| Author | Title | Year | Journal/Proceedings | arXiv | PDF/URL |
|---|---|---|---|---|---|
| Nualart, D. and Swanson, J. | Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion II | 2013 | 1303.0892 | ||
| Abstract: The purpose of this paper is to provide a complete description the convergence in distribution of two subsequences of the signed cubic variation of the fractional Brownian motion with Hurst parameter $H = 1/6$. | |||||
BibTeX:
@unpublished{Nualart2013,
author = {David Nualart and Jason Swanson},
title = {Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion II},
year = {2013},
note = {Preprint, arxiv:1303.0892}
}
|
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| Burdzy, K., Nualart, D. and Swanson, J. | Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion | 2013 | Probability Theory and Related Fields, pp. 1-36 | 1210.1560 | PDF URL |
| Abstract: The purpose of this paper is to study the convergence in distribution of two subsequences of the signed cubic variation of the fractional Brownian motion with Hurst parameter $H=1/6$. We prove that, under some conditions on both subsequences, the limit is a two-dimensional Brownian motion whose components may be correlated and we find explicit formulae for its covariance function. | |||||
BibTeX:
@article{Burdzy2013,
author = {Burdzy, Krzysztof and Nualart, David and Swanson, Jason},
title = {Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion},
journal = {Probability Theory and Related Fields},
publisher = {Springer-Verlag},
year = {2013},
pages = {1-36},
url = {http://dx.doi.org/10.1007/s00440-013-0511-2},
doi = {http://dx.doi.org/10.1007/s00440-013-0511-2}
}
|
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| Swanson, J. | The Calculus of Differentials for the Weak Stratonovich Integral | 2013 | Springer Proceedings in Mathematics & Statistics Vol. 34 Malliavin Calculus and Stochastic Analysis, pp. 95-111 |
1103.0341 | PDF URL |
| Abstract: The weak Stratonovich integral is defined as the limit, in law, of Stratonovich-type symmetric Riemann sums. We derive an explicit expression for the weak Stratonovich integral of $f(B)$ with respect to $g(B)$, where $B$ is a fractional Brownian motion with Hurst parameter 1/6, and $f$ and $g$ are smooth functions. We use this expression to derive an Itô-type formula for this integral. As in the case where $g$ is the identity, the Itô-type formula has a correction term which is a classical Itô integral, and which is related to the so-called signed cubic variation of $g(B)$. Finally, we derive a surprising formula for calculating with differentials. We show that if $dM = X dN$, then $Z dM$ can be written as $ZX dN$ minus a stochastic correction term which is again related to the signed cubic variation. | |||||
BibTeX:
@incollection{Swanson2013,
author = {Swanson, Jason},
title = {The Calculus of Differentials for the Weak Stratonovich Integral},
booktitle = {Malliavin Calculus and Stochastic Analysis},
publisher = {Springer US},
year = {2013},
volume = {34},
pages = {95-111},
url = {http://dx.doi.org/10.1007/978-1-4614-5906-4_5},
doi = {http://dx.doi.org/10.1007/978-1-4614-5906-4_5}
}
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| Swanson, J. | Fluctuations of the empirical quantiles of independent Brownian motions | 2011 | Stochastic Process. Appl. Vol. 121(3), pp. 479-514 |
0812.4102 | PDF URL |
| Abstract: We consider iid Brownian motions, $B_j(t)$, where $B_j(0)$ has a rapidly decreasing, smooth density function $f$. The empirical quantiles, or pointwise order statistics, are denoted by $B_j:n(t)$, and we consider a sequence $Q_n(t) = B_j(n):n(t)$, where $j(n)/ntoal0,1)$. This sequence converges in probability to $q(t)$, the $-quantile of the law of $B_j(t)$. We first show convergence in law in $C[0,$ of $F_n=n^1/2(Q_n-q)$. We then investigate properties of the limit process $F$, including its local covariance structure, and Hölder-continuity and variations of its sample paths. In particular, we find that $F$ has the same local properties as fBm with Hurst parameter $H=1/4$. | |||||
BibTeX:
@article{Swanson2011,
author = {Swanson, Jason},
title = {Fluctuations of the empirical quantiles of independent Brownian motions},
journal = {Stochastic Process. Appl.},
year = {2011},
volume = {121},
number = {3},
pages = {479--514},
url = {http://dx.doi.org/10.1016/j.spa.2010.11.012},
doi = {http://dx.doi.org/10.1016/j.spa.2010.11.012}
}
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| Burdzy, K., Pal, S. & Swanson, J. | Crowding of Brownian spheres | 2010 | ALEA Lat. Am. J. Probab. Math. Stat. Vol. 7, pp. 193-205 |
1002.1057 | PDF URL |
| Abstract: We study two models consisting of reflecting one-dimensional Brownian ``particles'' of positive radius. We show that the stationary empirical distributions for the particle systems do not converge to the harmonic function for the generator of the individual particle process, unlike in the case when the particles are infinitely small. | |||||
BibTeX:
@article{Burdzy2010,
author = {Burdzy, Krzysztof and Pal, Soumik and Swanson, Jason},
title = {Crowding of Brownian spheres},
journal = {ALEA Lat. Am. J. Probab. Math. Stat.},
year = {2010},
volume = {7},
pages = {193--205},
doi = {http://dx.doi.org/10.1016/j.spa.2010.11.012}
}
|
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| Burdzy, K. & Swanson, J. | A change of variable formula with Itô correction term | 2010 | Ann. Probab. Vol. 38(5), pp. 1817-1869 |
0802.3356 | PDF URL |
| Abstract: We consider the solution $u(x,t)$ to a stochastic heat equation. For fixed $x$, the process $F(t)=u(x,t)$ has a nontrivial quartic variation. It follows that $F$ is not a semimartingale, so a stochastic integral with respect to $F$ cannot be defined in the classical Itô sense. We show that for sufficiently differentiable functions $g(x,t)$, a stochastic integral $int g(F(t),t)dF(t)$ exists as a limit of discrete, midpoint style Riemann sums, where the limit is taken in distribution in the Skorohod space of cadlag functions. Moreover, we show that this integral satisfies a change of variables formula with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent of $F$. | |||||
BibTeX:
@article{Burdzy2010a,
author = {Burdzy, Krzysztof and Swanson, Jason},
title = {A change of variable formula with Itô correction term},
journal = {Ann. Probab.},
year = {2010},
volume = {38},
number = {5},
pages = {1817--1869},
url = {http://dx.doi.org/10.1214/09-AOP523},
doi = {http://dx.doi.org/10.1214/09-AOP523}
}
|
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| Nourdin, I., Réveillac, A. & Swanson, J. | The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter $1/6$ | 2010 | Electron. J. Probab. Vol. 15, pp. no. 70, 2117-2162 |
1006.4238 | PDF URL |
| Abstract: Let $B$ be a fractional Brownian motion with Hurst parameter $H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums for $int g(B(s))dB(s)$ do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of càdlàg functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent of $B$. | |||||
BibTeX:
@article{Nourdin2010,
author = {Nourdin, Ivan and Réveillac, Anthony and Swanson, Jason},
title = {The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter $1/6$},
journal = {Electron. J. Probab.},
year = {2010},
volume = {15},
pages = {no. 70, 2117--2162},
url = {http://www.math.washington.edu/~ejpecp/}
}
|
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| Ott, T.J. & Swanson, J. | Asymptotic behavior of a generalized TCP congestion avoidance algorithm | 2007 | J. Appl. Probab. Vol. 44(3), pp. 618-635 |
math/0608476 | PDF URL |
| Abstract: The
Transmission Control Protocol (TCP) is a Transport Protocol used in the
Internet. In tepOtt2005, a more general class of candidate Transport
Protocols called ``protocols in the TCP Paradigm'' is introduced. The
long run objective of studying this class is to find protocols with
promising performance characteristics. This paper studies Markov chain
models derived from protocols in the TCP Paradigm.
Protocols in the TCP Paradigm, as TCP, protect the network from congestion by decreasing the ``Congestion Window'' (the amount of data allowed to be sent but not yet acknowledged) when there is packet loss or packet marking, and increasing it when there is no loss. When loss of different packets are assumed to be independent events and the probability $p$ of loss is assumed to be constant, the protocol gives rise to a Markov chain $W_n$, where $W_n$ is the size of the congestion window after the transmission of the $n$-th packet. For a wide class of such Markov chains, we prove weak convergence results, after appropriate rescaling of time and space, as $p$. The limiting processes are defined by stochastic differential equations. Depending on certain parameter values, the stochastic differential equation can define an Ornstein-Uhlenbeck process or can be driven by a Poisson process. |
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BibTeX:
@article{Ott2007,
author = {Ott, Teunis J. and Swanson, Jason},
title = {Asymptotic behavior of a generalized TCP congestion avoidance algorithm},
journal = {J. Appl. Probab.},
year = {2007},
volume = {44},
number = {3},
pages = {618--635},
url = {http://dx.doi.org/10.1239/jap/1189717533},
doi = {http://dx.doi.org/10.1239/jap/1189717533}
}
|
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| Swanson, J. | Variations of the solution to a stochastic heat equation | 2007 | Ann. Probab. Vol. 35(6), pp. 2122-2159 |
math/0601007 | PDF URL |
| Abstract: We consider the solution to a stochastic heat equation. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time, $F(t)$, has a nontrivial quartic variation. This process, therefore, has infinite quadratic variation and is not a semimartingale. It follows that the classical Itô calculus does not apply. Motivated by heuristic ideas about a possible new calculus for this process, we are led to study modifications of the quadratic variation. Namely, we modify each term in the sum of the squares of the increments so that it has mean zero. We then show that these sums, as functions of $t$, converge weakly to Brownian motion. | |||||
BibTeX:
@article{Swanson2007,
author = {Swanson, Jason},
title = {Variations of the solution to a stochastic heat equation},
journal = {Ann. Probab.},
year = {2007},
volume = {35},
number = {6},
pages = {2122--2159},
url = {http://dx.doi.org/10.1214/009117907000000196},
doi = {http://dx.doi.org/10.1214/009117907000000196}
}
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| Swanson, J. | Weak convergence of the scaled median of independent Brownian motions | 2007 | Probab. Theory Related Fields Vol. 138(1-2), pp. 269-304 |
math/0507524 | PDF URL |
| Abstract: We consider the median of $n$ independent Brownian motions, denoted by $M_n(t)$, and show that $nM_n$ converges weakly to a centered Gaussian process. The chief difficulty is establishing tightness, which is proved through direct estimates on the increments of the median process. An explicit formula is given for the covariance function of the limit process. The limit process is also shown to be Hölder continuous with exponent $ for all $1/4$. | |||||
BibTeX:
@article{Swanson2007a,
author = {Swanson, Jason},
title = {Weak convergence of the scaled median of independent Brownian motions},
journal = {Probab. Theory Related Fields},
year = {2007},
volume = {138},
number = {1-2},
pages = {269--304},
url = {http://dx.doi.org/10.1007/s00440-006-0024-3},
doi = {http://dx.doi.org/10.1007/s00440-006-0024-3}
}
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| Ott, T.J. & Swanson, J. | Stationarity of some processes in transport protocols | 2006 | SIGMETRICS Perform. Eval. Rev. Vol. 34(3), pp. 30-32 |
PDF URL |
|
| Abstract: This
note establishes stationarity of a number of stochastic processes of
interest in the study of Transport Protocols. For many of the processes
studied in this note stationarity had been established before, but for
one class the result is new. For that class, it was counterintuitive
that stationarity was hard to prove. This note also explains why that
class offered such stiff resistance.
The stationarity is proven using Liapunov functions, without first proving tightness by proving boundedness of moments. After the 2006 MAMA workshop simple conditions for existence of such moments were obtained and were added to this note. |
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BibTeX:
@article{Ott2006,
author = {Ott, Teunis J. and Swanson, Jason},
title = {Stationarity of some processes in transport protocols},
journal = {SIGMETRICS Perform. Eval. Rev.},
publisher = {ACM},
year = {2006},
volume = {34},
number = {3},
pages = {30--32},
doi = {http://doi.acm.org/10.1145/1215956.1215969}
}
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