G ( t ) G (cid:0) Λ( t ) , µ ( t ) (cid:1) (cid:33) > . (19) Proof.
The same argument as in Proposition . in Karatzas and Ruf (2017) applies. .3. Sufficient conditions for arbitrage relative to the market In Karatzas and Ruf (2017), Theorem 5.1 and Theorem 5.2 give sufficient conditions forstrong arbitrage relative to the market to exist for both additively and multiplicatively gen-erated portfolios, respectively. These results still hold for a regular / Lyapunov function G : W → [0 , ∞ ) under specific conditions.To be consistent with the conditions of arbitrage relative to the market in (3), wenormalize G (cid:0) Λ(0) , µ (0) (cid:1) = 1 such that both of the wealth processes in (16) and (19)have initial values . This normalization is guaranteed by replacing G with G + 1 when G (cid:0) Λ(0) , µ (0) (cid:1) = 0 , or with
G/G (cid:0)
Λ(0) , µ (0) (cid:1) when G (cid:0) Λ(0) , µ (0) (cid:1) > . Theorem 3.
Fix a Lyapunov function G : W → [0 , ∞ ) with G (cid:0) Λ(0) , µ (0) (cid:1) = 1 . Forsome real number T ∗ > , suppose that P (cid:2) Γ G ( T ∗ ) > (cid:3) = 1 . Then the additively generated trading strategy ϕ ( · ) of Definition 5 is strong arbitragerelative to the market over every time horizon [0 , T ] with T ≥ T ∗ .Proof. Use the same reasoning as in the proof of Theorem 5.1 in Karatzas and Ruf (2017).
Theorem 4.
Assume that | Λ( · ) | is uniformly bounded. Fix a regular function G : W → [0 , ∞ ) with G (cid:0) Λ(0) , µ (0) (cid:1) = 1 . For some real numbers T ∗ > , suppose that we can findan ε = ε ( T ∗ ) > such that P (cid:2) Γ G ( T ∗ ) > ε (cid:3) = 1 . Then there exists a constant c = c ( T ∗ , ε ) > such that the trading strategy ψ ( c ) ( · ) ,generated multiplicatively by the regular function G ( c ) = G + c c as in Definition 6, is strong arbitrage relative to the market over the time horizon [0 , T ∗ ] .Moreover, if G is a Lyapunov function, then ψ ( c ) ( · ) is also a strong relative arbitrage overevery time horizon [0 , T ] with T ≥ T ∗ .Proof. See the proof of Theorem 5.2 in Karatzas and Ruf (2017). Note that G (cid:0) Λ( · ) , µ ( · ) (cid:1) is uniformly bounded thanks to the assumptions.
5. Data source and processing
We start this section by describing the data used in the next section, where severaltrading strategies are implemented. Then we discuss the method to process the data. .1. Data source and description We shall consider a market consisting of all stocks in the S&P 500 index. We are inter-ested in the beginning of day and the end of day market weights of each of these stocks.To calculate these market weights accurately (according to the method in Subsection 5.2),we make use of two time series: the daily market values (market capitalizations, whichexclude all the dividend payments) and the daily return indexes (used to consider the ef-fect of reinvestment of dividend payments) of the corresponding component stocks in theS&P 500 index. Both of these time series are available at the end of each trading day.The data of the market values and return indexes is downloaded from DataStream . Thefirst day, for which the data is available on DataStream, is September 29 th , 1989. Sincethen there are in total 1140 constituents that have belonged to the S&P 500 index. A listof stocks in the S&P 500 index is also attainable on DataStream. In particular, for eachmonth, we derive the list of constituents of the index at the last day of this month. Fora constituent delisted from the index in that month, we keep it in our portfolio providedthat the constituent still remains in the market till the end of that month. However, we getrid of it from our portfolio on the same day when the constituent does no longer exist inthe market, usually due to mergers and acquisitions, bankruptcies, etc. For a constituentnewly added to the index in that month, we put it into our portfolio from the first day ofthe following month. Theoretically, trading strategies vary continuously in time, while in the empirical analy-sis a daily trading frequency is used. The following procedure illustrates how we examinethe gains and losses in our portfolio relative to the market portfolio.We discretize the time horizon as t < t < · · · < t N − = T , where N is the totalnumber of trading days. • The transaction on day t l , for all l ∈ { , · · · , N − } , is made at the beginningof day ( t l ), taking the beginning of day t l market weights µ ( t l ) as inputs. Thesemarket weights µ ( t l ) are computed by µ i ( t l ) = MV i ( t l )Σ( t l ) , i ∈ { , · · · , d } , where MV i ( t l ) is the market value of stock i at the beginning of day t l , which isassumed to be equal to the market value attainable at the end of the last tradingday t l − , and Σ( t l ) = (cid:80) dj =1 MV j ( t l ) denotes the total market capitalization at thebeginning of day t l . • The theoretical (non-self-financing) trading strategy throughout t l , denoted by θ ( t l ) , is computed based on either (7) or (17), taking µ ( t l ) as inputs. Denotethe implemented (self-financing) trading strategy corresponding to θ ( t l ) by φ ( t l ) .Then V φ ( t l ) , the beginning of day t l wealth of the portfolio corresponding to φ ( t l ) ,is given by V φ ( t ) = V φ ( t l − )Σ( t l − )Σ( t ) . (20) DataStream, operated by Thomson Reuters, is a financial time series database; see https://financial.thomsonreuters.com/en/products/tools-applications/trading-investment-tools/datastream-macroeconomic-analysis.html. his is based on the assumption that the real portfolio wealth does not changeovernight. In (20), V φ ( t l − ) and Σ( t l − ) are the end of day t l − portfolio wealthand total market capitalization, respectively, computed at t l − (thus already knownat t l ). • To derive the implemented (self-financing) trading strategy φ ( t l ) corresponding to θ ( t l ) , we compute the number C ( t l ) = d (cid:88) j =1 θ j ( t l ) µ j ( t l ) − V φ ( t l ) . (21)Then φ ( t l ) is derived by φ i ( t l ) = θ i ( t l ) − C ( t l ) , i ∈ { , · · · , d } . (22)This guarantees V φ ( t l ) = (cid:80) di =1 φ i ( t l ) µ i ( t l ) . • At the end of day t l , the return indexes of the stocks for t l are available, and thetotal returns TR( t l ) are computed through dividing the return indexes of t l withthe return indexes of t l − . Then the end of day t l implied market values MV( t l ) ,which take the dividend payments into consideration, are given by MV i ( t l ) = MV i ( t l )TR i ( t l ) , i ∈ { , · · · , d } . The end of day t l modified total market capitalization Σ( t l ) and market weights µ ( t l ) are calculated similarly as Σ( t l ) and µ ( t l ) , with MV( t l ) replaced by MV( t l ) . • The end of day t l portfolio wealth is then computed by V φ ( t l ) = d (cid:88) j =1 φ j ( t l ) µ j ( t l ) . Note that we have V φ ( t l ) = V φ ( t l ) + d (cid:88) j =1 θ j ( t l ) (cid:0) µ j ( t l ) − µ j ( t l ) (cid:1) . (23)In particular, at the beginning of day t , all of the above steps are still applied, except thatwe have V φ ( t ) = 1 instead of (20) due to Definition 2.
6. Examples and empirical results
In this section, several examples of portfolio generating functions are empirically stud-ied.
Example 4.
Define the generalized entropy function G ( λ, x ) = λ d (cid:88) i =1 x i log (cid:18) x i (cid:19) , λ ∈ R + , x ∈ ∆ d + , ith values in (0 , λ log d ) , for fixed λ > . Suppose that µ ( · ) takes values in ∆ d + and that Λ( · ) is (0 , ∞ ) -valued.From (9) we have Γ G ( · ) = d (cid:88) i =1 (cid:90) · µ i ( t ) log µ i ( t )dΛ( t ) + 12 d (cid:88) i =1 (cid:90) · Λ( t ) d (cid:104) µ i , µ i (cid:105) ( t ) µ i ( t ) . (24)Then G is a Lyapunov function for Λ( · ) and µ ( · ) provided that Γ G ( · ) is non-decreasing.One sufficient condition for this to hold is that Λ( · ) is non-increasing.From (15), the trading strategy ϕ ( · ) , generated additively by G , has components ϕ i ( · ) = Γ G ( · ) − Λ( · ) log µ i ( · ) , i ∈ { , · · · , d } . (25)Using (16), the corresponding wealth process V ϕ ( · ) = G (cid:0) Λ( · ) , µ ( · ) (cid:1) + Γ G ( · ) is strictlypositive if G is a generalized Lyapunov function.For the multiplicative generation, G is required to be bounded away from zero. Onesufficient condition for this to hold is that Λ( · ) is bounded away from and the market isdiverse on [0 , ∞ ) , i.e., there exists (cid:15) > such that G (cid:0) Λ( t ) , µ ( t ) (cid:1) ≥ Λ( t ) (cid:15) , for all t ≥ (see Proposition 2.3.2 in Fernholz (2002)). Then from (18), the trading strategy ψ ( · ) ,generated multiplicatively by G , has components ψ i ( · ) = − Λ( · ) log µ i ( · ) exp (cid:32)(cid:90) · dΓ G ( t ) G (cid:0) Λ( t ) , µ ( t ) (cid:1) (cid:33) , i ∈ { , · · · , d } . The corresponding wealth process V ψ ( · ) is given in (19).Now, let us discuss sufficient conditions for the existence of arbitrage relative to themarket. For the Lyapunov function G , let us consider G = GG (cid:0) Λ(0) , µ (0) (cid:1) (26)normalized to have initial value , together with the non-decreasing process Γ G ( · ) = Γ G ( · ) G (cid:0) Λ(0) , µ (0) (cid:1) . (27)Then from Theorem 3, if P (cid:2) Γ G ( T ∗ ) > (cid:3) = P (cid:2) Γ G ( T ∗ ) > G (cid:0) Λ(0) , µ (0) (cid:1)(cid:3) = 1 , then the trading strategy ϕ ( · ) /G (cid:0) Λ(0) , µ (0) (cid:1) , generated additively by G , is strong relativearbitrage over every time horizon [0 , T ] with T ≥ T ∗ .Similarly, from Theorem 4, if P (cid:2) Γ G ( T ∗ ) > ε (cid:3) = P (cid:2) Γ G ( T ∗ ) > G (cid:0) Λ(0) , µ (0) (cid:1) (1 + ε ) (cid:3) = 1 , then the trading strategy ψ ( c ) ( · ) , generated multiplicatively by G ( c ) = G + cG (cid:0) Λ(0) , µ (0) (cid:1) + c , (28) igure 1. The case Λ( · ) = 1 .Figure 2. Wealth process V ϕ ( · ) with Λ( · ) adeterministic exponential. Figure 3. Wealth process V ϕ ( · ) with Λ( · ) an exponential of the quadratic variation of µ ( · ) . for some sufficiently large c > , is strong relative arbitrage over every time horizon [0 , T ] with T ≥ T ∗ .Figure 1 presents Γ G ( · ) given in (27) and the wealth processes V ϕ ( · ) and V ψ (0) ( · ) , withfinite variation process Λ( · ) = 1 . As we can observe from the figure, both V ϕ ( · ) and V ψ (0) ( · ) have been increasing since the year 2000.Figure 2 and Figure 3 display the wealth processes V ϕ ( · ) corresponding to two differentgroups of Λ( · ) . Namely, for all l ∈ { , · · · , N } , in Figure 2, the wealth processes V ϕ ( · ) corresponding to Λ( t l ) = exp (10 − l ) and Λ( t l ) = exp ( − − l ) are plotted; in Figure 3,the wealth processes V ϕ ( · ) corresponding to Λ( t l ) = exp (cid:32) d (cid:88) j =1 (cid:104) µ j , µ j (cid:105) ( t l ) (cid:33) and Λ( t l ) = exp (cid:32) − d (cid:88) j =1 (cid:104) µ j , µ j (cid:105) ( t l ) (cid:33) are plotted. The constants − and are chosen such that, with these forms, thedaily changes of both G (cid:0) Λ( · ) , µ ( · ) (cid:1) and Γ G ( · ) are roughly at the same level of magnitude.Hence, in (16), neither part on the right hand side dominates the other.As we can observe from the figures, choosing Λ( · ) increasing seems to lead to a betterperformance than choosing Λ( · ) constant, which again seems to be better than choosing igure 4. Integration process E ( · ) withcomponents given by (30) . Figure 5. Process (cid:80) di =1 ( µ i ∧ . · ) asa measure of the market diversification de-gree in the S&P 500 market. Λ( · ) decreasing. We attribute the reason behind this observation to the state of marketdiversification as follows.Observe that (23) yields V ϕ ( t l ) = V ϕ ( t l ) + 1 G (cid:0) Λ(0) , µ (0) (cid:1) Λ( t l ) D ( t l ) , l ∈ { , · · · , N } , (29)where D ( t l ) is given by D ( t l ) = d (cid:88) j =1 − log µ j ( t l ) (cid:0) µ j ( t l ) − µ j ( t l ) (cid:1) . (30)The value D ( t l ) can be considered as an indicator of the direction of changes in mar-ket weights from the beginning to the end of date t l . The value D ( t l ) will be positive(negative), if market weights are shifted from companies with large (small) beginningof day market weights to companies with small (large) beginning of day market weightsthroughout date t l . We consider a simple example to better understand why this is thecase.Fix d = 2 and assume that µ ( t l ) > µ ( t l ) . Then D ( t l ) = − log µ ( t l ) (cid:0) µ ( t l ) − µ ( t l ) (cid:1) − log µ ( t l ) (cid:0) µ ( t l ) − µ ( t l ) (cid:1) = ( − log µ ( t l ) + log µ ( t l )) (cid:0) µ ( t l ) − µ ( t l ) (cid:1) holds due to the fact that (cid:0) µ ( t l ) − µ ( t l ) (cid:1) = − (cid:0) µ ( t l ) − µ ( t l ) (cid:1) . Hence, D ( t l ) > ifand only if µ ( t l ) < µ ( t l ) , i.e., the market weight of the company with larger beginningof day market weight decreases, while the market weight of the company with smallerbeginning of day market weight increases.Hence, a positive D ( · ) indicates an enhancement in market diversification, while D ( · ) being negative actually implies a reduction in market diversification. Figure 4 plots thecumulative process E ( · ) = (cid:80) · t l = t D ( t l ) . The process E ( · ) is increasing (decreasing)whenever D ( · ) is positive (negative). From Figure 4 we can observe that after a slightincrease from the year 1991 to the year 1995, E ( · ) keeps declining till the year 2000.Then E ( · ) rises up in the long run from the year 2000 until now. he behavior of the process E ( · ) is in line with another measurement of the marketdiversification. More precisely, let us consider the process (cid:80) di =1 ( µ i ∧ . · ) . Note thatthe value .
002 = 1 / , which is roughly the number of constituents in the portfolio.This process is a measure of the market diversification, as it goes up when the marketweights of small companies become larger, i.e., the market diversification is strengthened.Figure 5 plots the process, which first grows from the year 1991 to the year 1995. Thenfrom the year 1995 to 2000, the process declines rapidly. This indicates that during thisperiod, the market diversification weakens. On the contrary, the market diversificationstrengthens afterwards until the year 2008, as the process goes up. Then the level ofmarket diversification remains within a relatively small range.As a result, according to (29), if the market presents a trend of increasing diversification,an increasing positive Λ( · ) helps to reinforce this effect, and further assists in pulling up V ϕ ( · ) , while a decreasing positive Λ( · ) is counteractive. On the other hand, if the marketpresents a trend of decreasing diversification, then a decreasing positive Λ( · ) helps toslow down the declining speed of V ϕ ( · ) , while an increasing positive Λ( · ) would makethe speed even faster. This is confirmed in Figure 2 and Figure 3, as from the year 1991to the year 1995 and from the year 2000 till now, an increasing positive Λ( · ) makes V ϕ ( · ) perform better, while from the year 1995 to the year 2000, V ϕ ( · ) corresponding to adecreasing positive Λ( · ) is slightly larger.Although an increasing positive Λ( · ) has positive effect on the portfolio performance V ϕ ( · ) whenever the market diversification strengthens, we are not allowed to choose Λ( · ) arbitrarily fast increasing. The reason is that Γ G ( · ) will become negative and decreaserapidly if the increments in Λ( · ) are sufficiently large, which will result in a negative ψ ( · ) given in (25).As for the multiplicative generation, the different choices of finite variation processesdo not change the wealth processes significantly. Indeed, according to (24), an increasing Λ( · ) may slow down the growth rate of Γ( · ) , or even turn Γ( · ) into a decreasing one.When applying (22) to (cid:101) ϑ ( · ) from (17), we have V ψ ( c ) ( t l ) = exp (cid:32)(cid:90) t l dΓ G ( t ) G (cid:0) Λ( t ) , µ ( t ) (cid:1) + c (cid:33) Λ( t l ) G (cid:0) Λ(0) , µ (0) (cid:1) + c D ( t l ) + V ψ ( c ) ( t l ) , for all l ∈ { , · · · , N } , with D ( · ) given in (30). In this example, according to the aboveequation, the positive effect in boosting V ψ ( c ) ( · ) contributed by an increasing positive Λ( · ) is counteracted more or less by the opposite impact the same Λ( · ) has on the exponentialpart. A similar analysis also applies to a decreasing positive Λ( · ) . Therefore, underthe above mentioned situation (market diversification increases in general), the differentchoices of a monotone Λ( · ) do not influence V ψ ( c ) ( · ) as much as they do on V ϕ ( · ) .Note that our process D ( · ) is related but not the same as the Bregman divergence D B,G (cid:2) µ ( t l ) | µ ( t l ) (cid:3) = Λ( t l ) D ( t l ) − (cid:0) G (cid:0) Λ( t l ) , µ ( t l ) (cid:1) − G (cid:0) Λ( t l ) , µ ( t l ) (cid:1)(cid:1) , defined in Definition 3.6 of Wong (2017). For its connection to optimal transport, we referto Wong (2017).The following example is motivated by Schied et al. (2016). xample 5. Consider the function G ( λ, x ) = (cid:32) d (cid:88) i =1 ( αx i + (1 − α ) λ i ) p (cid:33) p , λ ∈ R d + , x ∈ ∆ d + , with constants α, p ∈ (0 , . Then G is concave.For fixed constant δ > , define the R d + -valued moving average process Λ( · ) by Λ i ( · ) = (cid:40) δ (cid:82) · µ i ( t )d t + δ (cid:82) ·− δ µ i (0)d t on [0 , δ ) δ (cid:82) ··− δ µ i ( t )d t on [ δ, ∞ ) , for all i ∈ { , · · · , d } .Write µ ( · ) = αµ ( · ) + (1 − α )Λ( · ) . Then by (9), Γ G ( · ) = − (1 − α ) d (cid:88) i =1 (cid:90) · (cid:32) G (cid:0) Λ( t ) , µ ( t ) (cid:1) µ i ( t ) (cid:33) − p dΛ i ( t ) − α (1 − p )2 d (cid:88) i,j =1 (cid:90) · (cid:32) G (cid:0) Λ( t ) , µ ( t ) (cid:1) µ i ( t ) µ j ( t ) (cid:33) − p (cid:80) dv =1 ( µ v ( t )) p d (cid:104) µ i , µ j (cid:105) ( t )+ α (1 − p )2 d (cid:88) i =1 (cid:90) · (cid:32) G (cid:0) Λ( t ) , µ ( t ) (cid:1) µ i ( t ) (cid:33) − p µ i ( t ) d (cid:104) µ i , µ i (cid:105) ( t ) . Notice that G is not Lyapunov in general.The trading strategies ϕ ( · ) and ψ ( · ) , generated additively and multiplicatively by G ,respectively, are given by ϕ i ( · ) = G (cid:0) Λ( · ) , µ ( · ) (cid:1) (cid:32) α ( µ i ( · )) p µ i ( · ) (cid:80) dv =1 ( µ v ( · )) p − d (cid:88) j =1 αµ j ( · ) (cid:0) µ j ( · ) (cid:1) p µ j ( · ) (cid:80) dv =1 ( µ v ( · )) p + 1 (cid:33) + Γ G ( · ) and ψ i ( · ) = (cid:0) ϕ i ( · ) − Γ G ( · ) (cid:1) exp (cid:32)(cid:90) · dΓ G ( t ) G (cid:0) Λ( t ) , µ ( t ) (cid:1) (cid:33) , for all i ∈ { , · · · , d } . The corresponding wealth processes V ϕ ( · ) and V ψ ( · ) can bederived from (16) and (19), respectively.Consider the normalized regular function G given in (26) and the corresponding process Γ G ( · ) given in (27). By Theorem 4, if P (cid:2) Γ G ( T ∗ ) > ε (cid:3) = P (cid:2) Γ G ( T ∗ ) > G (cid:0) Λ(0) , µ (0) (cid:1) (1 + ε ) (cid:3) = 1 , then the trading strategy ψ ( c ) ( · ) , generated multiplicatively by G ( c ) given in (28) for somesufficiently large c > , is strong relative arbitrage over the time horizon [0 , T ∗ ] .To simulate the relative performance of the portfolio, we use the parameters δ = 250 days and p = 0 . . Figure 6 shows Γ G ( · ) and the wealth processes V ϕ ( · ) and V ψ (0) ( · ) without the effect of the moving average part, i.e., α = 1 . In this case, G is Lyapunov. igure 6. The case δ = 250 days, p = 0 . and α = 1 . Figure 7. The case δ = 250 days, p = 0 . and α = 0 . . The relative performance of the portfolio is similar to that in Example 4, when the finitevariation process is chosen to be constant. Figure 7 presents the case when α = 0 . . Itcan be observed that Γ G ( · ) increases slower when the moving average part is considered.Compared with the case that the moving average part is not included, the wealth processes V ϕ ( · ) and V ψ (0) ( · ) also take smaller values in the long run. This is due to the fact thatwhen α decreases, the volatility of µ ( · ) decreases as well. In this case, we trade slower,and the gains and losses will also be relatively less.
7. Conclusion
Karatzas and Ruf (2017) build a simple and intuitive structure by interpreting the port-folio generating functions G initiated by Fernholz (1999, 2001, 2002) as Lyapunov func-tions. They formulate conditions for the existence of strong arbitrage relative to themarket over appropriate time horizons. The purpose of this paper is to investigate thedependence of the portfolio generating functions G on an extra R m -valued, progressive,continuous process Λ( · ) of finite variation on [0 , T ] , for all T ≥ .The results of this paper are illuminated by several examples and shown to work onempirical data using stocks from the S&P 500 index. The effects that different choicesof Λ( · ) have on the portfolio wealths are analyzed. Provided that the market undergos anexplicit trend of either increasing or decreasing market diversification, certain choices of Λ( · ) are better than others. A. Proofs of Theorems 1 and 2
A.1. Preliminaries
Before providing the proof of Theorem 1, we discuss some technical details.Recall the open set W from (4) and consider a continuous function g : W → R . Definea function g : R m + d → R by g ( z ) = (cid:40) g ( z ) , if z ∈ W , if z / ∈ W . ext, consider the family of functions ( g n ,n ) n ,n ∈ N with g n ,n : W → R given by g n ,n ( λ, x ) = (cid:90) R d η n ( y ) (cid:90) R m η n ( u ) g ( λ − u, x − y )d u d y, (31)for all ( λ, x ) ∈ W , with g n ,n ( λ, x ) = 0 whenever the right hand side of (31) is notdefined. Here in (31), for z ∈ R l and n ∈ N , η n ( z ) = (cid:40) βn l exp (cid:16) n (cid:107) z (cid:107) − (cid:17) , if (cid:107) z (cid:107) < n , if (cid:107) z (cid:107) ≥ n (32)is used with the normalization constant β = (cid:18)(cid:90) R l exp (cid:18) (cid:107) y (cid:107) − (cid:19) d y (cid:19) − , independent of n . Lemma 1.
Let V denote any closed subset of W . Consider a continuous function g : W → R and the mollification ( g n ,n ) n ,n ∈ N of g defined as in (31) .(i) We have lim n ↑∞ lim n ↑∞ g n ,n = g. (ii) For n , n ∈ N large enough, g n ,n ∈ C ∞ ( V ) .(iii) If there exists a constant L = L ( V ) ≥ such that, for all ( λ , x ) , ( λ , x ) ∈ V , | g ( λ , x ) − g ( λ , x ) | ≤ L (cid:107) λ − λ (cid:107) , then, for n , n ∈ N large enough and all ( λ, x ) ∈ V , we have (cid:12)(cid:12)(cid:12)(cid:12) ∂g n ,n ∂λ v ( λ, x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ L, v ∈ { , · · · , m } . (iv) If g ∈ C , , then, for all ( λ, x ) ∈ W , we have lim n ↑∞ lim n ↑∞ ∂g n ,n ∂x i ( λ, x ) = ∂g∂x i ( λ, x ) , i ∈ { , · · · , d } . (v) If g ∈ C , and if there exists a constant L = L ( V ) ≥ such that, for all ( λ, x ) , ( λ, x ) ∈ V , (cid:13)(cid:13)(cid:13)(cid:13) ∂g∂x ( λ, x ) − ∂g∂x ( λ, x ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ L (cid:107) x − x (cid:107) , then, for n , n ∈ N large enough and all ( λ, x ) ∈ V , we have (cid:12)(cid:12)(cid:12)(cid:12) ∂ g n ,n ∂x i ∂x j ( λ, x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ L, i, j ∈ { , · · · , d } . Proof.
For (i) and (ii), see Theorem 6 in Appendix C of Evans (1998). or (iii), observe that, for each n , n ∈ N large enough and all v ∈ { , · · · , m } , (31)yields (cid:12)(cid:12)(cid:12)(cid:12) ∂g n ,n ∂λ v ( λ, x ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) lim δ → g n ,n ( λ + δ e v , x ) − g n ,n ( λ, x ) δ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) lim δ → δ (cid:90) R d η n ( y ) (cid:90) R m η n ( u ) (cid:0) g ( λ + δ e v − u, x − y ) − g ( λ − u, x − y ) (cid:1) d u d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim δ → δ (cid:90) R d η n ( y ) (cid:90) R m η n ( u ) | g ( λ + δ e v − u, x − y ) − g ( λ − u, x − y ) | d u d y ≤ lim δ → δ δL (cid:90) R d η n ( y ) (cid:90) R m η n ( u )d u d y = L, for all ( λ, x ) ∈ V , where e v is the unit vector in the v -th dimension.For (iv), apply the dominated convergence theorem and (i) to ∂g∂x i , for all i ∈ { , · · · , d } .For (v), similarly to (iii), for each n , n ∈ N large enough and all i, j ∈ { , · · · , d } ,we have (cid:12)(cid:12)(cid:12)(cid:12) ∂ g n ,n ∂x i ∂x j ( λ, x ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim δ → ∂g n ,n ∂x i ( λ, x + δ e j ) − ∂g n ,n ∂x i ( λ, x ) δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) lim δ → δ (cid:90) R d η n ( y ) (cid:90) R m η n ( u ) (cid:18) ∂g∂x i ( λ − u, x + δ e j − y ) − ∂g∂x i ( λ − u, x − y ) (cid:19) d u d y (cid:12)(cid:12)(cid:12)(cid:12) ≤ L, for all ( λ, x ) ∈ V , where for the second equality we apply the dominated convergencetheorem.The following lemma is an extension of Lemma 2 in Bouleau (1981). For a continuousfunction g : W → R , consider its corresponding mollification ( g n ,n ) n ,n ∈ N defined asin (31). Lemma 2.
If a continuous function g : W → R is concave in its second argument, then lim n ↑∞ lim n ↑∞ ∂g n ,n ∂x i = f i , i ∈ { , · · · , d } , for some measurable function f i : W → R , bounded on any compact V ⊂ W .Proof.
With the notation in (32), we have η n ( z ) = n l η ( nz ) , z ∈ R l , n ∈ N . For ( λ, x ) ∈ W and n ∈ N large enough, the definition of g n ,n in (31), the dominated onvergence theorem, and Lemma 1 yield lim n ↑∞ ∂g n ,n ∂x i ( λ, x ) = lim n ↑∞ (cid:90) R d ∂η n ∂x i ( x − y ) (cid:90) R m η n ( u ) g ( λ − u, y )d u d y = (cid:90) R d ∂η n ∂x i ( x − y ) lim n ↑∞ (cid:90) R m η n ( u ) g ( λ − u, y )d u d y = (cid:90) R d ∂η n ∂x i ( x − y ) g ( λ, y )d y = − (cid:90) R d ∂η n ∂y i ( y ) g ( λ, x − y )d y = (cid:90) R d n ∂η ∂y i ( y ) g (cid:18) λ, x + yn (cid:19) d y = (cid:90) R d ∂η ∂y i ( y ) n (cid:18) g (cid:18) λ, x + yn (cid:19) − g ( λ, x ) (cid:19) d y. Note that the last equality holds due to the fact that (cid:90) R d ∂η ∂y i ( y )d y = 0 . Next, for all ( λ, x ) ∈ W and y ∈ R d , define the one-sided directional partial derivativeas ∇ g ( λ, x ; y ) = lim n ↑∞ g ( λ, x + y/n ) − g ( λ, x )1 /n . Such ∇ g exists according to Theorem 23.1 in Rockafellar (1970). Since g is concavein the second argument, it is locally Lipschitz in its second argument on W (see Theo-rem 10.4 in Rockafellar (1970)). Hence, for each compact V ⊂ W , there exists a constant L = L ( V ) ≥ such that ∇ g ( λ, x ; y ) ≤ L , for all y ∈ R d and ( λ, x ) in the interior of V .The statement now follows with f i ( λ, x ) = (cid:90) R d ∇ g ( λ, x ; y ) ∂η ∂y i ( y )d y, for all ( λ, x ) ∈ W , by the dominated convergence theorem. Lemma 3.
Assume that µ ( · ) has Doob-Meyer decomposition µ ( · ) = µ (0) + M ( · ) + V ( · ) ,where M ( · ) is a d -dimensional continuous local martingale and V ( · ) is a d -dimensionalfinite variation process with M (0) = V (0) = 0 . Moreover, suppose that,(i) for some open V ⊂ W , we have (cid:0) Λ( · ) , µ ( · ) (cid:1) = (cid:0) Λ( · ∧ τ ) , µ ( · ∧ τ ) (cid:1) , where τ = inf (cid:8) t ≥ (cid:0) Λ( t ) , µ ( t ) (cid:1) / ∈ V (cid:9) ; (ii) for some constant κ ≥ , we have d (cid:88) i =1 (cid:18) (cid:104) M i , M i (cid:105) ( ∞ ) + (cid:90) ∞ d | V i ( t ) | (cid:19) + m (cid:88) v =1 (cid:90) ∞ d | Λ v ( t ) | ≤ κ < ∞ . (33) onsider two families of bounded functions ( h i ) i ∈{ , ··· ,d } and ( h n ,n i ) n ,n ∈ N ,i ∈{ , ··· ,d } with h i , h n ,n i : V → R . If lim n ↑∞ lim n ↑∞ h n ,n i = h i , i ∈ { , · · · , d } , then there exist two random subsequences (cid:0) n k (cid:1) k ∈ N and (cid:0) n k (cid:1) k ∈ N with lim k ↑∞ n k = ∞ =lim k ↑∞ n k such that lim k ↑∞ (cid:90) t d (cid:88) i =1 h n k ,n k i (cid:0) Λ( u ) , µ ( u ) (cid:1) d µ i ( u ) = (cid:90) t d (cid:88) i =1 h i (cid:0) Λ( u ) , µ ( u ) (cid:1) d µ i ( u ) , a.s. , for all t ≥ .Proof. Fix i ∈ { , · · · , d } and write Θ n ,n i ( · ) = h n ,n i (cid:0) Λ( · ) , µ ( · ) (cid:1) − h i (cid:0) Λ( · ) , µ ( · ) (cid:1) . By (33) and the bounded convergence theorem, we have lim n ↑∞ lim n ↑∞ (cid:90) ∞ (cid:0) Θ n ,n i ( t ) (cid:1) d (cid:104) M i , M i (cid:105) ( t ) = 0 , a.s. , and lim n ↑∞ lim n ↑∞ (cid:18)(cid:90) ∞ | Θ n ,n i ( t ) | d | V i ( t ) | (cid:19) = 0 , a.s.Therefore, by the bounded convergence theorem again, we have E (cid:20) lim n ↑∞ lim n ↑∞ (cid:90) ∞ (cid:0) Θ n ,n i ( t ) (cid:1) d (cid:104) M i , M i (cid:105) ( t ) (cid:21) = lim n ↑∞ lim n ↑∞ E (cid:20)(cid:90) ∞ (cid:0) Θ n ,n i ( t ) (cid:1) d (cid:104) M i , M i (cid:105) ( t ) (cid:21) = lim n ↑∞ lim n ↑∞ E (cid:34)(cid:18)(cid:90) ∞ Θ n ,n i ( t )d M i ( t ) (cid:19) (cid:35) , by Itˆo’s isometry, and n ↑∞ lim n ↑∞ E (cid:34)(cid:18)(cid:90) ∞ | Θ n ,n i ( t ) | d | V i ( t ) | (cid:19) (cid:35) . (34)Since (cid:82) · Θ n ,n i ( t )d M i ( t ) is a uniformly integrable martingale (as it is a local martingalewith bounded quadratic variation), Doob’s submartingale inequality yields E (cid:34)(cid:18) sup t ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t Θ n ,n i ( u )d M i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:35) ≤ E (cid:34)(cid:18)(cid:90) ∞ Θ n ,n i ( t )d M i ( t ) (cid:19) (cid:35) , which implies n ↑∞ lim n ↑∞ E (cid:34)(cid:18) sup t ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t Θ n ,n i ( u )d M i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:35) . (35) herefore, we have lim n ↑∞ lim n ↑∞ E (cid:34)(cid:18) sup t ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t Θ n ,n i ( u )d µ i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:35) ≤ lim n ↑∞ lim n ↑∞ E (cid:34) (cid:18) sup t ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t Θ n ,n i ( u )d M i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + (cid:18) sup t ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t Θ n ,n i ( u )d V i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:35) ≤ , where the second inequality holds due to (34) and (35).Write E n ,n i = E (cid:34)(cid:18) sup t ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t Θ n ,n i ( u )d µ i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (cid:35) , n , n ∈ N , and E i = lim n ↑∞ lim n ↑∞ E n ,n i . For each n ∈ N , denote E n i = lim n ↑∞ E n ,n i . Then we can find a subsequence (cid:0) n ( n ) (cid:1) n ∈ N of N with n ( n ) ↑ ∞ as n ↑ ∞ such that, for each n ∈ N , (cid:12)(cid:12)(cid:12) E n ( n ) ,n i − E n i (cid:12)(cid:12)(cid:12) ≤ n . Since (cid:12)(cid:12)(cid:12) E n ( n ) ,n i − E i (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) E n ( n ) ,n i − E n i (cid:12)(cid:12)(cid:12) + | E n i − E i |≤ n + | E n i − E i | → as n ↑ ∞ , we have lim n ↑∞ E n ( n ) ,n i = E i = 0 . This implies lim n ↑∞ sup t ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t d (cid:88) i =1 h n ( n ) ,n i (cid:0) Λ( u ) , µ ( u ) (cid:1) d µ i ( u ) − (cid:90) t d (cid:88) i =1 h i (cid:0) Λ( u ) , µ ( u ) (cid:1) d µ i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 in L . Since convergence in L implies almost sure convergence of a subsequence, we canfind a random subsequence (cid:0) n k (cid:1) k ∈ N of N with n k ↑ ∞ as k ↑ ∞ and write n k = n (cid:0) n k (cid:1) such that lim k ↑∞ sup t ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t d (cid:88) i =1 h n k ,n k i (cid:0) Λ( u ) , µ ( u ) (cid:1) d µ i ( u ) − (cid:90) t d (cid:88) i =1 h i (cid:0) Λ( u ) , µ ( u ) (cid:1) d µ i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , a.s. This implies the assertion. Lemma 4.
Fix l ∈ N ; let Λ( · ) be an l -dimensional continuous process of finite variation;let (cid:0) Υ u,n ( · ) (cid:1) u ∈{ , ··· ,l } ,n ∈ N be a family of processes with (cid:0) Υ u,n ( · ) (cid:1) n ∈ N uniformly bounded,for each u ∈ { , · · · , l } ; and let (cid:0) Θ n ( · ) (cid:1) n ∈ N be a sequence of non-decreasing continuousprocesses. Define H n ( · ) = (cid:90) · l (cid:88) u =1 Υ u,n ( t )dΛ u ( t ) + Θ n ( · ) , n ∈ N . If lim n ↑∞ H n ( · ) = H ( · ) , a.s., then H ( · ) is of finite variation. roof. The following steps are partially inspired by the proof of Lemma 3.3 in Jaber et al.(2016).Since (cid:0) Υ ,n ( · ) (cid:1) n ∈ N is uniformly bounded, the Koml´os theorem (see Theorem 1.3 inDelbaen and Schachermayer (1999)) yields the following. For each n ∈ N , there existsa convex combination Υ ,n ( · ) ∈ Conv (cid:0) Υ ,k ( · ) , k ≥ n (cid:1) such that (cid:0) Υ ,n ( · ) (cid:1) n ∈ N convergesto some adapted bounded process Υ ( · ) . More precisely, for each n ∈ N , we can findsome random integer N n ≥ and (cid:0) w kn (cid:1) n ≤ k ≤ N n ⊂ [0 , such that N n (cid:88) k = n w kn = 1 and Υ ,n ( · ) = N n (cid:88) k = n w kn Υ ,k ( · ) . For each n ∈ N , define H n ( · ) = N n (cid:88) k = n w kn H n ( · ) , Θ n ( · ) = N n (cid:88) k = n w kn Θ k ( · ) , and Υ u,n ( · ) = N n (cid:88) k = n w kn Υ u,k ( · ) , for all u ∈ { , · · · , l } .Since lim n ↑∞ H n ( · ) = H ( · ) , a.s., we have (cid:12)(cid:12) H n ( · ) − H ( · ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N n (cid:88) k = n w kn H k ( · ) − H ( · ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N n (cid:88) k = n w kn | H k ( · ) − H ( · ) | → as n ↑ ∞ , which implies lim n ↑∞ H n ( · ) = H ( · ) , a.s. Besides, Θ n ( · ) is non-decreasing, asit is a convex combination of non-decreasing processes.Since (cid:0) Υ ,n ( · ) (cid:1) n ∈ N is also uniformly bounded, by the Koml´os theorem again, for each n ∈ N , there exists another convex combination Υ ,n ( · ) ∈ Conv (cid:0) Υ ,k ( · ) , k ≥ n (cid:1) suchthat (cid:0) Υ ,n ( · ) (cid:1) n ∈ N converges to some adapted bounded process Υ ( · ) . With the same con-vex combination for each n ∈ N , define Υ u,n ( · ) , for all u ∈ { , , · · · , l } , H n ( · ) , andsimilarly Θ n ( · ) . In particular, (cid:0) Υ ,n ( · ) (cid:1) n ∈ N still converges to Υ ( · ) , as for each n ∈ N , Υ ,n ( · ) is a convex combination of processes that converge to Υ ( · ) . Similarly, we have lim n ↑∞ H n ( · ) = H ( · ) , a.s. Moreover, Θ n ( · ) is non-decreasing.Iteratively, we construct sequences of processes (cid:0) Υ u,n ( · ) (cid:1) n ∈ N , · · · , (cid:0) Υ lu,n ( · ) (cid:1) n ∈ N , foreach u ∈ { , · · · , l } , and processes H n ( · ) , · · · , H ln ( · ) and Θ n ( · ) , · · · , Θ ln ( · ) in the samemanner. In particular, (cid:0) Υ lu,n ( · ) (cid:1) n ∈ N converges to some adapted bounded process Υ u , foreach u ∈ { , · · · , l } , and we have lim n ↑∞ H ln ( · ) = H ( · ) , a.s. Moreover, Θ ln ( · ) is non-decreasing.By the dominated convergence theorem, we have lim n ↑∞ (cid:90) · l (cid:88) u =1 Υ lu,n ( t )dΛ u ( t ) = (cid:90) · l (cid:88) u =1 Υ u ( t )dΛ u ( t ) , a.s. , which is of finite variation. Therefore, we have H ( · ) = lim n ↑∞ H ln ( · ) = (cid:90) · l (cid:88) u =1 Υ u ( t )dΛ u ( t ) + lim n ↑∞ Θ ln ( · ) , a.s. ince Θ ln ( · ) is non-decreasing and converges, it is of finite variation, which implies theassertion. A.2. Proof of Theorem 1
Proof of Theorem 1.
Assume that the semimartingale µ ( · ) has the Doob-Meyer decom-position µ ( · ) = µ (0) + M ( · ) + V ( · ) , where M ( · ) is a d -dimensional continuous localmartingale and V ( · ) is a d -dimensional finite variation process with M (0) = V (0) = 0 .Let ( W n ) n ∈ N be a non-decreasing sequence of open sets such that the closure of W n isin W , for all n ∈ N . For each κ ∈ N , we consider the stopping time τ κ = inf (cid:40) t ≥ (cid:0) Λ( t ) , µ ( t ) (cid:1) / ∈ W κ or d (cid:88) i,j =1 |(cid:104) M i , M j (cid:105)| ( t ) + d (cid:88) i =1 (cid:90) t d | V i ( u ) | + m (cid:88) v =1 (cid:90) t d | Λ v ( u ) | ≥ κ (cid:41) (36)with inf {∅} = ∞ . Since (cid:0) Λ( · ) , µ ( · ) (cid:1) ∈ W , we have lim κ ↑∞ τ κ = ∞ , a.s. As (cid:83) κ ∈ N { τ κ >t } = Ω , for all t ≥ , to prove that G is regular (Lyapunov), it is equivalent to show that G is regular (Lyapunov) for Λ ( · ∧ τ κ ) and µ ( · ∧ τ κ ) , for all κ ∈ N . Hence, without lossof generality, let us assume that (cid:0) Λ( · ) , µ ( · ) (cid:1) = (cid:0) Λ( · ∧ τ κ ) , µ ( · ∧ τ κ ) (cid:1) , for some κ ∈ N .Without loss of generality, assume that a ij ( · ) is a predictable and uniformly boundedprocess, for all i, j ∈ { , · · · , d } , such that (cid:104) µ i , µ j (cid:105) ( t ) = (cid:90) t a ij ( u )d A ( u ) ≤ κ, t ≥ , where A ( · ) = (cid:80) di =1 (cid:104) µ i , µ i (cid:105) ( · ) . Here, the equality holds according to the Ku-nita–Watanabe theorem and the inequality due to (36).Now, consider a mollification ( G n ,n ) n ,n ∈ N of G defined as in (31). By Lemma 1,Itˆo’s lemma applied to G n ,n yields G n ,n (cid:0) Λ( t ) , µ ( t ) (cid:1) = G n ,n (cid:0) Λ(0) , µ (0) (cid:1) + (cid:90) t d (cid:88) i =1 ∂G n ,n ∂x i (cid:0) Λ( u ) , µ ( u ) (cid:1) d µ i ( u )+ (cid:90) t Υ ,n ,n ( u )d A ( u ) + (cid:90) t m (cid:88) v =1 Υ v,n ,n ( u )dΛ v ( u ) , (37)for all t ≥ , where Υ ,n ,n ( t ) = 12 d (cid:88) i,j =1 ∂ G n ,n ∂x i ∂x j (cid:0) Λ( t ) , µ ( t ) (cid:1) a ij ( t ) and Υ v,n ,n ( t ) = ∂G n ,n ∂λ v (cid:0) Λ( t ) , µ ( t ) (cid:1) , for all v ∈ { , · · · , m } .For all ( λ, x ) ∈ W and i ∈ { , · · · , d } , if (bi) holds, Lemma 1(iii) yields lim n ↑∞ lim n ↑∞ ∂G n ,n ∂x i ( λ, x ) = ∂G∂x i ( λ, x ); f (bii) holds, Lemma 2 yields lim n ↑∞ lim n ↑∞ ∂G n ,n ∂x i ( λ, x ) = f i ( λ, x ) , for some measurable function f i . Then according to Lemma 3, we can find random sub-sequences (cid:0) n k (cid:1) k ∈ N and (cid:0) n k (cid:1) k ∈ N with lim k ↑∞ n k = ∞ = lim k ↑∞ n k such that, if we write G k = G n k ,n k , we have lim k ↑∞ (cid:90) t d (cid:88) i =1 ∂G k ∂x i (cid:0) Λ( u ) , µ ( u ) (cid:1) d µ i ( u ) = F (cid:0) Λ( t ) , µ ( t ) (cid:1) , a.s. , (38)for all t ≥ , where F (cid:0) Λ( t ) , µ ( t ) (cid:1) = (cid:40)(cid:82) t (cid:80) di =1 ∂G∂x i (cid:0) Λ( u ) , µ ( u ) (cid:1) d µ i ( u ) , if (bi) holds (cid:82) t (cid:80) di =1 f i (cid:0) Λ( u ) , µ ( u ) (cid:1) d µ i ( u ) , if (bii) holds . To proceed, write H k ( t ) = G k (cid:0) Λ(0) , µ (0) (cid:1) − G k (cid:0) Λ( t ) , µ ( t ) (cid:1) + (cid:90) t d (cid:88) i =1 ∂G k ∂x i (cid:0) Λ( u ) , µ ( u ) (cid:1) d µ i ( u ) , for all k ∈ N , and H ( t ) = G (cid:0) Λ(0) , µ (0) (cid:1) − G (cid:0) Λ( t ) , µ ( t ) (cid:1) + F (cid:0) Λ( t ) , µ ( t ) (cid:1) , for all t ≥ . Then, (37) with respect to the random subsequences (cid:0) n k (cid:1) k ∈ N and (cid:0) n k (cid:1) k ∈ N is of the form H k ( t ) = − (cid:90) t Υ ,k ( u )d A ( u ) − (cid:90) t m (cid:88) v =1 Υ v,k ( u )dΛ v ( u ) , t ≥ . Note that by Lemma 1(i) and (38), lim k ↑∞ H k ( t ) = H ( t ) , a.s., for all t ≥ .A measurable function DG in Condition of Definition 3 is chosen with components D i G (cid:0) λ, x (cid:1) = (cid:40) ∂G∂x i ( λ, x ) , if (bi) holds f i ( λ, x ) , if (bii) holds , i ∈ { , · · · , d } . Then, as Γ G ( · ) = H ( · ) according to (8), it is enough to show that H ( · ) is of finite variationin the following four cases. Case 1.
Assume that (ai) and (bi) hold. Then by Lemma 1, the processes (cid:0) Υ ,k ( · ) (cid:1) k ∈ N and (cid:0) Υ v,k ( · ) (cid:1) v ∈{ , ··· ,m } ,k ∈ N are uniformly bounded. With l = m + 1 , Λ v ( · ) = Λ v ( · ) and (cid:0) Υ v,k ( · ) (cid:1) k ∈ N = (cid:0) Υ v,k ( · ) (cid:1) k ∈ N , for all v ∈ { , · · · , m } , Λ m +1 ( · ) = A ( · ) , (cid:0) Υ m +1 ,k ( · ) (cid:1) k ∈ N = (cid:0) Υ ,k ( · ) (cid:1) k ∈ N , and (cid:0) Θ k ( · ) (cid:1) k ∈ N = 0 , Lemma 4 yields that H ( · ) is of finite variation oncompact sets. Case 2. ssume that (ai) and (bii) hold. By Lemma 1(iii), the processes (cid:0) Υ v,k ( · ) (cid:1) v ∈{ , ··· ,m } ,k ∈ N are uniformly bounded. Since G is concave in the second argument, for each k ∈ N , G k is also concave in the second argument. As a consequence, the matrix ∇ G k = (cid:18) ∂ G k ∂x i ∂x j (cid:19) i,j ∈{ , ··· ,d } is negative semidefinite. Note that the matrix-valued process a ( · ) = (cid:0) a ij ( · ) (cid:1) i,j ∈{ , ··· ,d } can be chosen to be symmetric and positive semidefinite. Hence, we can find a matrix-valued process σ ( · ) = (cid:0) σ ij ( · ) (cid:1) i,j ∈{ , ··· ,d } such that a ( · ) = σ ( · ) σ (cid:48) ( · ) , which yields a ij ( · ) = (cid:80) dl =1 σ il ( · ) σ jl ( · ) , for all i, j ∈ { , · · · , d } . In this case, d (cid:88) i,j =1 ∂ G k ∂x i ∂x j (cid:0) Λ( t ) , µ ( t ) (cid:1) a ij ( t ) = d (cid:88) i,j =1 ∂ G k ∂x i ∂x j (cid:0) Λ( t ) , µ ( t ) (cid:1) d (cid:88) l =1 σ il ( t ) σ jl ( t )= d (cid:88) l =1 σ l ( t ) ∇ G k (cid:0) Λ( t ) , µ ( t ) (cid:1) σ (cid:48) l ( t ) ≤ , for all t ≥ , where σ l ( · ) is the l -th row of σ ( · ) . Hence, Υ ,k ( t ) ≤ , for all t ≥ , whichimplies that the processes Θ k ( · ) = − (cid:90) · Υ ,k ( t )d A ( t ) , k ∈ N , are non-decreasing. Similar to Case 1, but now with l = m , Lemma 4 yields again that H ( · ) is of finite variation. Case 3.
Assume that (aii) and (bi) hold. By Lemma 1(v), the process (cid:0) Υ ,k ( · ) (cid:1) k ∈ N is uniformlybounded. As G is non-increasing in the v -th dimension of the first argument, so is G k , forall v ∈ { , · · · , m } . Therefore, Υ v,k ( t ) ≤ , for all t ≥ , as Λ( · ) is non-decreasing inthe v -th dimension, for all v ∈ { , · · · , m } . This implies that the processes Θ k ( · ) = − (cid:90) · m (cid:88) v =1 Υ v,k ( t )dΛ v ( t ) , k ∈ N , are non-decreasing. Similar to above, Lemma 4 implies that H ( · ) is of finite variation. Case 4.
Assume that (aii) and (bii) hold. With Θ k ( · ) = − (cid:90) · Υ ,k ( t )d A ( t ) − (cid:90) · m (cid:88) v =1 Υ v,k ( t )dΛ v ( t ) , k ∈ N , Lemma 4 implies again that H ( · ) is of finite variation. It is clear that G is Lyapunov. A.3. Proof of Theorem 2
Proof of Theorem 2.
The following steps are partially inspired by the proof of Theo-rem 3.8 in Karatzas and Ruf (2017). According to Theorem 2.3 in Banner and Ghomrasni l ∈ { , · · · , d } , one can find a measurable function h l : ∆ d → (0 , anda finite variation process B l ( · ) with B l (0) = 0 such that µ l ( · ) = µ l (0) + (cid:90) · d (cid:88) i =1 h l (cid:0) µ ( t ) (cid:1) { µ ( l ) ( t )= µ i ( t ) } d µ i ( t ) + B l ( · ) . (39)Since G is regular for Λ( · ) and µ ( · ) , by Definition 3, there exist a measurable function D G and a finite variation process Γ G ( · ) such that G (cid:0) Λ( · ) , µ ( · ) (cid:1) = G (cid:0) Λ(0) , µ (0) (cid:1) + (cid:90) · d (cid:88) l =1 D l G (cid:0) Λ( t ) , µ ( t ) (cid:1) d µ l ( t ) − Γ G ( · ) . (40)By (39), we have (cid:90) · d (cid:88) l =1 D l G (cid:0) Λ( t ) , µ ( t ) (cid:1) d µ l ( t ) = (cid:90) · d (cid:88) l =1 D l G (cid:0) Λ( t ) , µ ( t ) (cid:1) h l (cid:0) µ ( t ) (cid:1) { µ ( l ) ( t )= µ i ( t ) } d µ i ( t )+ (cid:90) · d (cid:88) l =1 D l G (cid:0) Λ( t ) , µ ( t ) (cid:1) d B l ( t ) . (41)Now consider the measurable function DG : W → R d with components D i G ( λ, x ) = d (cid:88) l =1 D l G (cid:0) λ, R ( x ) (cid:1) h l ( x ) { x ( l ) = x i } , i ∈ { , · · · , d } , and the finite variation process Γ G ( · ) = Γ G ( · ) − (cid:90) · d (cid:88) l =1 D l G (cid:0) Λ( t ) , µ ( t ) (cid:1) d B l ( t ) . Then (40) and (41), together with G ( λ, x ) = G (cid:0) λ, R ( x ) (cid:1) , yield (8), i.e., G is regular for Λ( · ) and µ ( · ) . A.4. An alternative proof for a special case
The proof technique of Theorem VII.31 in Dellacherie and Meyer (1982) suggests analternative argument for the case that conditions (ai) and (bii) in Theorem 1 hold. Wesummarize these ideas in the following result.
Theorem 5.
If a function f : W → R is locally Lipschitz in the first argument andconcave in the second argument, then the process f (cid:0) Λ( · ) , µ ( · ) (cid:1) is a semimartingale.Proof. Assume that the semimartingale µ ( · ) has the Doob-Meyer decomposition µ ( · ) = µ (0) + M ( · ) + V ( · ) , where M ( · ) is a d -dimensional continuous local martingale and V ( · ) is a d -dimensional finite variation process with M (0) = V (0) = 0 .Let ( W n ) n ∈ N be a non-decreasing sequence of open sets such that the closure of W n isin W , for all n ∈ N . For each κ ∈ N , we consider the stopping time τ κ given in (36). ithout loss of generality, let us assume again that (cid:0) Λ( · ) , µ ( · ) (cid:1) = (cid:0) Λ( · ∧ τ κ ) , µ ( · ∧ τ κ ) (cid:1) ,for some κ ∈ N .Since f is locally Lipschitz in both arguments (see Theorem 10.4 in Rockafellar(1970)), we can find a Lipschitz constant L such that, for all s, t ≥ with s ≤ t , wehave (cid:12)(cid:12) f (cid:0) Λ( t ) , µ ( t ) (cid:1) − f (cid:0) Λ( s ) , µ (0) + M ( t ) + V ( s ) (cid:1)(cid:12)(cid:12) ≤ L (cid:32) m (cid:88) v =1 (cid:12)(cid:12) Λ v ( t ) − Λ v ( s ) (cid:12)(cid:12) + d (cid:88) i =1 (cid:12)(cid:12) V i ( t ) − V i ( s ) (cid:12)(cid:12)(cid:33) ≤ L (cid:32) m (cid:88) v =1 (cid:90) ts (cid:12)(cid:12) dΛ v ( u ) (cid:12)(cid:12) + d (cid:88) i =1 (cid:90) ts (cid:12)(cid:12) d V i ( u ) (cid:12)(cid:12)(cid:33) . (42)Let Z ( · ) = − f (cid:0) Λ( · ) , µ ( · ) (cid:1) + L (cid:32) m (cid:88) v =1 (cid:90) · (cid:12)(cid:12) dΛ v ( t ) (cid:12)(cid:12) + d (cid:88) i =1 (cid:90) · (cid:12)(cid:12) d V i ( t ) (cid:12)(cid:12)(cid:33) , then Z ( · ) is bounded. Hence we have E [ Z ( t ) − Z ( s ) |F ( s )] = E (cid:2) f (cid:0) Λ( s ) , µ ( s ) (cid:1) − f (cid:0) Λ( s ) , µ (0) + M ( t ) + V ( s ) (cid:1) |F ( s ) (cid:3) + E (cid:34) f (cid:0) Λ( s ) , µ (0) + M ( t ) + V ( s ) (cid:1) − f (cid:0) Λ( t ) , µ ( t ) (cid:1) + L (cid:32) m (cid:88) v =1 (cid:90) ts (cid:12)(cid:12) dΛ v ( u ) (cid:12)(cid:12) + d (cid:88) i =1 (cid:90) ts (cid:12)(cid:12) d V i ( u ) (cid:12)(cid:12)(cid:33) (cid:12)(cid:12)(cid:12) F ( s ) (cid:35) ≥ E (cid:2) f (cid:0) Λ( s ) , µ ( s ) (cid:1) − f (cid:0) Λ( s ) , µ (0) + M ( t ) + V ( s ) (cid:1) |F ( s ) (cid:3) ≥ , where the first inequality is by (42) and the second inequality holds by Jensen’s inequality.Therefore, Z ( · ) is a submartingale, which makes f (cid:0) Λ( · ) , µ ( · ) (cid:1) a semimartingale. References
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