Projections of scaled Bessel processs

We investigate the effects of noise reinforcement on a Bessel process of dimension d∈(0,2), and more specifically on the asymptotic behavior of its additive functionals. This leads us to introduce a local time process and its inverse. We identify the latter as an increasing self-similar (time-homogeneous) Markov process, and from this, several explicit results can be deduced.

This paper derives a unified framework for portfolio optimization, derivative pricing, financial modeling, and risk measurement. It is based on the natural assumption that investors prefer more rather than less, in the sense that given two portfolios with the same diffusion coefficient value, the one with the higher drift is preferred. Each such investor is shown to hold an efficient portfolio in the sense of Markowitz with units in the market portfolio and the savings account. The market portfolio of investable wealth is shown to equal a combination of the growth optimal portfolio (GOP) and the savings account. In this setup the capital asset pricing model follows without the use of expected utility functions, Markovianity, or equilibrium assumptions. The expected increase of the discounted value of the GOP is shown to coincide with the expected increase of its discounted underlying value. The discounted GOP has the dynamics of a time transformed squared Bessel process of dimension four. The time transformation is given by the discounted underlying value of the GOP. The squared volatility of the GOP equals the discounted GOP drift, when expressed in units of the discounted GOP. Risk‐neutral derivative pricing and actuarial pricing are generalized by the fair pricing concept, which uses the GOP as numeraire and the real‐world probability measure as pricing measure. An equivalent risk‐neutral martingale measure does not exist under the derived minimal market model.

Pitman transforms and Brownian motion in the interval viewed as an affine alcove

This chapter is a self-contained introduction to the theory of stochastic integration with respect to continuous semimartingales. Apart from some simple facts on the predictable σ-field (which are included with proofs in Section 1.2), we do not need any other result from the general theory of processes. The Doob—Meyer decomposition of the square of a continuous (local) martingale is given by constructing the quadratic variation process directly. Once this is done, the stochastic integral is defined and its properties are obtained in a natural fashion, using arguments common in the theory of the Lebesgue integral, namely, the monotone class theorem and dominated convergence theorem. We first give the definition of the stochastic integral with respect to Brownian motion and then deal with the case of continuous semimartingales in stages. In the last section, we define the integral with respect to a general (r.c.l.l.) semimartingale. However, this time we state without proof some key results, including the Doob—Meyer decomposition. We show that these results, three to be precise, enable us to define the integral in the general case and derive the properties of the integral following the route adopted earlier.KeywordsBrownian MotionDominate Convergence TheoremQuadratic VariationStochastic IntegrationLocal MartingaleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

D. Gilat has shown that any non-negative submartingale (X′, ℱ′.) is equal in law to the absolute value of a martingale (M, ℱ.). This result may be strenthened so that the pairs (X′,ℱ′.) and (¦M¦,ℱ.) are synonomous. In this paper the question of uniqueness of M is considered. Conditions on a local martingale (M, ℱ.) are found that lead to an explicit formula for the finite-dimensional distributions of M in terms of the Doob-Meyer decomposition of the local martingale X′. In many cases of interest the conditions on M are unnecessary. For example, if X′ is the pth power of an Ito integral it is shown that ℒ(M) is unique if p> 1 but not in general if p=1.

The infinitesimal generator of a one-dimensional strictly alpha -stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order alpha and one obtains the fractional Laplacian in the case of symmetric stable processes. Using this relationship, we compute the inverse of the infinitesimal generator on Lizorkin space, from which we can recover the potential if alpha in (0,1) and the recurrent potential if alpha in (1,2). The inverse of the infinitesimal generator is expressed in terms of a linear combination of (right and left) Riemann-Liouville fractional integrals of order alpha . One can then state a class of functions that give semimartingales when applied to strictly stable processes and state a Meyer-Itô theorem with a non-zero (occupational) local time term, providing a generalization of the Tanaka formula given by Tsukada [1]. This result is used to find a Doob-Meyer (or semimartingale) decomposition for |X_t - x|^{gamma } with X a recurrent strictly stable process of index alpha and gamma in (alpha -1,alpha ), generalizing the work of Engelbert and Kurenok [2] to the asymmetric case.

Let M(X) be a family of all equivalent local martingale measures for some locally bounded d-dimensional process X, and V be a positive process. Main result of the paper (Theorem 2.1) states that the process V is a supermartingale whatever Q in M(X), if and only if this process admits the following decomposition: V_t = V_0 + \int_0^t H_s dX_s - C_t, t>= 0, where H is an integrand for X, and C is an adapted increasing process. We call such a representation the optional because, in contrast to Doob-Meyer decomposition, it generally exists only with an adapted (optional) process C. We apply this decomposition to the problem of hedging European and American style contingent claims in a setting of incomplete security markets.

It is well known that upward conditioned Brownian motion is a three-dimensional Bessel process, and that a downward conditioned Bessel process is a Brownian motion. We give a simple proof for this result, which generalizes to any continuous local martingale and clarifies the role of finite versus infinite time in this setting. As a consequence, we can describe the law of regular diffusions that are conditioned upward or downward.

ABSTRACTThe invariance formulas for stopping times of squared Bessel processes with positive index are presented. For a stopping time τ satisfying some relatively mild assumptions and every positive Borel function f, we have the invariance formulawhere and γμ is a gamma random variable independent of σ(τ, R(τ)). The second invariance formula iswhere K is a constant and Δμ is an explicitly given function. The applications of invariance formulas are presented. In particular, new results on the distribution of a killed squared Bessel process, a conditional inverse local time, and the first hitting time of a graph by a squared Bessel process are delivered.

For any integer n = 1, 2, ⋯, limiting laws, as t→∞, for a Bessel process with dimension d (0 < d < 2) penalised by the nth-ranked length of its excursions up to t, or up to the last zero before t, or again up to the first zero after t, are shown to exist, and are characterized.Under these limiting laws Q (n), the canonical process admits a last zero g, and the sequence of the normalized ranked lengths of its excursions up to g, is described in terms of the Poisson-Dirichlet distribution studies, e.g., by Pitman-Yor [PY5]. As n → ∞, Q (n) is shown to converge to Q (∞) the distribution of the Bessel process penalised by an adequate function of its local time at 0, and the sequence of the normalized ranked lengths of the excursions up to g under Q (∞) is then precisely the Poisson-Dirichlet distribution PD1 − d/2 , 0 studied in Pitman-Yor [PY5].KeywordsRanked lengths of excursionsPoisson-Dirichlet distributionsBessel meandersPenalisations

In this note we consider local time of a regular, transient diffusion as a density of a occupation measure, on the one hand, and as a dual predictable projection, on the other hand. The essential tool in our discussion is the Doob-Meyer decomposition for submartingales.

A new global isomorphism theorem is obtained that expresses the local\ntimes of transient regular diffusions under $P^{x,y}$, in terms of related\nGaussian processes. This theorem immediately gives an explicit description\nof the local times of diffusions in terms of $0$th order squared Bessel\nprocesses similar to that of Eisenbaum and Ray's classical description\nin terms of certain randomized fourth order squared Bessel processes. The\nproofs given are very simple. They depend on a new version of Kac's lemma\nfor $h$-transformed Markov processes and employ little more than standard\nlinear algebra. The global isomorphism theorem leads to an elementary\nproof of the\nMarkov property of the local times of diffusions and to other recent\nresults about the local times of general strongly symmetric Markov\nprocesses. The new version of Kac's lemma gives simple, short proofs of\nDynkin's isomorphism theorem and an unconditioned isomorphism theorem due\nto Eisenbaum.

Lévy discovered that the fraction of time a standard one-dimensional Brownian motion B spends positive before time t has arcsine distribution, both for t a fixed time when Bt ≠ 0 almost surely, and for t an inverse local time, when Bt = 0 almost surely. This identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of B away from 0. Similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator, for instance a Bessel process of dimension d for 0 < d < 2.

Squares of Bessel processes and Ray-Knight theorems for Brownian local times