Trading Strategies Generated by Path-dependent Functionals of Market Weights

# Trading Strategies Generated by Path-dependent Functionals of Market Weights

Ioannis Karatzas Department of Mathematics, Columbia University, New York, NY 10027 (E-mail: ik@math.columbia.edu), and Intech Investment Management, One Palmer Square, Suite 441, Princeton, NJ 08542 (E-mail: ikaratzas@intechjanus.com).    Donghan Kim Department of Mathematics, Columbia University, New York, NY 10027 (E-mail: dk2571@columbia.edu).
###### Abstract

Almost twenty years ago, E.R. Fernholz introduced portfolio generating functions which can be used to construct a variety of portfolios, solely in the terms of the individual companies’ market weights. I. Karatzas and J. Ruf recently developed another methodology for the functional construction of portfolios, which leads to very simple conditions for strong relative arbitrage with respect to the market. In this paper, both of these notions of functional portfolio generation are generalized in a pathwise, probability-free setting; portfolio generating functions are substituted by path-dependent functionals, which involve the current market weights, as well as additional bounded-variation functions of past and present market weights. This generalization leads to a wider class of functionally-generated portfolios than was heretofore possible, and yields improved conditions for outperforming the market portfolio over suitable time-horizons.

Keywords and Phrases: Stochastic portfolios, pathwise functional Itô formula, trading strategies, functional generation, regular functionals, strong relative arbitrage.

## 1 Introduction

The concept of ‘functionally generated portfolios’ was introduced by Fernholz (1999, 2002) and has been one of the essential components of stochastic portfolio theory; see Fernholz and Karatzas (2009) for an overview. Portfolios generated by appropriate functions of the individual companies’ market weights have wealth dynamics which can be expressed solely in terms of these weights, and do not involve any stochastic integration. Constructing such portfolios does not require any statistical estimation of parameters, or any optimization. Completely observable quantities such as the current values of ‘market weights’, whose temporal evolution is modeled in terms of continuous semimartingales, are the only ingredients for building these portfolios. Once this structure has been discerned, the mathematics underpinning its construction involves just a simple application of Itô’s rule. Then the goal is to find such portfolios that outperform a reference portfolio, for example, the market portfolio.

Karatzas and Ruf (2017) recently found a new way for the functional generation of trading strategies, which they call ‘additive generation’ as opposed to Fernholz’s ‘multiplicative generation’ of portfolios. This new methodology weakens the assumptions on the market model: asset prices and market weights are continuous semimartingales, and trading strategies are constructed from ‘regular’ functions of the semimartingales without the help of stochastic calculus. Trading strategies generated in this additive way require simpler conditions for strong relative arbitrage with respect to the market over appropriate time horizons; see also Fernholz et al. (2018).

Along a different, but related, development, Föllmer (1981) has shown that Itô calculus can be developed ‘path by path’, without any probability structure. Then Dupire (2009), and Cont and Fournié (2010, 2013) introduced an associated functional Itô calculus. This new type of stochastic calculus extends the classical Itô formula to functionals depending on the entire history of the path, not only on its current value. Versions of this pathwise Itô calculus have recently been applied to various fields in mathematical finance, in particular, to stochastic portfolio theory. Schied et al. (2016) developed path-dependent functional version of Fernholz’s ‘master formula’ for portfolios which are generated multiplicatively by functionals of the entire past evolution of the market weights.

We also present a new sufficient condition for strong relative arbitrage via additively generated trading strategies. The existing sufficient condition in Karatzas and Ruf (2017) requires the generating function to be ‘Lyapunov’, or the corresponding ‘Gamma functional’ to be nondecreasing. In contrast, the new sufficient condition in this paper depends on the intrinsic nondecreasing structure of the generating function itself. This new condition shows that trading strategies outperforming the market portfolio can be generated from a much richer collection of functions, or of functionals depending on the market weights and on an additional argument of finite variation. We give some interesting examples of such additively generated trading strategies and empirical analysis of them.

This paper is organized as follows: Section 2 presents the pathwise functional Itô calculus that will be needed for our purposes. Section 3 first defines trading strategies and regular functionals, then discusses how to generate trading strategies from regular functionals in ways both additive and multiplicative. Section 4 gives sufficient conditions for such trading strategies to generate strong relative arbitrage with respect to the market. Section 5 builds trading strategies depending on current and past values of market weights, in contexts where the use of the pathwise functional Itô calculus is essential. Section 6 gives some examples of trading strategies additively generated from entropic functionals and corresponding sufficient conditions for strong arbitrage. Section 7 contains empirical results of portfolios discussed in Section 6. Finally, Section 8 concludes.

## 2 Pathwise functional Itô calculus

In what follows, we let be a -valued continuous function, representing a -dimensional vector of assets whose values change over time, and is the value of the th asset at time . As in Schied et al. (2016), we require that the components of admit continuous covariations in the pathwise sense with respect to a given, refining sequence of partitions of . The sequence is such that each partition satisfies , for each , as well as , and the mesh size of decreases to zero on each compact interval as . We fix such a sequence of partitions for the remainder of this paper. We denote by the successor of a given in , i.e.,

 t′n=min{u∈Tn|u>t}. (2.1)

With this notation, we define the pathwise covariation of and on the interval , denoted by for any , as the limit of the sequence

 ∑s∈Tns≤t(Xi(s′n)−Xi(s))(Xj(s′n)−Xj(s)),n∈N.

We assume that this limit is finite, and that the resulting mapping is continuous. We also define the pathwise quadratic variation of by as usual.

Here, we need to emphasize that the existence of the pathwise covariations and quadratic variations for the components of depends heavily on the choice of the sequence of partitions. Example 5.3.2. in Cont (2016), and the arguments following this example, illustrate this fact. Also, we note that the existence of the pathwise covariations and quadratic variations is required for Itô’s formula to hold in a pathwise sense.

We will consider only the finite time-horizon case, so we fix for the remainder of this paper. For an open subset of a Euclidean space, we denote by the space of right continuous -valued functions with left limits on . As usual, we denote by the space of continuous -valued functions; whereas stands for the space of those functions in which are of bounded variation, and stands for the space of continuous functions in .

Let be an open set of , and an open subset of . Here, is the dimension of our vector function mentioned earlier, and is the dimension for some additional vector function of finite variation on compact time-intervals. For in , and for in , we denote the distance

 d∞((X,A),(~X,~A)):=supu∈[0,T]|X(u)−~X(u)|+supu∈[0,T]|A(u)−~A(u)|.

Next, we state a pathwise functional Itô formula, as well as some definitions and notation based on Appendix of Schied et al. (2016).

### 2.1 Non-anticipative functional

A functional is called non-anticipative, if holds for each . Here we denote by the function, or “path”, stopped at , for any given path defined on . We use the term “path” instead of function, to emphasize that it represents an evolution of value which changes over time. We present some concepts regarding a non-anticipative functional , acting on a space of such paths, as follows:

1. is called left-continuous if, for any given and for any given , there exists such that, for all and with , we have

 |F(t,X,A)−F(~t,~X,~A)|<ϵ.
2. is called boundedness-preserving if, for any given compact sets and , there exists a constant such that

 |F(t,X,A)|≤CK,L

holds for all .

### 2.2 Horizonal and vertical derivatives

We present now two notions of differentiability for a given non-anticipative functional . Intuitively, the first of these notions, horizontal differentiation, concerns the arguments that correspond to functions of bounded variation; we set here . The second notion, vertical differentiation, concerns the arguments that correspond to general, right continuous functions with left limits.

###### Definition 2.1 (Horizontal derivative).

A non-anticipative functional is called horizontally differentiable if, for each , the limits

 D0F(t,X,A):=limh↓0F(t,Xt−h,At−h)−F(t−h,Xt−h,At−h)h DkF(t,X,A):=limh↓0F(t,Xt−h,At−h1,⋯,Atk,⋯,At−hm)−F(t,Xt−h,At−h)Ak(t)−Ak(t−h),k=1,⋯,m,

exist and are finite. We use here the convention . Then, the horizontal derivative of at is given by the -dimensional vector

 DF(t,X,A)=(D0F(t,X,A),D1F(t,X,A),⋯,DmF(t,X,A))′,

and the map

 DF:[0,T]×D([0,T],U)×BV([0,T],W) →Rm+1 (t,X,A) ↦DF(t,X,A)

defines a non-anticipative vector-valued functional called the horizontal derivative of , with the convention .

We note that our definition of horizontal derivative is based on the left-hand limit as in Schied et al. (2016), and is different from that of Dupire (2009) and Cont and Fournié (2010). Thus, only the past evolution of the underlying path is relevant for this definition, while no assumptions on future values are imposed.

Now let us fix , and , and define the vertical perturbation of the stopped path , as the right-continuous path obtained by shifting the value of the path stopped at by the amount on the interval , i.e.,

 Xt,h(u):=Xt(u)+h\mathbbm1[t,T](u),0≤u≤T.
###### Definition 2.2 (Vertical derivative).

A non-anticipative functional is called vertically differentiable at if the map is differentiable at . In this case, the ith vertical partial derivative of the functional at is defined as

 ∂iF(t,X,A) :=limθ→0F(t,Xt,θei,A)−F(t,Xt,A)θ =limθ→0F(t,Xt,θei,A)−F(t,X,A)θ,i=1,⋯,d.

Here is the ith unit vector in , and the last equality holds because is non-anticipative. The corresponding gradient is denoted by

and is called the vertical derivative of at . If is vertically differentiable at every triple in , then the mapping

 ∇XF:[0,T]×D([0,T],U)×BV([0,T],W) →Rd (t,X,A) ↦∇XF(t,X,A)

defines a non-anticipative functional with values in , called the vertical derivative of .

We can iterate the operations of horizontal and vertical differentiation, to define higher-order horizontal and vertical derivatives as long as the functional admits horizontal and vertical derivatives. In particular, we define the mixed vertical derivatives

 ∂i,jF:=∂i(∂jF),1≤i,j≤d.
###### Definition 2.3.

We denote by the set of all non-anticipative functionals which satisfy the following conditions.

1. is continuous at fixed times , uniformly in over compact intervals. That is, for all and , there exists such that for all with , we have

 |F(t,X,A)−F(t,~X,~A)|<ϵ.
2. is j-times horizontally differentiable and k-times vertically differentiable.

3. and all its horizontal and vertical derivatives are left-continuous and boundedness-preserving.

### 2.3 Pathwise functional Itô formula

We are in a position now, to state the celebrated functional change-of-variable formula, in the form of Schied et al. (2016) or Schied and Voloshchenko (2016). The proof can be found in Schied and Voloshchenko (2016). As before, we let be a fixed refining sequence of partitions of , and suppose a continuous function admits finite covariations , along the sequence of partitions .

###### Theorem 2.4 (Pathwise functional Itô formula).

Let and be given functions, recall the notation of (2.1), define the -th piecewise-constant approximation of by

 Xn(t):=∑s∈TnX(s′n)\mathbbm1[s,s′n)(t)+X(T)\mathbbm1{T}(t),0≤t≤T,

and let . Then, for any functional , the pathwise Itô integral along , namely

 ∫T0∇XF(s,X,A)dX(s):=limn→∞∑s∈Tn∇XF(s,Xs−n,A)⋅(X(s′n)−X(s)),

exists; and with , we have the expansion

 F(T,X,A)−F(0,X,A)= ∫T0∇XF(s,X,A)dX(s)+m∑k=0∫T0DkF(s,X,A)dAk(s) +12d∑i,j=1∫T0∂2i,jF(s,X,A)d⟨Xi,Xj⟩(s).

## 3 Trading strategies generated by path-dependent functionals

As in the previous section, let be a -valued continuous function which admits continuous covariations with respect to a refining sequence of partitions of and be an additional vector function of finite variation. For the purpose of this section, the components of will denote the value processes of tradable assets, and eventually stand for the vector of market weights in an equity market. At the same time, the components of will model the evolution of an observable, but non-tradable quantity related to the market weights. We have the following definition of trading strategy with respect to the pair in the manner of Karatzas and Ruf (2017).

For the pair of a -dimensional function and an -dimensional function , suppose that is a -dimensional function with a representation

 ϑi(⋅)=Θi(⋅,X,A),i=1,⋯,d,

for a vector of non-anticipative functionals, for which we can define an integral with respect to ; we write , to express this. We shall say that is a trading strategy with respect to if it is ‘self-financed’, in the sense that

 Vϑ(⋅,X)−Vϑ(0,X)=∫⋅0d∑i=1ϑi(t)dXi(t) (3.1)

holds. Here and in what follows,

 Vϑ(t,X):=d∑i=1ϑi(t)Xi(t),0≤t≤T (3.2)

denotes the value process of the strategy at time .

The interpretation here, is that stands for the “number of shares” invested in asset at time . If is the price of this asset, then is the dollar amount invested in asset at time , and is the total value of investment across all assets.

The preceding Itô formula in Theorem 2.4 suggests that integrands of the special form , for some non-anticipative functional , play a very important role for integrators that admit finite covariations , along an appropriate nested sequence of partitions. This gives rise to the following definition.

Let be a -dimensional function in , and be an -dimensional function in . A -dimensional trading strategy in is called an admissible trading strategy for the pair , if there exists a non-anticipative functional in the space , such that

 ϑ(t)=∇XF(t,X,A),0≤t≤T. (3.3)

If is an admissible trading strategy for , the last integral of (3.1) above can be viewed as either the usual vector Itô integral (when is a continuous vector semimartingale in a probabilistic setting), or as the pathwise Itô integral (in the context of our Theorem 2.4).

In the following, we will define a regular functional for the -dimensional continuous function and the -dimensional function in in a manner similar to that of Karatzas and Ruf (2017).

###### Definition 3.3 (Regular functional).

We say that a non-anticipative functional in is regular for the pair of a -dimensional continuous function and a function , if the continuous function

 ΓG(t):=G(0,X,A)−G(t,X,A)+∫t0d∑i=1ϑi(s)dXi(s),0≤t≤T (3.4)

has finite variation on compact intervals of . Here, is the function of (3.3) right above, with components

 ϑi(t):=∂iG(t,X,A),i=1,⋯,d,0≤t≤T. (3.5)
###### Remark 3.4.

In order to define a pathwise functional Itô integral and be able to use pathwise functional Itô formulas, we need a sufficiently smooth (in general, at least ) functional, and an integrand which can be cast in the form of a vertical derivative of this functional. Thus, thanks to the above definition, we can always apply the pathwise functional Itô formula (Theorem 2.4) to the functional as in Definition 3.3 above, and get another expression for the so-called “Gamma functional” in (3.4); namely,

 ΓG(t)=−m∑k=0∫t0DkG(s,X,A)dAk(s)−12d∑i,j=1∫t0∂2i,jG(s,X,A)d⟨Xi,Xj⟩(s). (3.6)

Here and are, respectively, the horizontal derivative and the second-order vertical derivative of at .

The difference in Definition 3.3 here, with Definition 3.1 of Karatzas and Ruf (2017), should be noted and stressed. In Karatzas and Ruf (2017), the integrand need not be the form of ‘gradient’ of a regular function . Here, we need the special structure of (3.5) for the integrand; this is the “price to pay” for being able to work in a pathwise, probability-free setting, without having to invoke the theory of rough paths.

### 3.1 Trading strategies depending on market weights

We place ourselves from now onward in a frictionless equity market with a fixed number of companies. For an open set with , we also consider a vector of continuous functions where represents the capitalization of the company at time . Here we take and allow to vanish at some time for all , but assume that the total capitalization does not vanish at any time . Then we define another vector of continuous functions that consists of the companies’ relative market weights

 μi(t):=Si(t)Σ(t)=Si(t)S1(t)+⋯+Sd(t),t∈[0,T],i=1,⋯,d. (3.7)

We also assume that the components of admit finite covariations , along a nested sequence of partitions of . In the following, we will consider only regular functionals of the form which depend on the vector of market weights and on some additional function . Examples of such functions appear in (4.3), (4.4).

###### Remark 3.5.

In order to simplify the expression of (3.6) and to do concave analysis in a manner analogous to that of Karatzas and Ruf (2017), we can make depend only on the function . For example, if we consider the Gibbs entropy function and set , elementary computations show that the first term on the right-hand side of (3.6) vanishes and we obtain ; this coincides with Example 5.3 of Karatzas and Ruf (2017).

We would like now to introduce an additively-generated trading strategy, starting from a regular functional. For this, we will need a result from Karatzas and Ruf (2017). For any given functional which is regular for the pair , where is the vector of market weights and an appropriate function in , we consider the vector with components

 ϑi:=∂iG(⋅,μ,A),i=1,⋯,d, (3.8)

as in (3.5) of the Definition 3.3, and the vector with components

 φi(t):=ϑi(t)−Qϑ(t)−C(0),i=1,⋯,d,0≤t≤T. (3.9)

Here,

 Qϑ(t):=Vϑ(t)−Vϑ(0)−∫t0d∑i=1ϑi(s)dμi(s) (3.10)

is the “defect of self-financibility” at time of the integrand in (3.8), and

 C(0):=d∑i=1∂iG(0,μ,A)μi(0)−G(0,μ,A) (3.11)

is the “defect of balance” at time of the regular functional . By analogy with Proposition 2.3 of Karatzas and Ruf (2017), the vector of (3.9), (3.8) defines a trading strategy with respect to .

We say that the trading strategy of the form (3.9), (3.8) is additively generated by the functional , which is assumed to be regular for the vector of market weights.

###### Proposition 3.7.

The trading strategy , generated additively as in (3.9) by a regular functional for the pair , where is the vector of market weights and , has value

 Vφ(t)=G(t,μ,A)+ΓG(t),0≤t≤T (3.12)

as in Definitions 3.1 and 3.3, and its components can be represented, for , in the form

 φi(t) =∂iG(t,μ,A)+ΓG(t)+G(t,μ,A)−d∑j=1μj(t)∂jG(t,μ,A) (3.13) =Vφ(t)+∂iG(t,μ,A)−d∑j=1μj(t)∂jG(t,μ,A).

We can think of in (3.4), (3.6) and (3.12), as expressing the “cumulative earnings” of the strategy around the “baseline” .

###### Proof.

The proof is exactly the same as the proof of Proposition 4.3 of Karatzas and Ruf (2017), if we change , there, into , . ∎

###### Remark 3.8.
1. When the functional in Proposition 3.7 satisfies the ‘balance’ condition,

 G(⋅,μ,A)=d∑j=1μj(⋅)∂jG(⋅,μ,A), (3.14)

the additively generated trading strategy in (3.13) takes the considerably simpler form

 φi(t)=∂iG(t,μ,A)+ΓG(t),i=1,⋯,d. (3.15)
2. For an additively generated trading strategy with strictly positive value process , the corresponding portfolio weights are defined as

 πi:=φiμiVφ=φiμi∑di=1φiμi,i=1,⋯,d,

or with the help of (3.12) and (3.13), as

 πi(t)=μi(t)(1+1G(t,μ,A)+ΓG(t)(∂iG(t,μ,A)−d∑j=1μj(t)∂jG(t,μ,A))), (3.16)

for .

### 3.3 Multiplicatively generated trading strategies

Next, we introduce the notion of multiplicatively generated trading strategy. We suppose that a functional is regular as in Definition 3.3 for the pair , where is the vector of market weights and is some additional function in , and that is locally bounded. This holds, for example, if is bounded away from zero. We consider the vector function defined by

 ηi:=ϑi×exp(∫⋅0dΓG(t)G(t,μ,A))=∂iG(⋅,μ,A)×exp(∫⋅0dΓG(t)G(t,μ,A)) (3.17)

in the notation of (3.4), (3.8) for . The integral here is well-defined, as is assumed to be locally bounded. Moreover, we have , since from Definition 3.1, and the exponential process is again locally bounded. As before, we turn this into a trading strategy by setting

 ψi:=ηi−Qη−C(0),i=1,⋯,d (3.18)

in the manner of (3.9), and with , defined as in (3.10) and (3.11).

###### Definition 3.9 (Multiplicative generation).

The trading strategy of (3.18), (3.17) is said to be multiplicatively generated by the functional .

###### Proposition 3.10.

The trading strategy , generated as in (3.18) by a given functional which is regular for the pair with vector of market weights and a suitable such that is locally bounded, has value process

 Vψ=G(⋅,μ,A)exp(∫⋅0dΓG(t)G(t,μ,A))>0 (3.19)

in the notation of (3.4). This strategy can be represented for in the form

 ψi(t)=Vψ(t)(1+1G(t,μ,A)(∂iG(t,μ,A)−d∑j=1μj(t)∂jG(t,μ,A))). (3.20)
###### Proof.

We follow the proof of Proposition 4.8 in Karatzas and Ruf (2017), using the pathwise functional Itô formula instead of the standard Itô formula. If we denote the exponential in (3.19) by , the pathwise functional Itô formula (Theorem 2.4) yields

 G(t,μ,A)K(t) =G(0,μ,A)K(0) +∫t0d∑i=1∂iG(s,μ,A)K(s)dμi(s)+∫t0K(s)dΓG(s) +∫t0m∑i=0DiG(s,μ,A)K(s)dAi(s) +12∫t0d∑i=1d∑j=1∂2i,jG(s,μ,A)K(s)d⟨μi,μj⟩(s) =G(0,μ,A)K(0) +∫t0d∑i=1∂iG(s,μ,A)K(s)dμi(s) =G(0,μ,A)K(0) +∫t0d∑i=1ηi(s)dμi(s) =G(0,μ,A)K(0) +∫t0d∑i=1ψi(s)dμi(s),0≤t≤T.

Here, the second equality uses the expression in (3.6), and the last equality relies on Proposition 2.3 of Karatzas and Ruf (2017). Since (3.19) holds at time zero, it follows that (3.19) holds at any time between and . The justification of (3.20) is exactly the same with that of Proposition 4.8 in Karatzas and Ruf (2017). ∎

###### Remark 3.11.
1. The multiplicatively generated trading strategy in (3.20) takes the simpler form

 ψi(t)=∂iG(t,μ,A)exp(∫t0dΓG(s)G(s,μ,A)),i=1,⋯,d (3.21)

when the functional in Proposition 3.10 is ‘balanced’ as in (3.14).

2. The portfolio weights corresponding to the multiplicatively generated trading strategy , are similarly defined as

 Πi:=ψiμiVψ=ψiμi∑di=1ψiμi,i=1,⋯,d,

and with the help of (3.19) and (3.20), takes the form

 Πi(t)=μi(t)(1+1G(t,μ,A)(∂iG(t,μ,A)−d∑j=1μj(t)∂jG(t,μ,A))),i=1,⋯,d.

For a functional that satisfies the “balance” condition (3.14), this simplifies to

 Πi(t)=μi(t) ∂iG(t,μ,A)G(t,μ,A),i=1,⋯,d.

## 4 Sufficient conditions for strong relative arbitrage

We consider the vector of market weights as in (3.7). For a given trading strategy with respect to the market weights , let us recall the value process from Definition 3.1. As we will always consider trading strategies with respect to market weights, we write instead of from now on. For some fixed , we say that is strong relative arbitrage with respect to the market over the time-horizon , if we have

 Vφ(t)≥0,∀ t∈[0,T∗], (4.1)

along with

 Vφ(T∗)>Vφ(0). (4.2)
###### Remark 4.1.

The notion of strong relative arbitrage defined above does not depend on any probability measure and is slightly more strict than the existing definition of strong relative arbitrage. The classical definition involves an underlying filtered probability space, and posits that the market weights should be continuous, adapted stochastic processes on this space. Also, there are two types of classical arbitrage; relative arbitrage and ‘strong’ relative arbitrage as in Definition 4.1 of Fernholz et al. (2018). In this old definition, an underlying probability measure is essential in defining this ‘weak’ version of relative arbitrage. However, if we say that is strong relative arbitrage when (4.2) holds for ‘every’ realization of , instead of ‘almost sure’ realization of , the notion of strong relative arbitrage can be established without referring to any probability structure. Since we constructed trading strategies in a pathwise, probability-free setting, the ‘strong’ version of relative arbitrage is a more appropriate concept of arbitrage for this paper, and we adopt the above strict definition of strong relative arbitrage from now on.

The value process of a trading strategy generated functionally, either additively or multiplicatively, admits a quite simple representation in terms of the generating functional and the derived functional as in (3.12) and (3.19). This simple representation provides in turn nice sufficient conditions for strong relative arbitrage with respect to the market; for example, as in Theorem 5.1 and Theorem 5.2 of Karatzas and Ruf (2017). In this section, we find such conditions on trading strategies generated by a pathwise functional which depends not only on the vector of market weights , but also on an additional finite-variation process related to . We also give a new sufficient condition leading to strong relative arbitrage for additively generated trading strategies, which is different from Theorem 5.1 of Karatzas and Ruf (2017).

Until now, we have not specified the -dimensional function , so it is time to consider some plausible candidates for this finite variation function. A first suitable candidate would be the -dimensional vector

 A=⟨μ⟩=(⟨μ1⟩,⟨μ2⟩,⋯,⟨μd⟩)′ (4.3)

of quadratic variation of market weights. We can also think about a more general candidate; namely, the -valued covariation process of market weights. Here, is the notation for symmetric positive matrices, and we will use double bracket to distinguish this -dimensional vector from (4.3): namely,

 A=⟨⟨μ⟩⟩,(A)i,j=⟨μi,μj⟩1≤i,j≤d. (4.4)

The advantage of choosing as in (4.4), is that we can match the integrators of the two integrals in (3.6), and the resulting expression for can then be cast as one integral.

There are many other functions of finite variation which can be candidates for the process . We list some examples below:

1. The moving average of defined by

 ¯μi(t):={1δ∫t0μi(s)ds+1δ∫0t−δμi(0)ds,t∈[0,δ),1δ∫tt−δμi(s)ds,t∈[δ,T],i=1,⋯,d.
2. The running maximum of the market weights with the components , and the running minimum of the market weights with the components for .

3. The local time process of the continuous semimartingale at the origin, for . We call this process the “collision local time of order ” for the ranked market weights

 μ(1):=maxiμi≥μ(2)≥⋯≥μ