A pr 2 01 7 Stratonovich representation of semimartingale rank processes

Suppose that X1, . . . , Xn are continuous semimartingales that are reversible and have nondegenerate crossings. Then the corresponding rank processes can be represented by generalized Stratonovich integrals, and this representation can be used to decompose the relative log-return of portfolios generated by functions of ranked market weights. Introduction For n ≥ 2, consider a family of continuous semimartingales X1, . . . , Xn defined on [0, T ] under the usual filtration F t , with quadratic variation processes 〈Xi〉. Let rt(i) be the rank of Xi(t), with rt(i) < rt(j) if Xi(t) > Xj(t) or if Xi(t) = Xj(t) and i < j. The corresponding rank processes X(1), . . . , X(n) are defined by X(rt(i))(t) = Xi(t). We shall show that if the Xi are reversible and have nondegenerate crossings, then the rank processes can be represented by dX(k)(t) = n ∑ i=1 {Xi(t)=X(k)(t)} ◦ dXi(t), a.s., (1) where ◦ d is the generalized Stratonovich integral developed by Russo and Vallois (2007). An Atlas model is a family of positive continuous semimartingales X1, . . . , Xn defined as an Itô integral on [0, T ] by d logXi(t) = ( − g + ng1{rt(i)=n} ) dt+ σ dWi(t), (2) where g and σ are positive constants and (W1, . . . ,Wn) is a Brownian motion (see Fernholz (2002)). Here the Xi represent the capitalizations of the companies in a stock market, and d logXi represents the log-return of the ith stock. We shall show the representation (1) is valid for the Atlas rank processes logX(k). In Fernholz (2016) it was shown that in a stock market with stocks represented by positive continuous semimartingales, under certain conditions the log-return of a portfolio can be decomposed into a structural process and a trading process, and for a portfolio generated by a C function of the market weight processes, these components correspond to the log-change in the generating function and the drift process (see Fernholz (2001)). The Stratonovich representation (1) allows us to extend this decomposition to portfolios generated by C functions of the ranked market weight processes in Atlas models. Itô integrals and Stratonovich integrals LetX and Y be continuous semimartingales on [0, T ] with the filtration F t . Then the Fisk-Stratonovich integral is defined by ∫ t 0 Y (s) ◦ dX(s) , ∫ t 0 Y (s) dX(s) + 1 2 〈Y,X〉t, (3) for t ∈ [0, T ], where the integral on the right hand side is the Itô integral and 〈X,Y 〉t is the cross variation of X and Y over [0, t] (see Karatzas and Shreve (1991)). The Fisk-Stratonovich integral is defined only for semimartingales, but in some cases can be extended to more general integrands. Following Russo and Vallois (2007), Definition 1, for a continuous semimartingale X and a locally integrable process Y , both defined on [0, T ], we define the forward integral, backward integral, and covariation process by ∫ t 0 Y (s) dX(s) , lim ε↓0 ∫ t 0 Y (s) X(s+ ε)−X(s) ε ds (4) INTECH, One Palmer Square, Princeton, NJ 08542. [email protected]. The author thanks Ioannis Karatzas and Mykhaylo Shkolnikov for their invaluable comments and suggestions regarding this research.