Academic literature on the topic 'Random time change of Brownian motion and symmetry'
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Journal articles on the topic "Random time change of Brownian motion and symmetry"
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Székely, B., and T. Szabados. "Strong approximation of continuous local martingales by simple random walks." Studia Scientiarum Mathematicarum Hungarica 41, no. 1 (March 2004): 101–26. http://dx.doi.org/10.1556/012.2004.41.1.6.
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The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walk, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for the independence of the random walks and the discrete time changes or equivalently, for the independence of the DDS Brownian motion and the quadratic variation is proved to be the symmetry of increments of the martingale given the past, which is a reformulation of an earlier result by Ocone [8].
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Lerche, Hans Rudolf, and Ilse Maahs. "Sequential Detection of Drift Change for Brownian Motion with Unknown Sign." gmj 15, no. 4 (December 2008): 713–30. http://dx.doi.org/10.1515/gmj.2008.713.
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Abstract We study tests of power one for the following change-point problem. Suppose one observes a process 𝑊 which is either a Brownian motion without drift or a Brownian motion that has zero drift up to a random time τ after which with equal probability the drift becomes either θ or –θ, where the value of θ > 0 is known. The distribution of τ is also assumed to be known. We search for a stopping time 𝑇* that minimizes an appropriate Bayes risk and give a solution that is asymptotically optimal, when the cost of observation tends to zero.
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Gwynne, Ewain, Jason Miller, and Scott Sheffield. "The Tutte Embedding of the Poisson–Voronoi Tessellation of the Brownian Disk Converges to $$\sqrt{8/3}$$-Liouville Quantum Gravity." Communications in Mathematical Physics 374, no. 2 (November 4, 2019): 735–84. http://dx.doi.org/10.1007/s00220-019-03610-5.
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Abstract Recent works have shown that an instance of a Brownian surface (such as the Brownian map or Brownian disk) a.s. has a canonical conformal structure under which it is equivalent to a $$\sqrt{8/3}$$8/3-Liouville quantum gravity (LQG) surface. In particular, Brownian motion on a Brownian surface is well-defined. The construction in these works is indirect, however, and leaves open a basic question: is Brownian motion on a Brownian surface the limit of simple random walk on increasingly fine discretizations of that surface, the way Brownian motion on $$\mathbb {R}^2$$R2 is the $$\epsilon \rightarrow 0$$ϵ→0 limit of simple random walk on $$\epsilon \mathbb {Z}^2$$ϵZ2? We answer this question affirmatively by showing that Brownian motion on a Brownian surface is (up to time change) the $$\lambda \rightarrow \infty $$λ→∞ limit of simple random walk on the Voronoi tessellation induced by a Poisson point process whose intensity is $$\lambda $$λ times the associated area measure. Among other things, this implies that as $$\lambda \rightarrow \infty $$λ→∞ the Tutte embedding (a.k.a. harmonic embedding) of the discretized Brownian disk converges to the canonical conformal embedding of the continuum Brownian disk, which in turn corresponds to $$\sqrt{8/3}$$8/3-LQG. Along the way, we obtain other independently interesting facts about conformal embeddings of Brownian surfaces, including information about the Euclidean shapes of embedded metric balls and Voronoi cells. For example, we derive moment estimates that imply, in a certain precise sense, that these shapes are unlikely to be very long and thin.
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BAYLY, PHILIP V., and LAWRANCE N. VIRGIN. "EXPERIMENTAL EVIDENCE OF DIFFUSIVE DYNAMICS AND “RANDOM WALKING” IN A SIMPLE DETERMINISTIC MECHANICAL SYSTEM: THE SHAKEN PENDULUM." International Journal of Bifurcation and Chaos 02, no. 04 (December 1992): 983–88. http://dx.doi.org/10.1142/s0218127492000586.
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An experimental model of a simple pendulum, harmonically shaken, displays chaotic dynamics. Moreover, in strongly excited chaotic regimes the time series of total angular displacement, which is rarely examined, wanders unboundedly, displaying a power spectrum which falls off as 1/fα over several decades. This behavior corresponds to deterministic diffusion, which has been found in simulations of nonlinear maps with periodic translational symmetry. The displacement time series obtained by sampling the pendulum displacement once per cycle is self-affine and quantitatively similar to Brownian motion.
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Henderson, Vicky, and Rafał Wojakowski. "On the equivalence of floating- and fixed-strike Asian options." Journal of Applied Probability 39, no. 2 (June 2002): 391–94. http://dx.doi.org/10.1239/jap/1025131434.
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There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating- and fixed-strike Asian options. The proof involves a change of numéraire and time reversal of Brownian motion. Symmetries are very useful in option valuation, and in this case the result allows the use of more established fixed-strike pricing methods to price floating-strike Asian options.
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Henderson, Vicky, and Rafał Wojakowski. "On the equivalence of floating- and fixed-strike Asian options." Journal of Applied Probability 39, no. 02 (June 2002): 391–94. http://dx.doi.org/10.1017/s0021900200022592.
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There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating- and fixed-strike Asian options. The proof involves a change of numéraire and time reversal of Brownian motion. Symmetries are very useful in option valuation, and in this case the result allows the use of more established fixed-strike pricing methods to price floating-strike Asian options.
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CRIENS, DAVID. "A NOTE ON REAL-WORLD AND RISK-NEUTRAL DYNAMICS FOR HEATH–JARROW–MORTON FRAMEWORKS." International Journal of Theoretical and Applied Finance 23, no. 03 (May 2020): 2050020. http://dx.doi.org/10.1142/s021902492050020x.
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We show that for time-inhomogeneous Markovian Heath–Jarrow–Morton models driven by an infinite-dimensional Brownian motion and a Poisson random measure an equivalent change of measure exists whenever the real-world and the risk-neutral dynamics can be defined uniquely and are related via a drift and a jump condition.
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FEDOTOV, SERGEI, and ABBY TAN. "LONG MEMORY STOCHASTIC VOLATILITY IN OPTION PRICING." International Journal of Theoretical and Applied Finance 08, no. 03 (May 2005): 381–92. http://dx.doi.org/10.1142/s0219024905003013.
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The aim of this paper is to present a stochastic model that accounts for the effects of a long-memory in volatility on option pricing. The starting point is the stochastic Black–Scholes equation involving volatility with long-range dependence. We define the stochastic option price as a sum of classical Black–Scholes price and random deviation describing the risk from the random volatility. By using the fact that the option price and random volatility change on different time scales, we derive the asymptotic equation for this deviation involving fractional Brownian motion. The solution to this equation allows us to find the pricing bands for options.
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Kendall, Wilfrid S. "Symbolic computation and the diffusion of shapes of triads." Advances in Applied Probability 20, no. 4 (December 1988): 775–97. http://dx.doi.org/10.2307/1427360.
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This paper introduces the use of symbolic computation (also known as computer algebra) in stochastic analysis and particularly in the Itô calculus. Two related examples are considered: the Clifford-Green theorem on random Gaussian triangles, and a generalization of the D. G. Kendall theorem on the kinematics of shape.The Clifford–Green theorem gives a remarkable characterization of the joint distribution of the squared-side-lengths of n independent Gaussian points in n-space, namely that this distribution is that of n independent exponential random variables conditioned to satisfy all the inequalities requisite if they are to arise as squared-side-lengths from a point-set in n-space. The D. G. Kendall theorem on the diffusion of shape identifies the statistics of the diffusion arising (under a time-change) as the shape of a triangle whose vertices diffuse by Brownian motion in 2-space or 3-space.Symbolic Itô calculus is used to give a new proof of the Clifford-Green theorem, and to generalize the D. G. Kendall theorem to the case of triangles in higher-dimensional space whose vertices diffuse either according to Brownian motion or according to an Ornstein–Uhlenbeck process.
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Kendall, Wilfrid S. "Symbolic computation and the diffusion of shapes of triads." Advances in Applied Probability 20, no. 04 (December 1988): 775–97. http://dx.doi.org/10.1017/s0001867800018371.
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This paper introduces the use of symbolic computation (also known as computer algebra) in stochastic analysis and particularly in the Itô calculus. Two related examples are considered: the Clifford-Green theorem on random Gaussian triangles, and a generalization of the D. G. Kendall theorem on the kinematics of shape. The Clifford–Green theorem gives a remarkable characterization of the joint distribution of the squared-side-lengths of n independent Gaussian points in n-space, namely that this distribution is that of n independent exponential random variables conditioned to satisfy all the inequalities requisite if they are to arise as squared-side-lengths from a point-set in n-space. The D. G. Kendall theorem on the diffusion of shape identifies the statistics of the diffusion arising (under a time-change) as the shape of a triangle whose vertices diffuse by Brownian motion in 2-space or 3-space. Symbolic Itô calculus is used to give a new proof of the Clifford-Green theorem, and to generalize the D. G. Kendall theorem to the case of triangles in higher-dimensional space whose vertices diffuse either according to Brownian motion or according to an Ornstein–Uhlenbeck process.
Dissertations / Theses on the topic "Random time change of Brownian motion and symmetry"
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Ouknine, Anas. "Μοdèles affines généralisées et symétries d'équatiοns aux dérivés partielles." Electronic Thesis or Diss., Normandie, 2024. http://www.theses.fr/2024NORMR085.
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Cette thèse se consacre à étudier les symétries de Lie d'une classe particulière d'équations différentielles partielles (EDP), désignée sous le nom d'équation de Kolmogorov rétrograde. Cette équation joue un rôle essentiel dans le cadre des modèles financiers, notamment en lien avec le modèle de Longstaff-Schwartz, qui est largement utilisé pour la valorisation des options et des produits dérivés.Dans un contexte plus générale, notre étude s'oriente vers l'analyse des symétries de Lie de l'équation de Kolmogorov rétrograde, en introduisant un terme non linéaire. Cette généralisation est significative, car l'équation ainsi modifiée est liée à une équation différentielle stochastique rétrograde et progressive (EDSRP) via la formule de Feynman-Kac généralisée (non linéaire). Nous nous intéressons également à l'exploration des symétries de cette équation stochastique, ainsi qu'à la manière dont les symétries de l'EDP sont connectées à celles de l'EDSRP.Enfin, nous proposons un recalcul des symétries de l'équation différentielle stochastique rétrograde (EDSR) et de l'EDSRP, en adoptant une nouvelle approche. Cette approche se distingue par le fait que le groupe de symétries qui opère sur le temps dépend lui-même du processus $Y$, qui constitue la solution de l'EDSR. Cette dépendance ouvre de nouvelles perspectives sur l'interaction entre les symétries temporelles et les solutions des équationsThis thesis is dedicated to studying the Lie symmetries of a particular class of partialdifferential equations (PDEs), known as the backward Kolmogorov equation. This equa-tion plays a crucial role in financial modeling, particularly in relation to the Longstaff-Schwartz model, which is widely used for pricing options and derivatives.In a broader context, our study focuses on analyzing the Lie symmetries of thebackward Kolmogorov equation by introducing a nonlinear term. This generalization issignificant, as the modified equation is linked to a forward backward stochastic differ-ential equation (FBSDE) through the generalized (nonlinear) Feynman-Kac formula.We also examine the symmetries of this stochastic equation and how the symmetriesof the PDE are connected to those of the BSDE.Finally, we propose a recalculation of the symmetries of the BSDE and FBSDE,adopting a new approach. This approach is distinguished by the fact that the symme-try group acting on time itself depends also on the process Y , which is the solutionof the BSDE. This dependence opens up new perspectives on the interaction betweentemporal symmetries and the solutions of the equations
Book chapters on the topic "Random time change of Brownian motion and symmetry"
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Zinn-Justin, Jean. "From random walk to critical dynamics." In From Random Walks to Random Matrices, 421–50. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198787754.003.0022.
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Chapter 22 studies stochastic dynamical equations, consistent with the detailed balance condition, which are generalized Langevin equations which describe a wide range of phenomena from Brownian motion to critical dynamics in continuous phase transitions. In the latter case, a dynamic action can be associated to the Langevin equation, which can be renormalized with the help of BRST symmetry. Dynamic renormalization group equations, describing critical dynamics, are then derived. Dynamic scaling follows, with a correlation time that exhibits critical slowing down governed by a dynamic exponent. In addition, Jarzinsky’s relation is derived in the case of a time–dependent driving force.
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Zinn-Justin, Jean. "Stochastic differential equations: Langevin, Fokker–Planck (FP) equations." In Quantum Field Theory and Critical Phenomena, 831–56. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780198834625.003.0034.
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This chapter is devoted to the study of Langevin equations, first order in time differential equations, which depend on a random noise, and which belong to a class of stochastic differential equations that describe diffusion processes, or random motion. From a Langevin equation, a Fokker–Planck (FP) equation for the probability distribution of the solutions, at given time, of the Langevin equation can be derived. It is shown that observables averaged over the noise can also be calculated from path integrals, whose integrands define automatically positive measures. The path integrals involve dynamic actions that have automatically a Becchi–Rouet–Stora–Tyutin (BRST) symmetry and, when the driving force derives from a potential, exhibit the simplest form of supersymmetry. In some cases, like Brownian motion on Riemannian manifolds, difficulties appear in the precise definition of stochastic equations, quite similar to the quantization problem encountered in quantum mechanics (QM). Time discretization provides one possible solution to the problem.
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Osorio, Roberto, and Lisa Borland. "Distributions of High-Frequency Stock-Market Observables." In Nonextensive Entropy. Oxford University Press, 2004. http://dx.doi.org/10.1093/oso/9780195159769.003.0023.
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Power laws and scaling are two features that have been known for some time in the distribution of returns (i.e., price fluctuations), and, more recently, in the distribution of volumes (i.e., numbers of shares traded) of financial assets. As in numerous examples in physics, these power laws can be understood as the asymptotic behavior of distributions that derive from nonextensive thermostatistics. Recent applications of the (Q-Gaussian distribution to returns of exchange rates and stock indices are extended here for individual U.S. stocks over very small time intervals and explained in terms of a feedback mechanism in the dynamics of price formation. In addition, we discuss some new empirical findings for the probability density of low volumes and show how the overall volume distribution is described by a function derived from q-exponentials. In March 1900 at the Sorbonne, a 30-year-old student—who had studied under Poincaré—submitted a doctoral thesis [2] that demonstrated an intimate knowledge of trading operations in the Paris Bourse. He proposed a probabilistic method to value some options on rentes, which were then the standard French government bonds. His work was based on the idea that rente prices evolved according to a random-walk process that resulted in a Gaussian distribution of price differences with a dispersion proportional to the square root of time. Although the importance of Louis Bachelier's accomplishment was not recognized by his contemporaries [24], it preceded by five years Einstein's famous independent, but mathematically equivalent, description of diffusion under Brownian motion. The idea of a Gaussian random-walk process (later preferably applied to logarithmic prices) eventually became one of the basic tenets of most twentieth-century quantitative works in finance, including the Black-Scholes [3] complete solution to the option-valuation problem—of which a special case had been solved by Bachelier in his thesis. In the times of the celebrated Black-Scholes solution, however, a change in perspective was already under way. Starting with the groundbreaking works of Mandelbrot [18] and Fama [11], it gradually became apparent that probability distribution functions of price changes of assets (including commodities, stocks, and bonds), indices, and exchange rates do not follow Bachelier's principle of Gaussian (or "normal") behavior.