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id stringlengths 40 40	title stringlengths 15 120	text stringlengths 41 3.14k	source stringclasses 2 values
a17be83e18d137169ea7a180bb1d1699142ff944	the signature can also be defined for geometric	the signature can also be defined for geometric rough paths. let x {\displaystyle \mathbf {x} } be a geometric rough path and let x (n) {\displaystyle \mathbf {x} (n)} be a sequence of paths with finite total variation such that	wikipedia
e6759690ff66105de1bcb5f55bf8cf2ea2508fae	let v: r n → l (r d,	let v: r n → l (r d, r n) {\displaystyle v:\mathbb {r} ^{n}\rightarrow l(\mathbb {r} ^{d},\mathbb {r} ^{n})} be functions that has at least ⌊ γ ⌋ {\displaystyle \lfloor \gamma \rfloor } derivatives and the ⌊ γ ⌋ {\displaystyle \lfloor \gamma \rfloor } -th derivatives are γ − ⌊ γ ⌋ {\displaystyle \gamma -\lfloor \gamma ...	wikipedia
f5fcc22163c19b8ce14c71dbaf0ec462dc5943ac	where x j {\displaystyle \mathbf {x} ^{j}} denotes	where x j {\displaystyle \mathbf {x} ^{j}} denotes the j {\displaystyle j} -th tensor component of x {\displaystyle \mathbf {x} }.let γ ≥ 1 {\displaystyle \gamma \geq 1}. let f: r d → r n {\displaystyle f:\mathbb {r} ^{d}\rightarrow \mathbb {r} ^{n}} be an ⌊ γ ⌋ {\displaystyle \lfloor \gamma \rfloor } -times differenti...	wikipedia
15399cd4ebfafc6bcf37412b52c67a44912b1dfd	let x = (1, x 1, …, x	let x = (1, x 1, …, x ⌊ p ⌋) {\displaystyle \mathbf {x} =(1,\mathbf {x} ^{1},\ldots,\mathbf {x} ^{\lfloor p\rfloor })} be a p {\displaystyle p} -geometric rough path satisfying the hölder condition that there exists m > 0 {\displaystyle m>0}, for all 0 ≤ s ≤ t ≤ 1 {\displaystyle 0\leq s\leq t\leq 1} and all j = 1,, 2, ...	wikipedia
1df748209cb8c94b0c0abd64c8fb9b8d1e6a6fdf	on r d {\displaystyle \mathbb {r} ^{d}}, a	on r d {\displaystyle \mathbb {r} ^{d}}, a γ {\displaystyle \gamma } - controlled path is a function y s = (y s 0, y s 1, …, y s ⌊ γ ⌋) {\displaystyle \mathbf {y} _{s}=(\mathbf {y} _{s}^{0},\mathbf {y} _{s}^{1},\ldots,\mathbf {y} _{s}^{\lfloor \gamma \rfloor })} such that y j: → l ((r d) ⊗ j + 1, r n) {\displaystyle \m...	wikipedia
5ee8864839e71355f25a6a39330b7990f771ab60	in fact, rough path theory gives the existence	in fact, rough path theory gives the existence and uniqueness of ϕ s, t (x) {\displaystyle \phi _{s,t}(x)} not only outside a null set independent of s {\displaystyle s}, t {\displaystyle t} and x {\displaystyle x} but also of the drift b {\displaystyle b} and the volatility σ {\displaystyle \sigma }.	wikipedia
a9437ad14307aff545ff7b5ad6b651b46b5a72fc	the universal limit theorem once again reduces this	the universal limit theorem once again reduces this problem to whether the brownian rough path b s, t {\displaystyle \mathbf {b_{s,t}} } exists and satisfies the multiplicative property that for all s ≤ u ≤ t {\displaystyle s\leq u\leq t},	wikipedia
518614d7816c9e01d5ad9fcd7f9d139297e24bc9	has a unique solution in the sense of	has a unique solution in the sense of rough path. a basic question in the theory of stochastic flow is whether the flow map ϕ s, t (x) {\displaystyle \phi _{s,t}(x)} exists and satisfy the cocyclic property that for all s ≤ u ≤ t {\displaystyle s\leq u\leq t},	wikipedia
37844322c3ce99593ef9b314b1e36fb35b2ca9c7	once again, let b t {\displaystyle b_{t}} be	once again, let b t {\displaystyle b_{t}} be a d {\displaystyle d} -dimensional brownian motion. assume that the drift term b {\displaystyle b} and the volatility term σ {\displaystyle \sigma } has sufficient regularity so that the stochastic differential equation	wikipedia
321d87209982c2f01c6c29b5fad3cd2e3b3c8090	the universal limit theorem guarantees that the itô	the universal limit theorem guarantees that the itô map sending the control path (t, ε b t) {\displaystyle (t,{\sqrt {\varepsilon }}b_{t})} to the solution x ε {\displaystyle x^{\varepsilon }} is a continuous map from the p {\displaystyle p} -variation topology to the p {\displaystyle p} -variation topology (and hence ...	wikipedia
0611c4eb0a6c273c73dcaa4546988863aab5289b	the freidlin wentzell's large deviation theory aims to	the freidlin wentzell's large deviation theory aims to study the asymptotic behavior, as ϵ → 0 {\displaystyle \epsilon \rightarrow 0}, of p {\displaystyle \mathbb {p} } for closed or open sets f {\displaystyle f} with respect to the uniform topology.	wikipedia
0dcc5eb835a855b592edc8050b5f800f2f698b29	let b t {\displaystyle b_{t}} be a d	let b t {\displaystyle b_{t}} be a d {\displaystyle d} -dimensional standard brownian motion. let b: r n → r d {\displaystyle b:\mathbb {r} ^{n}\rightarrow \mathbb {r} ^{d}} and σ: r n → l (r d, r n) {\displaystyle \sigma :\mathbb {r} ^{n}\rightarrow l(\mathbb {r} ^{d},\mathbb {r} ^{n})} be twice-differentiable functio...	wikipedia
b30eff20b5c390942d14430b353dcecccfa10679	there are, in particular, many results on the	there are, in particular, many results on the solution to differential equation driven by fractional brownian motion that have been proved using a combination of malliavin calculus and rough path theory. in fact, it has been proved recently that the solution to controlled differential equation driven by a class of gaus...	wikipedia
fa4f08a9f86e81c9972b1a6c61d68de478e603e0	provided that the multidimensional stochastic process x t	provided that the multidimensional stochastic process x t {\displaystyle x_{t}} can be almost surely enhanced as a rough path and that the drift b {\displaystyle b} and the volatility σ {\displaystyle \sigma } are sufficiently smooth (see the section on the universal limit theorem).	wikipedia
dcb190d0c17d5472593aeb6c5aecde75baa92df3	in fact any enhancement of brownian motion as	in fact any enhancement of brownian motion as a geometric rough path will give rise a calculus that satisfies this classical product rule. itô calculus does not come directly from enhancing brownian motion as a geometric rough path, but rather as a branched rough path.	wikipedia
3cb15ad47172007605066f6ac8791a9c99756021	note that every stochastic process (even if it	note that every stochastic process (even if it is a deterministic path) can have more than one (in fact, uncountably many) possible enhancements. different enhancements will give rise to different solutions to the controlled differential equations. in particular, it is possible to enhance brownian motion to a geometric...	wikipedia
69d1a47f389aa0f427ee8aa723ffe328ea5bd1b0	is a p {\displaystyle p} -geometric rough path,	is a p {\displaystyle p} -geometric rough path, then x s, t {\displaystyle \mathbf {x} _{s,t}} is an enhancement of the process x {\displaystyle x}. once an enhancement has been chosen, the machinery of rough path theory will allow one to make sense of the controlled differential equation	wikipedia
755bf6f7522a17542b7443f275e3b3a177b74235	in general, let (x t) t ≥ 0	in general, let (x t) t ≥ 0 {\displaystyle (x_{t})_{t\geq 0}} be a r d {\displaystyle \mathbb {r} ^{d}} -valued stochastic process. if one can construct, almost surely, functions (s, t) → x s, t j ∈ (r d) ⊗ j {\displaystyle (s,t)\rightarrow \mathbf {x} _{s,t}^{j}\in {\big (}\mathbb {r} ^{d}{\big)}^{\otimes j}} so that	wikipedia
07d9f70e1a6f48dab2f21046b88fba1c74f74ecc	converges almost surely in the p {\displaystyle p}	converges almost surely in the p {\displaystyle p} -variation metric to a p {\displaystyle p} -geometric rough path for 1 h < p {\displaystyle {\frac {1}{h}}<p}. this limiting geometric rough path can be used to make sense of differential equations driven by fractional brownian motion with hurst parameter h > 1 4 {\dis...	wikipedia
484922b85a721e16687b89268caecc6e081133b0	more generally, let b h (t) {\displaystyle b_{h}(t)}	more generally, let b h (t) {\displaystyle b_{h}(t)} be a multidimensional fractional brownian motion (a process whose coordinate components are independent fractional brownian motions) with h > 1 4 {\displaystyle h>{\frac {1}{4}}}. if b h m (t) {\displaystyle b_{h}^{m}(t)} is the m {\displaystyle m} -th dyadic piecewi...	wikipedia
2a6998ab507a0b2657a422487666db4fed44481a	let (b t) t ≥ 0 {\displaystyle (b_{t})_{t\geq	let (b t) t ≥ 0 {\displaystyle (b_{t})_{t\geq 0}} be a multidimensional standard brownian motion. let ∘ {\displaystyle \circ } denote the stratonovich integration. then	wikipedia
23fe4d3e8bd8c0c32d5b9c3ae72624e73e68078e	the theorem can be interpreted as saying that	the theorem can be interpreted as saying that the solution map (aka the itô-lyons map) Φ: g Ω p (r d) → g Ω p (r e) {\displaystyle \phi:g\omega _{p}(\mathbb {r} ^{d})\to g\omega _{p}(\mathbb {r} ^{e})} of the rde (⋆) {\displaystyle (\star)} is continuous (and in fact locally lipschitz) in the p {\displaystyle p} -varia...	wikipedia
d02b75856cee84d99ffdc64fa0abaf8e5ac11b73	then y (n) {\displaystyle \mathbf {y} (n)} converges	then y (n) {\displaystyle \mathbf {y} (n)} converges in the p {\displaystyle p} -variation metric to a p {\displaystyle p} -geometric rough path y {\displaystyle \mathbf {y} }.	wikipedia
35c13395adcacc371447f64ff321111d02957951	suppose that x (n) {\displaystyle \mathbf {x} (n)}	suppose that x (n) {\displaystyle \mathbf {x} (n)} converges in the p {\displaystyle p} -variation metric to a p {\displaystyle p} -geometric rough path x {\displaystyle \mathbf {x} } as n → ∞ {\displaystyle n\to \infty }. let (v j i) j = 1, …, d i = 1, …, n {\displaystyle (v_{j}^{i})_{j=1,\ldots,d}^{i=1,\ldots,n}} be ...	wikipedia
1b48661100d1a437c6c6e1114498caabdb61d37d	a continuous function x: △ 0, 1 →	a continuous function x: △ 0, 1 → t (⌊ p ⌋) (r d) {\displaystyle \mathbf {x} :\triangle _{0,1}\rightarrow t^{(\lfloor p\rfloor)}(\mathbb {r} ^{d})} is a p {\displaystyle p} -geometric rough path if there exists a sequence of paths with finite total variation x (1), x (2), … {\displaystyle x(1),x(2),\ldots } such that	wikipedia
22791fb0248ed9cead65e045ad028e0ebf413c90	where the supremum is taken over all finite	where the supremum is taken over all finite partitions { 0 = t 0 < t 1 < ⋯ < t n = 1 } {\displaystyle \{0=t_{0}<t_{1}<\cdots <t_{n}=1\}} of {\displaystyle }.	wikipedia
dcdb3bc8cc09a9ee483c5d80fd24fa0c5e607c3f	let △ 0, 1 {\displaystyle \triangle _{0,1}} be	let △ 0, 1 {\displaystyle \triangle _{0,1}} be the simplex { (s, t): 0 ≤ s ≤ t ≤ 1 } {\displaystyle \{(s,t):0\leq s\leq t\leq 1\}}. let p ≥ 1 {\displaystyle p\geq 1}. let x {\displaystyle \mathbf {x} } and y {\displaystyle \mathbf {y} } be continuous maps △ 0, 1 → t (⌊ p ⌋) (r d) {\displaystyle \triangle _{0,1}\to t^{(...	wikipedia
6a655b8b764463ce154951b2ca5f12c1dd8b79a9	rough paths are paths taking values in the	rough paths are paths taking values in the truncated free tensor algebra (more precisely: in the free nilpotent group embedded in the free tensor algebra), which this section now briefly recalls. the tensor powers of r d {\displaystyle \mathbb {r} ^{d}}, denoted (r d) ⊗ n {\displaystyle {\big (}\mathbb {r} ^{d}{\big)}^...	wikipedia
b2e2032010e0b77f21843a52fd0f69c9d56ec62e	then y (n) {\displaystyle y(n)} converges to y	then y (n) {\displaystyle y(n)} converges to y {\displaystyle y} in the p {\displaystyle p} -variation metric. this continuity property and the deterministic nature of solutions makes it possible to simplify and strengthen many results in stochastic analysis, such as the freidlin-wentzell's large deviation theory as we...	wikipedia
8fbd5731c463189a359f5e9bec580da8c499d3a0	where the control, the continuous path x t	where the control, the continuous path x t {\displaystyle x_{t}} taking values in a banach space, need not be differentiable nor of bounded variation. a prevalent example of the controlled path x t {\displaystyle x_{t}} is the sample path of a wiener process. in this case, the aforementioned controlled differential equ...	wikipedia
589ba29aa5924567d6e64faad9835bcdf51a8445	rough path theory is focused on capturing and	rough path theory is focused on capturing and making precise the interactions between highly oscillatory and non-linear systems. it builds upon the harmonic analysis of l.c. young, the geometric algebra of k.t. chen, the lipschitz function theory of h. whitney and core ideas of stochastic analysis. the concepts and the...	wikipedia
4fc5be80ec6760a3ba597d29688d77dbed051c7d	in stochastic analysis, a rough path is a	in stochastic analysis, a rough path is a generalization of the notion of smooth path allowing to construct a robust solution theory for controlled differential equations driven by classically irregular signals, for example a wiener process. the theory was developed in the 1990s by terry lyons. several accounts of the ...	wikipedia
c965fb9c2ab341720199bb7236170e202656fc68	donald eugene hanrahan was a professional basketball forward	donald eugene hanrahan was a professional basketball forward who spent one season in the national basketball association (nba) as a member of the indianapolis olympians during the 1952–53 season. he attended loyola university of chicago.	wikipedia
5ffcdcd7fd6d39696c041a59eacf84b06ff84e6a	donald eugene hanrahan (february 6, 1929 – february	donald eugene hanrahan (february 6, 1929 – february 8, 2010) was a professional basketball forward who spent one season in the national basketball association (nba) as a member of the indianapolis olympians during the 1952–53 season. he attended loyola university of chicago.	wikipedia
03e333cf53b59fb6daee239f685886ecb75803a4	sunspots are temporary spots on the sun's surface	sunspots are temporary spots on the sun's surface that are darker than the surrounding area. they are regions of reduced surface temperature caused by concentrations of magnetic flux that inhibit convection. sunspots appear within active regions, usually in pairs of opposite magnetic polarity. their number varies accor...	wikipedia
696bd0b57ed4f02ba3209615a515eabe4de5ca29	in 1947, g. e. kron proposed that starspots	in 1947, g. e. kron proposed that starspots were the reason for periodic changes in brightness on red dwarfs. since the mid-1990s, starspot observations have been made using increasingly powerful techniques yielding more and more detail: photometry showed starspot growth and decay and showed cyclic behavior similar to ...	wikipedia
340afe268828c6bb45d394f7e44dad918a92bdc6	sunspots themselves, in terms of the magnitude of	sunspots themselves, in terms of the magnitude of their radiant-energy deficit, have a weak effect on solar flux. the total effect of sunspots and other magnetic processes in the solar photosphere is an increase of roughly 0.1% in brightness of the sun in comparison with its brightness at the solar-minimum level. this ...	wikipedia
3423fe1ff563a84db0265c0edf0f18e6f2f2a5bd	solar activity (and the solar cycle) have been	solar activity (and the solar cycle) have been implicated as a factor in global warming. the first possible example of this is the maunder minimum period of low sunspot activity which occurred during the little ice age in europe. however, detailed studies from multiple paleoclimate indicators show that the lower northe...	wikipedia
9f0e57742324ae6d67a1a384b25ec530d4c9133b	due to their correlation with other kinds of	due to their correlation with other kinds of solar activity, sunspots can be used to help predict space weather, the state of the ionosphere, and conditions relevant to short-wave radio propagation or satellite communications. high sunspot activity is celebrated by members of the amateur radio community as a harbinger ...	wikipedia
d0ff101e29bae908b071b23a69923210b67048f8	since looking directly at the sun with the	since looking directly at the sun with the naked eye permanently damages human vision, amateur observation of sunspots is generally conducted using projected images, or directly through protective filters. small sections of very dark filter glass, such as a #14 welder's glass, are effective. a telescope eyepiece can pr...	wikipedia