Article

# On stochastic calculus related to financial assets without semimartingales

(Impact Factor: 1.19). 02/2011; 135(6). DOI: 10.1016/j.bulsci.2011.06.008
Source: arXiv

ABSTRACT

This paper does not suppose a priori that the evolution of the price of a
financial asset is a semimartingale. Since possible strategies of investors are
self-financing, previous prices are forced to be finite quadratic variation
processes. The non-arbitrage property is not excluded if the class
$\mathcal{A}$ of admissible strategies is restricted. The classical notion of
martingale is replaced with the notion of $\mathcal{A}$-martingale. A calculus
related to $\mathcal{A}$-martingales with some examples is developed. Some
applications to no-arbitrage, viability, hedging and the maximization of the
utility of an insider are expanded. We finally revisit some no arbitrage
conditions of Bender-Sottinen-Valkeila type.

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Available from: Francesco Russo, Feb 23, 2014
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• "[43] and [44] Russo and Vallois initiated a theory of stochastic integration via regularization procedures. In later years this was further developed by them and several other authors (see [21] [42] [34] [48] [18] [6] [20], and also the lecture notes [46] and its references). The regularization procedure is connected to the celebrated forward and backward integrals which can be used to integrate with respect to more general processes than semimartingales. "
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ABSTRACT: In this paper we study the forward integral of operator-valued processes with respect to a cylindrical Brownian motion. In particular, we provide conditions under which the approximating sequence of processes of the forward integral, converges to the stochastic integral process with respect to Sobolev norms of smoothness alpha < 1/2. This result will be used to derive a new integration by parts formula for the forward integral.
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• "There are also other no-arbitrage results in related setups, see Bender et al. (2008), Bender (2011) and Coviello et al. (2011). The concept of wealth functionals is needed when defining allowed strategies. "
##### Article: Robust Hedging and Pathwise Calculus
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ABSTRACT: We study the connections of two different pathwise hedging approaches. These approaches are Bender-Sottinen-Valkeila (BSV) by Bender et al. (2008, Pricing by hedging and no-arbitrage beyond semimartingales, finance and stochastics, 12(4), pp. 441-468.) and Cont and Fournié (CF) by Cont and Fournié (2010, Change of variable formulas for non-anticipative functionals on path space, Journal of Functional Analysis, 259(4), pp. 1043-1072; in press, Functional Ito calculus and stochastic integral representation of martingales, Annals of probability). We prove that both approaches give the same pathwise hedges, whenever both of the strategies exist. We also prove BSV-type robust replication result for CF strategies.
Applied Mathematical Finance 10/2011; 20(3). DOI:10.1080/1350486X.2012.725978
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• "If X is a martingale we recall that d − X s = dX s . This was proved in [5], Proposition 4.30. The aim of Proposition 6.13 is to construct effectively such functions ν 1 , . . . "
##### Article: Generalized covariation and extended Fukushima decompositions for Banach valued processes. Application to windows of Dirichlet processes
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ABSTRACT: This paper concerns a class of Banach valued processes which have finite quadratic variation. The notion introduced here generalizes the classical one, of M\'etivier and Pellaumail which is quite restrictive. We make use of the notion of $\chi$-covariation which is a generalized notion of covariation for processes with values in two Banach spaces $B_{1}$ and $B_{2}$. $\chi$ refers to a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$. We investigate some $C^{1}$ type transformations for various classes of stochastic processes admitting a $\chi$-quadratic variation and related properties. If $\X^1$ and $\X^2$ admit a $\chi$-covariation, $F^i: B_i \rightarrow \R$, $i = 1, 2$ are of class $C^1$ with some supplementary assumptions then the covariation of the real processes $F^1(\X^1)$ and $F^2(\X^2)$ exist. A detailed analysis will be devoted to the so-called window processes. Let $X$ be a real continuous process; the $C([-\tau,0])$-valued process $X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$, is called {\it window} process. Special attention is given to transformations of window processes associated with Dirichlet and weak Dirichlet processes. In fact we aim to generalize the following properties valid for $B=\R$. If $\X=X$ is a real valued Dirichlet process and $F:B \rightarrow \R$ of class $C^{1}(B)$ then $F(\X)$ is still a Dirichlet process. If $\X=X$ is a weak Dirichlet process with finite quadratic variation, and $F: C^{0,1}([0,T]\times B)$ is of class $C^{0,1}$, then $[ F(t, \X_t) ]$ is a weak Dirichlet process. We specify corresponding results when $B=C([-\tau,0])$ and $\X=X(\cdot)$. This will consitute a significant Fukushima decomposition for functionals of windows of (weak) Dirichlet processes. As applications, we give a new technique for representing path-dependent random variables.
Infinite Dimensional Analysis Quantum Probability and Related Topics 05/2011; 15(2). DOI:10.1142/S0219025712500075 · 0.73 Impact Factor