A note on arbitrage‐free pricing of forward contracts in energy markets

Arbitrage theory is used to price forward (futures) contracts in energy markets, where the underlying assets are non‐tradeable. The method is based on the so‐called ‘fitting of the yield curve’ technique from interest rate theory. The spot price dynamics of Schwartz is generalized to multidimensional correlated stochastic processes with Wiener and Lévy noise. Findings are illustrated with examples from oil and electricity markets.     

Mixed Brownian–fractional Brownian model: absence of arbitrage and related topics

We consider a mixed Brownian–fractional-Brownian model of a financial market. The class of self-financing strategies is restricted to Markov-type smooth functions. It is proved that such strategies satisfy a parabolic equation that can be reduced to heat equation. Then it is proved that the mixed model is arbitrage-free. Finally, the capital of the model is presented as the limit of a sequence of semimartingales.     

Spectral characterization of the quadratic variation of mixed Brownian–fractional Brownian motion

Dzhaparidze and Spreij (Stoch Process Appl, 54:165–174, 1994) showed that the quadratic variation of a semimartingale can be approximated using a randomized periodogram. We show that the same approximation is valid for a special class of continuous stochastic processes. This class contains both semimartingales and non-semimartingales. The motivation comes partially from the recent work by Bender et al. (Finance Stoch, 12:441–468, 2008), where it is shown that the quadratic variation of the log-returns determines the hedging strategy.     

Hölder conditions for the local times of multiscale fractional Brownian motion

We establish estimates for the local and uniform moduli of continuity of local times of multiscale fractional Brownian motion $\{X_{\rho }(t),t?0\}$. We also give Chung's form of the law of the iterated logarithm for $X_{\rho }$, this proves that the results on local times are sharp up to multiplicative constant. To cite this article: R. Guerbaz, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

On Itô s formula for multidimensional Brownian motion

Consider a d-dimensional Brownian motion X = (X ¹,…,X ^d ) and a function F which belongs locally to the Sobolev space W ^1,2. We prove an extension of It? s formula where the usual second order terms are replaced by the quadratic covariations [f _k (X), X ^k ] involving the weak first partial derivatives f _k of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), X ^k ] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation results for forward and backward stochastic integrals. Received: 16 March 1998 / Revised version: 4 April 1999     

Quasi sure large deviation for increments of fractional Brownian motion in Hölder norm

In this paper, we first prove Schilder's theorem in Hölder norm (0 ≤ α <1) with respect to Cr,p-capacity. Then, based on this result, we further prove a sharpening of large deviation principle for increments of fractional Brownian motion for C_r,p-capacity in the stronger topology.     

Power law Pólya’s urn and fractional Brownian motion

We introduce a natural family of random walks $S_n$ on $\mathbb{Z }$ that scale to fractional Brownian motion. The increments $X_n := S_n - S_{n-1} \in \{\pm 1\}$ have the property that given $\{ X_k : k < n \}$ , the conditional law of $X_n$ is that of $X_{n - k_n}$ , where $k_n$ is sampled independently from a fixed law $\mu $ on the positive integers. When $\mu $ has a roughly power law decay (precisely, when $\mu $ lies in the domain of attraction of an $\alpha $ -stable subordinator, for $0<\alpha <1/2$ ) the walks scale to fractional Brownian motion with Hurst parameter $\alpha + 1/2$ . The walks are easy to simulate and their increments satisfy an FKG inequality. In a sense we describe, they are the natural “fractional” analogues of simple random walk on $\mathbb{Z }$ .     

Stratonovich’s signatures of Brownian motion determine Brownian sample paths

The signature of Brownian motion in $\mathbb R ^{d}$ over a running time interval $[0,T]$ is the collection of all iterated Stratonovich path integrals along the Brownian motion. We show that, in dimension $d\ge 2$ , almost all Brownian motion sample paths (running up to time $T$ ) are determined by their signature over $[0,T]$ .     

Equidistribution and Brownian motion on the Sierpiński gasket

We introduce several concepts of discrepancy for sequences on the Sierpiski gasket. Furthermore a law of iterated logarithm for the discrepancy of trajectories of Brownian motion is proved. The main tools for this result are regularity properties of the heat kernel on the Sierpiski gasket. Some of the results can be generalized to arbitrary nested fractals in the sense of T. Lindstrøm.With 2 FiguresDedicated to Prof. Edmund Hlawka on the occasion of his 80^th birthdayThe authors are supported by the Austrian Science Foundation project Nr. P10223-PHY and by the Austrian-Italian scientific cooperation program project Nr. 39     

A nonstandard construction of Lévy Brownian motion

Summary A nonstandard construction of Lévy Brownian motion on ^d is presented, which extends R.M. Anderson's nonstandard representation of Brownian motion. It involves a nonstandard construction of white noise and gives as a classical corollary a new white noise integral representation of Lévy Brownian motion. Moreover, a new invariance principle can be deduced in a similar way as Donsker's invariance principles follows from Anderson's construction.     

Exit times from cones in ℝ n of Brownian motion

Summary For a fairly general class of cones inn dimensions (n3) we determine the corresponding distributions of Brownian first exit times. Asymptotic results may then be read off.This paper is a generalization of the author's Ph.D. dissertation completed in May 1984 at the Massachusetts Institute of Technology under the supervision of Professor R.M. DudleyThis research was supported in part by NSF grant DMS-8301367     

On a ogawa–type integral with application to the fractional brownian motion

We construct a deterministic Ogawa–type integral with respect to a continuous function that, in particular, can be a trajectory of the Fractional Brownian motion. This integral is related with the Stratonovich integral and with the integrals introduced by Ciesielski et altri and Zähle. We give a sufficient condition for the integrability of a function in this sense, that does not imply its continuity. Under this sufficient condition, we obtain a Besov regularity property of the indefinite integral. We also study the stochastic Ogawa integral for stochastic processes when integrate with respect to the Fractional Brownian motion of Hurst parameter H ∈ (1/2, 1)     

An Erdös–Révész type law of the iterated logarithm for reflected fractional Brownian motion

Let \(B_{H}=\{B_{H}(t):t\in \mathbb R\}\) be a fractional Brownian motion with Hurst parameter H ∈ (0,1). For the stationary storage process \(Q_{B_{H}}(t)=\sup _{-\infty <s\le t}(B_{H}(t)-B_{H}(s)-(t-s))\), t ≥ 0, we provide a tractable criterion for assessing whether, for any positive, non-decreasing function f, \( {\mathbb P(Q_{B_{H}}(t) > f(t)\, \text { i.o.})}\) equals 0 or 1. Using this criterion we find that, for a family of functions f _p (t), such that \(z_{p}(t)=\mathbb P(\sup _{s\in [0,f_{p}(t)]}Q_{B_{H}}(s)>f_{p}(t))/f_{p}(t)=\mathcal C(t\log ^{1-p} t)^{-1}\), for some \(\mathcal C>0\), \({\mathbb P(Q_{B_{H}}(t) > f_{p}(t)\, \text { i.o.})= 1_{\{p\ge 0\}}}\). Consequently, with \(\xi _{p} (t) = \sup \{s:0\le s\le t, Q_{B_{H}}(s)\ge f_{p}(s)\}\), for p ≥ 0, \(\lim _{t\to \infty }\xi _{p}(t)=\infty \) and \(\limsup _{t\to \infty }(\xi _{p}(t)-t)=0\) a.s. Complementary, we prove an Erdös–Révész type law of the iterated logarithm lower bound on ξ _p (t), i.e., \(\liminf _{t\to \infty }(\xi _{p}(t)-t)/h_{p}(t) = -1\) a.s., p > 1; \(\liminf _{t\to \infty }\log (\xi _{p}(t)/t)/(h_{p}(t)/t) = -1\) a.s., p ∈ (0,1], where h _p (t) = (1/z _p (t))p loglog t.     

On the Collision Local Time of Fractional Brownian Motions

In this paper, the existence and smoothness of the collision local time are proved for two independent fractional Brownian motions, through L2 convergence and Chaos expansion. Furthermore, the regularity of the collision local time process is studied.