LAN property for sde s with additive fractional noise and continuous time observation
|
|
- Junior Horn
- 5 years ago
- Views:
Transcription
1 LAN property for sde s with additive fractional noise and continuous time observation Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Samy Tindel (Purdue University) Vlad s 6th birthday, Université du Maine, 7th October, 215 Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
2 The Ornstein-Uhlenbeck process dx t = θx t dt + db t, t [, τ], θ >. B t is a standard Brownian motion. Let ˆθ τ be the MLE of θ from the continuous observation of X in [, τ]. Then, it is well-known and that lim ˆθ τ = θ τ a.s. L(P θ ) lim τ(ˆθτ θ) = N (, 2θ). τ where P θ is the probability law of the solution in the space C(R +; R). Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
3 The LAN propety for the Ornstein-Uhlenbeck process The parametric statistical model {P θ, θ Θ} satisfies the LAN property at θ Θ with rate τ since for any u R, as τ : log ( dp τ θ+ u τ dp τ θ ) ( L(P θ ) un, 1 ) u2 2θ 4θ, where P τ θ is probability law of the solution in the space C([, τ]; R)). The local log likelihood ratio is asymptotically normal, with a locally constant covariance matrix and a mean equal to minus one half the variance. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
4 Consequence of the LAN property Minimax Theorem : Let (ˆθ τ ) τ be a family of estimators of the parameter θ. Then [ τ(ˆθ τ θ ) 2] 2θ. lim δ lim inf τ sup E θ θ θ <δ In particular, the MLE is asymptotic minimax efficient. The LAN property is an important tool in order to quantify the identifiability of a system. Started by Le Cam 6. Parallel theory to Cramér-Rao bound. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
5 LAN property for ergodic diffusions Consider a non-linear d-dimensional ergodic diffusion dx t = b(x t; θ)dt + σ(x t)db t, t [, τ], θ Θ R q. Under regularity, ellipticity, and ergodic assumptions, for any θ Θ and u R q, as τ : ( dp τ ) θ+ u L(P τ θ ) log u T N (, Γ(θ)) 1 2 ut Γ(θ)u, dp τ θ where X is the ergodic limit of X, and Γ(θ) = E θ [ θ b(x; θ) T σ 1 (X) T σ 1 (X) θ b(x; θ)]. Proof : Girsanov s theorem, CLT for martingales and ergodicity. Consequence : Minimax theorem : lim δ lim inf τ sup E θ θ θ <δ [ τ(ˆθ τ θ ) 2] Γ(θ) 1. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
6 The fractional Ornstein-Uhlenbeck process X t = θ t Xs ds + Bt, t [, τ], θ >. B t fractional Brownian motion with Hurst parameter H > 1/2. P θ is the probability law of the solution in the space C λ (R +; R), for any λ < H. Let ˆθ τ be the MLE of θ from the continuous observation of X in [, τ]. Then, it is well-known and that lim ˆθ τ = θ τ a.s. L(P θ ) lim τ(ˆθτ θ) = N (, 2θ). τ This suggests that the LAN property holds with the same rate τ. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
7 Ergodic sde s with additive fractional noise t X t = x + b(x s; θ) ds + d σ j B j t, j=1 t [, τ]. θ Θ, where Θ is compactly embedded in R q. ergodicity condition : b(x; θ) b(y; θ), x y α x y 2. ˆb is the Jacobian matrix θ b. assumptions : xb, xxb, x ˆb, xx ˆb bounded, b, ˆb linear growth, ˆb Lipschitz in θ and x, and σ invertible. The solution converges for t a.s. to a unique stationary process (X t, t ). P τ θ is the probability laws of the solution in the spaces C λ ([, τ]; R d ), for any λ < H. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
8 The LAN property Theorem : For any θ Θ and u R q, as τ, ( dp τ ) θ+ u L(P τ θ ) log u T N (, Γ(θ)) 1 2 ut Γ(θ)u, dp τ θ where the matrix Γ(θ) is defined by Γ(θ) = R 2 + E θ [(ˆb(X ; θ) ˆb(X r1 ; θ)) T (σ 1 ) T σ 1 (ˆb(X ; θ) ˆb(X r2 ; θ))] dr r 1/2+H 1 r 1/2+H 1 dr 2. 2 Remark : The efficiency of the MLE in the fractional Ornstein-Uhlenbeck case remains open... Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
9 Steps of the proof of the LAN property Use the representation of the fbm given in Hairer 5 (introduced by Mandelbrot and Van Ness 68) which is suitable to get the desired ergodic properties. Apply Girsanov s theorem for the fbm following Moret and Nualart 2. Handle the singularities popping out the fractional derivatives in the Girsanov exponent. Get ergodic results in Hölder type norms for our process X. In order to apply a CLT for Brownian martingales, we use Malliavin calculus techniques : derive concentration properties for the Girsanov exponents by means of a Poincaré type inequality (Üstunel 95), which needs to conviniently upper bound some Malliavin derivatives. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
10 Mendelbrot and Van Ness representation of fbm Let W be a two sided Wiener process, then the following defines a two-sided fbm : for any t R : B t = c H R ( r) H 1/2 [dw t+r dw r ] = c H { [ ( (r t)) H 1/2 ( r) H 1/2] dw r t ( (r t)) H 1/2 dw r }. Abstract Wiener space : (B, H, P), where B = { f C(R; R d ); } f t 1 + t <, P is the law of our fbm, and h is an element of the Cameron-Martin space H iff there exists an element X h in the first chaos such that h t = E[B t X h ], and h H = X h L 2 (Ω). Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
11 Properties of the SDE Proposition : There exists a unique continuous pathwise solution on any arbitrary interval [, τ] such that : The map X : (x, B) R d C([, τ]; R d ) C([, τ]; R d ) is locally Lipschitz continuous. For any θ Θ, p 1, and s, t, E [ X t p] c p, and E [ δx st p] k p t s ph, where δ denotes the increment. For all ε (, H) there exists a random variable Z ε p 1 L p (Ω) such that a.s. X t Z ε (1 + t) 2ε, and δx st Z ε (1 + t) 2ε t s H ε, uniformly for s t. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
12 Ergodic properties of the SDE Garrido-Atienza, Kloeden and Neuenkirch 9 : Shift operators θ t : Ω Ω : θ tω( ) = ω( + t) ω(t), t R, ω Ω. The shifted process (B s(θ t )) s R is still a d-dimensional fractional Brownian motion and for any integrable random variable F : Ω R for P-almost all ω Ω. 1 lim τ τ τ F(θ t(ω)) dt = E[F], Theorem : There exists a random variable X : Ω R d such that lim t Xt(ω) X(θtω) = for P-almost all ω Ω. Moreover, we have E[ X p ] < for all p 1. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
13 Ergodic properties of the SDE Theorem : For any θ Θ and any f C 1 (R d ; R) such that ( f (x) + xf (x) c 1 + x N), x R d, we have 1 τ lim f (X τ t) dt = E[f (X)], τ P-a.s. Proposition : Let α (, H). There exists a random variable Z admitting moments of any order such that for all s t [ ] X t X t Z e cs and δ X X Z e cs (t s) α. st Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
14 Operator that transforms W into B Proposition : For w C c (R) and H (, 1), set [K H w] t = c H R ( r) H 1/2 [ẇ t+r ẇ r ] dr. Then : (i) There exists a constant c H such that ( ) c H [I H 1/2 + w] t [I H 1/2 + w], for H > 1 2 [K H w] t = ( ) c H [D 1/2 H + w] t [D 1/2 H + w], for H < 1, 2 where [D α +ϕ] t = c α ϕ t ϕ t r dr, and [I+ϕ] α R + r 1+α t = c α R + ϕ t r r α 1 dr. (ii) For H > 1/2, K H can be extended as an isometry from L 2 (R) to I H 1/2 + (L 2 (R)). (iii) There exists a constant c H such that K 1 H = c H K 1 H. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
15 Girsanov s transformation Proposition : For a given θ Θ, onsider the d-dimensional process Q t = t σ 1 b(x s; θ) ds + B t. Then Q is a d-dimensional fractional Brownian motion under the probability P θ defined by d P θ dp θ [,τ] = e L, with L = τ σ 1 [D H 1/2 + b(x; θ)] u, dw u + 1 τ σ 1 [D H 1/2 + b(x; θ)] u 2 du. 2 Proof : show that D H 1/2 + b(x; θ) is well defined on [, τ], and Novikov s condition : there exists λ > such that sup E θ [exp t [,τ] ( λ t σ 1 [D H 1/2 + b(x; θ)] s 2 ds )] <. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
16 Proof of the LAN property Step 1 : Apply Girsanov s theorem. Fix θ Θ, and set θ τ = θ + τ 1/2 u. Then ( dp τ ) τ θτ log = σ 1 ([D H 1/2 dp τ + b(x; θ τ )] t [D H 1/2 + b(x; θ)] t), dw t θ 1 2 τ σ 1 ([D H 1/2 + b(x; θ τ )] t [D H 1/2 + b(x; θ)] t) 2 dt. Step 2 : Linearize this relation : add and substract the d-dimensional vector [D H 1/2 + ˆb(X; θ)] t(θ τ θ) = 1 [D H 1/2 τ + ˆb(X; θ)] t, where ˆb = θ b. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
17 Step 2 : linearization where I 1 = 1 τ τ I 2 = I 3 = τ τ ( dp τ ) θτ log = I dp τ 1 I 2 1 θ 2 I 3 I 4, σ 1 [D H 1/2 + ˆb(X; θ)] tu, dw t 1 τ 2τ σ 1 ([D H 1/2 + b(x; θ τ )] t [D H 1/2 σ 1 ([D H 1/2 + b(x; θ τ )] t [D H 1/2 σ 1 [D H 1/2 + ˆb(X; θ)] tu 2 dt + b(x; θ)] t [D H 1/2 + ˆb(X; θ)] t(θ τ θ)), dw t + b(x; θ)] t [D H 1/2 + ˆb(X; θ)] t(θ τ θ)) 2 dt I 4 = τ σ 1 ([D H 1/2 + b(x; θ τ )] t [D H 1/2 + b(x; θ)] t [D H 1/2 + ˆb(X; θ)] t(θ τ θ)), σ 1 [D H 1/2 + ˆb(X; θ)] t(θ τ θ) dt. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
18 Remaining steps of the proof Step 3 : Main contribution to our log-likelihood : we show that as τ 1 τ τ σ 1 [D H 1/2 + ˆb(X; θ)] tu 2 P dt θ u T Γ(θ)u. Together with multivariate central limit theorem for Brownian martingales implies that as τ I 1 L(P θ ) u T N (, Γ(θ)) 1 2 ut Γ(θ)u. Step 4 : Negligible contributions : We show that the terms I 2, I 3 and I 4 converge to zero in P θ -probability as τ. For I 3 apply Taylor s expansion and some computations, I 3 is the quadratic variation of the martingale I 2, and by Cauchy-Schwarz inequality, I 4 is bounded by I 3. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
19 Proof of Step 3 Step 3a : We have that 1 τ τ σ 1 [D H 1/2 + ˆb(X; θ)] tu 2 dt 1 τ Jτ (X) = 1 τ where (ˆb(X t; θ) N t(x) = ˆb(X t r ; θ))u dr = N R + r H+1/2 1,t (X) + N 2,t (X) = t (ˆb(X t; θ) ˆb(X t r ; θ))u r H+1/2 dr + where we have set X t = x for all t. t τ σ 1 N t(x) 2 dt, (ˆb(X t; θ) ˆb(x ; θ))u r H+1/2 We denote by J τ (X), N t(x), N 1,t (X), N 2,t (X) the same quantities with X replaced by X. dr Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
20 Proof of Step 3 Step 3b : We show that N t(x) N t(x) is of order t η Z with η > and Z p 1 L p (Ω). Hence J τ (X) J τ (X) is of order τ 1 2η, which is a negligible term on the scale τ. This allows us to consider the limiting behavior of J τ (X) instead of J τ (X). Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
21 Proof of Step 3 Step 3c : This step is devoted to reduce our computations to an evaluation for the expected value. We show that 1 lim τ τ E θ [ J τ (X) E θ [J τ (X)] ] =, Poincaré type inequality : Let F : B R be a functional in D 1,2. Then, This reduces to show that E [ F E[F] ] π E [ DF H] 2 [ E θ DJτ (X) lim H τ τ ] =. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
22 Proof of Step 3 Proposition :For all t >, X t belongs to D 1,2, and the Malliavin derivative satisfies that ( DX t H c exp α t ), 2 uniformly in t R +. Moreover, for u v, ( D(δX uv) H c 1 exp α u ) (v u) H/2, 2 uniformly in u and v. Idea of proof : Derive contraction properties of the map h X h, h H, where X h is the solution to our SDE driven by B + h : ( Xt h X t c exp αt ) h H, 2 uniformly in t R +. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
23 Proof of Step 3 Step 3d : We are now reduced to the analysis of the quantity E θ [J τ (X)]. This is equal to u T Ψu where Ψ equals the matrix τ E θ [(ˆb(Y t; θ) dt ˆb(Y t r1 ; θ)) T (σ 1 ) T σ 1 (ˆb(Y t; θ) ˆb(Y t r2 ; θ))] dr R 2 + r 1/2+H 1 r 1/2+H 1 dr 2 2 By stationarity of Y, the expected value does not depend on t and ( ) ( ) E θ [(ˆb(Y ; θ) ˆb(Y r1 ; θ)) T (σ 1 ) T σ 1 (ˆb(Y ; θ) ˆb(Y r2 ; θ))] r1 H 1 r2 H 1 We obtain that Ψ is a convergent integral and E θ [J τ (Y )] = τu T Γ(θ)u. Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
24 BON ANNIVERSAIRE VLAD!!!! Eulalia Nualart (Universitat Pompeu Fabra) 7th October, / 24
LAN property for ergodic jump-diffusion processes with discrete observations
LAN property for ergodic jump-diffusion processes with discrete observations Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Arturo Kohatsu-Higa (Ritsumeikan University, Japan) &
More informationConvergence to equilibrium for rough differential equations
Convergence to equilibrium for rough differential equations Samy Tindel Purdue University Barcelona GSE Summer Forum 2017 Joint work with Aurélien Deya (Nancy) and Fabien Panloup (Angers) Samy T. (Purdue)
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationRough paths methods 4: Application to fbm
Rough paths methods 4: Application to fbm Samy Tindel Purdue University University of Aarhus 2016 Samy T. (Purdue) Rough Paths 4 Aarhus 2016 1 / 67 Outline 1 Main result 2 Construction of the Levy area:
More informationJoint Parameter Estimation of the Ornstein-Uhlenbeck SDE driven by Fractional Brownian Motion
Joint Parameter Estimation of the Ornstein-Uhlenbeck SDE driven by Fractional Brownian Motion Luis Barboza October 23, 2012 Department of Statistics, Purdue University () Probability Seminar 1 / 59 Introduction
More informationMalliavin calculus and central limit theorems
Malliavin calculus and central limit theorems David Nualart Department of Mathematics Kansas University Seminar on Stochastic Processes 2017 University of Virginia March 8-11 2017 David Nualart (Kansas
More informationRough paths methods 1: Introduction
Rough paths methods 1: Introduction Samy Tindel Purdue University University of Aarhus 216 Samy T. (Purdue) Rough Paths 1 Aarhus 216 1 / 16 Outline 1 Motivations for rough paths techniques 2 Summary of
More informationMaximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments
Austrian Journal of Statistics April 27, Volume 46, 67 78. AJS http://www.ajs.or.at/ doi:.773/ajs.v46i3-4.672 Maximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments Yuliya
More informationMean-field SDE driven by a fractional BM. A related stochastic control problem
Mean-field SDE driven by a fractional BM. A related stochastic control problem Rainer Buckdahn, Université de Bretagne Occidentale, Brest Durham Symposium on Stochastic Analysis, July 1th to July 2th,
More informationI forgot to mention last time: in the Ito formula for two standard processes, putting
I forgot to mention last time: in the Ito formula for two standard processes, putting dx t = a t dt + b t db t dy t = α t dt + β t db t, and taking f(x, y = xy, one has f x = y, f y = x, and f xx = f yy
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationAn adaptive numerical scheme for fractional differential equations with explosions
An adaptive numerical scheme for fractional differential equations with explosions Johanna Garzón Departamento de Matemáticas, Universidad Nacional de Colombia Seminario de procesos estocásticos Jointly
More informationSome Tools From Stochastic Analysis
W H I T E Some Tools From Stochastic Analysis J. Potthoff Lehrstuhl für Mathematik V Universität Mannheim email: potthoff@math.uni-mannheim.de url: http://ls5.math.uni-mannheim.de To close the file, click
More informationTopics in fractional Brownian motion
Topics in fractional Brownian motion Esko Valkeila Spring School, Jena 25.3. 2011 We plan to discuss the following items during these lectures: Fractional Brownian motion and its properties. Topics in
More informationIntegral representations in models with long memory
Integral representations in models with long memory Georgiy Shevchenko, Yuliya Mishura, Esko Valkeila, Lauri Viitasaari, Taras Shalaiko Taras Shevchenko National University of Kyiv 29 September 215, Ulm
More informationLeast Squares Estimators for Stochastic Differential Equations Driven by Small Lévy Noises
Least Squares Estimators for Stochastic Differential Equations Driven by Small Lévy Noises Hongwei Long* Department of Mathematical Sciences, Florida Atlantic University, Boca Raton Florida 33431-991,
More informationFrom Fractional Brownian Motion to Multifractional Brownian Motion
From Fractional Brownian Motion to Multifractional Brownian Motion Antoine Ayache USTL (Lille) Antoine.Ayache@math.univ-lille1.fr Cassino December 2010 A.Ayache (USTL) From FBM to MBM Cassino December
More informationp 1 ( Y p dp) 1/p ( X p dp) 1 1 p
Doob s inequality Let X(t) be a right continuous submartingale with respect to F(t), t 1 P(sup s t X(s) λ) 1 λ {sup s t X(s) λ} X + (t)dp 2 For 1 < p
More informationA Class of Fractional Stochastic Differential Equations
Vietnam Journal of Mathematics 36:38) 71 79 Vietnam Journal of MATHEMATICS VAST 8 A Class of Fractional Stochastic Differential Equations Nguyen Tien Dung Department of Mathematics, Vietnam National University,
More informationBernardo D Auria Stochastic Processes /10. Notes. Abril 13 th, 2010
1 Stochastic Calculus Notes Abril 13 th, 1 As we have seen in previous lessons, the stochastic integral with respect to the Brownian motion shows a behavior different from the classical Riemann-Stieltjes
More informationNonparametric Drift Estimation for Stochastic Differential Equations
Nonparametric Drift Estimation for Stochastic Differential Equations Gareth Roberts 1 Department of Statistics University of Warwick Brazilian Bayesian meeting, March 2010 Joint work with O. Papaspiliopoulos,
More informationHarnack Inequalities and Applications for Stochastic Equations
p. 1/32 Harnack Inequalities and Applications for Stochastic Equations PhD Thesis Defense Shun-Xiang Ouyang Under the Supervision of Prof. Michael Röckner & Prof. Feng-Yu Wang March 6, 29 p. 2/32 Outline
More informationA Short Introduction to Diffusion Processes and Ito Calculus
A Short Introduction to Diffusion Processes and Ito Calculus Cédric Archambeau University College, London Center for Computational Statistics and Machine Learning c.archambeau@cs.ucl.ac.uk January 24,
More informationAn Introduction to Malliavin calculus and its applications
An Introduction to Malliavin calculus and its applications Lecture 3: Clark-Ocone formula David Nualart Department of Mathematics Kansas University University of Wyoming Summer School 214 David Nualart
More informationRepresenting Gaussian Processes with Martingales
Representing Gaussian Processes with Martingales with Application to MLE of Langevin Equation Tommi Sottinen University of Vaasa Based on ongoing joint work with Lauri Viitasaari, University of Saarland
More informationA Concise Course on Stochastic Partial Differential Equations
A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original
More informationLecture 22 Girsanov s Theorem
Lecture 22: Girsanov s Theorem of 8 Course: Theory of Probability II Term: Spring 25 Instructor: Gordan Zitkovic Lecture 22 Girsanov s Theorem An example Consider a finite Gaussian random walk X n = n
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationGaussian Processes. 1. Basic Notions
Gaussian Processes 1. Basic Notions Let T be a set, and X : {X } T a stochastic process, defined on a suitable probability space (Ω P), that is indexed by T. Definition 1.1. We say that X is a Gaussian
More informationOn Stochastic Adaptive Control & its Applications. Bozenna Pasik-Duncan University of Kansas, USA
On Stochastic Adaptive Control & its Applications Bozenna Pasik-Duncan University of Kansas, USA ASEAS Workshop, AFOSR, 23-24 March, 2009 1. Motivation: Work in the 1970's 2. Adaptive Control of Continuous
More informationMinimum L 1 -norm Estimation for Fractional Ornstein-Uhlenbeck Type Process
isid/ms/23/25 September 11, 23 http://www.isid.ac.in/ statmath/eprints Minimum L 1 -norm Estimation for Fractional Ornstein-Uhlenbeck Type Process B. L. S. Prakasa Rao Indian Statistical Institute, Delhi
More informationSome functional (Hölderian) limit theorems and their applications (II)
Some functional (Hölderian) limit theorems and their applications (II) Alfredas Račkauskas Vilnius University Outils Statistiques et Probabilistes pour la Finance Université de Rouen June 1 5, Rouen (Rouen
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More informationThe Wiener Itô Chaos Expansion
1 The Wiener Itô Chaos Expansion The celebrated Wiener Itô chaos expansion is fundamental in stochastic analysis. In particular, it plays a crucial role in the Malliavin calculus as it is presented in
More informationDensities for the Navier Stokes equations with noise
Densities for the Navier Stokes equations with noise Marco Romito Università di Pisa Universitat de Barcelona March 25, 2015 Summary 1 Introduction & motivations 2 Malliavin calculus 3 Besov bounds 4 Other
More informationMULTIDIMENSIONAL WICK-ITÔ FORMULA FOR GAUSSIAN PROCESSES
MULTIDIMENSIONAL WICK-ITÔ FORMULA FOR GAUSSIAN PROCESSES D. NUALART Department of Mathematics, University of Kansas Lawrence, KS 6645, USA E-mail: nualart@math.ku.edu S. ORTIZ-LATORRE Departament de Probabilitat,
More informationExercises. T 2T. e ita φ(t)dt.
Exercises. Set #. Construct an example of a sequence of probability measures P n on R which converge weakly to a probability measure P but so that the first moments m,n = xdp n do not converge to m = xdp.
More informationSolution for Problem 7.1. We argue by contradiction. If the limit were not infinite, then since τ M (ω) is nondecreasing we would have
362 Problem Hints and Solutions sup g n (ω, t) g(ω, t) sup g(ω, s) g(ω, t) µ n (ω). t T s,t: s t 1/n By the uniform continuity of t g(ω, t) on [, T], one has for each ω that µ n (ω) as n. Two applications
More informationEstimates for the density of functionals of SDE s with irregular drift
Estimates for the density of functionals of SDE s with irregular drift Arturo KOHATSU-HIGA a, Azmi MAKHLOUF a, a Ritsumeikan University and Japan Science and Technology Agency, Japan Abstract We obtain
More informationOn the Goodness-of-Fit Tests for Some Continuous Time Processes
On the Goodness-of-Fit Tests for Some Continuous Time Processes Sergueï Dachian and Yury A. Kutoyants Laboratoire de Mathématiques, Université Blaise Pascal Laboratoire de Statistique et Processus, Université
More informationHölder continuity of the solution to the 3-dimensional stochastic wave equation
Hölder continuity of the solution to the 3-dimensional stochastic wave equation (joint work with Yaozhong Hu and Jingyu Huang) Department of Mathematics Kansas University CBMS Conference: Analysis of Stochastic
More informationThe stochastic heat equation with a fractional-colored noise: existence of solution
The stochastic heat equation with a fractional-colored noise: existence of solution Raluca Balan (Ottawa) Ciprian Tudor (Paris 1) June 11-12, 27 aluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic
More informationItô s formula. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Itô s formula Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Itô s formula Probability Theory
More informationPathwise volatility in a long-memory pricing model: estimation and asymptotic behavior
Pathwise volatility in a long-memory pricing model: estimation and asymptotic behavior Ehsan Azmoodeh University of Vaasa Finland 7th General AMaMeF and Swissquote Conference September 7 1, 215 Outline
More informationOn rough PDEs. Samy Tindel. University of Nancy. Rough Paths Analysis and Related Topics - Nagoya 2012
On rough PDEs Samy Tindel University of Nancy Rough Paths Analysis and Related Topics - Nagoya 2012 Joint work with: Aurélien Deya and Massimiliano Gubinelli Samy T. (Nancy) On rough PDEs Nagoya 2012 1
More informationOn continuous time contract theory
Ecole Polytechnique, France Journée de rentrée du CMAP, 3 octobre, 218 Outline 1 2 Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs (Static) Principal-Agent Problem
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationStochastic Differential Equations.
Chapter 3 Stochastic Differential Equations. 3.1 Existence and Uniqueness. One of the ways of constructing a Diffusion process is to solve the stochastic differential equation dx(t) = σ(t, x(t)) dβ(t)
More informationBessel-like SPDEs. Lorenzo Zambotti, Sorbonne Université (joint work with Henri Elad-Altman) 15th May 2018, Luminy
Bessel-like SPDEs, Sorbonne Université (joint work with Henri Elad-Altman) Squared Bessel processes Let δ, y, and (B t ) t a BM. By Yamada-Watanabe s Theorem, there exists a unique (strong) solution (Y
More information-variation of the divergence integral w.r.t. fbm with Hurst parameter H < 1 2
/4 On the -variation of the divergence integral w.r.t. fbm with urst parameter < 2 EL ASSAN ESSAKY joint work with : David Nualart Cadi Ayyad University Poly-disciplinary Faculty, Safi Colloque Franco-Maghrébin
More informationMathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )
Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical
More informationConvoluted Brownian motions: a class of remarkable Gaussian processes
Convoluted Brownian motions: a class of remarkable Gaussian processes Sylvie Roelly Random models with applications in the natural sciences Bogotá, December 11-15, 217 S. Roelly (Universität Potsdam) 1
More informationBayesian Regularization
Bayesian Regularization Aad van der Vaart Vrije Universiteit Amsterdam International Congress of Mathematicians Hyderabad, August 2010 Contents Introduction Abstract result Gaussian process priors Co-authors
More informationThe Continuity of SDE With Respect to Initial Value in the Total Variation
Ξ44fflΞ5» ο ffi fi $ Vol.44, No.5 2015 9" ADVANCES IN MATHEMATICS(CHINA) Sep., 2015 doi: 10.11845/sxjz.2014024b The Continuity of SDE With Respect to Initial Value in the Total Variation PENG Xuhui (1.
More informationStochastic Differential Equations
Chapter 5 Stochastic Differential Equations We would like to introduce stochastic ODE s without going first through the machinery of stochastic integrals. 5.1 Itô Integrals and Itô Differential Equations
More informationMA8109 Stochastic Processes in Systems Theory Autumn 2013
Norwegian University of Science and Technology Department of Mathematical Sciences MA819 Stochastic Processes in Systems Theory Autumn 213 1 MA819 Exam 23, problem 3b This is a linear equation of the form
More informationMultiple points of the Brownian sheet in critical dimensions
Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang Ecole Polytechnique Fédérale de Lausanne Based on joint work with: Carl Mueller Multiple points of the Brownian sheet in critical
More informationON A SDE DRIVEN BY A FRACTIONAL BROWNIAN MOTION AND WITH MONOTONE DRIFT
Elect. Comm. in Probab. 8 23122 134 ELECTRONIC COMMUNICATIONS in PROBABILITY ON A SDE DRIVEN BY A FRACTIONAL BROWNIAN MOTION AND WIT MONOTONE DRIFT BRAIM BOUFOUSSI 1 Cadi Ayyad University FSSM, Department
More informationThe concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.
The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes
More informationAsymptotic Properties of an Approximate Maximum Likelihood Estimator for Stochastic PDEs
Asymptotic Properties of an Approximate Maximum Likelihood Estimator for Stochastic PDEs M. Huebner S. Lototsky B.L. Rozovskii In: Yu. M. Kabanov, B. L. Rozovskii, and A. N. Shiryaev editors, Statistics
More informationEstimation of arrival and service rates for M/M/c queue system
Estimation of arrival and service rates for M/M/c queue system Katarína Starinská starinskak@gmail.com Charles University Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics
More informationAsymptotical distribution free test for parameter change in a diffusion model (joint work with Y. Nishiyama) Ilia Negri
Asymptotical distribution free test for parameter change in a diffusion model (joint work with Y. Nishiyama) Ilia Negri University of Bergamo (Italy) ilia.negri@unibg.it SAPS VIII, Le Mans 21-24 March,
More informationGAUSSIAN PROCESSES; KOLMOGOROV-CHENTSOV THEOREM
GAUSSIAN PROCESSES; KOLMOGOROV-CHENTSOV THEOREM STEVEN P. LALLEY 1. GAUSSIAN PROCESSES: DEFINITIONS AND EXAMPLES Definition 1.1. A standard (one-dimensional) Wiener process (also called Brownian motion)
More informationApplications of Ito s Formula
CHAPTER 4 Applications of Ito s Formula In this chapter, we discuss several basic theorems in stochastic analysis. Their proofs are good examples of applications of Itô s formula. 1. Lévy s martingale
More informationON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS
PORTUGALIAE MATHEMATICA Vol. 55 Fasc. 4 1998 ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS C. Sonoc Abstract: A sufficient condition for uniqueness of solutions of ordinary
More informationarxiv: v2 [math.pr] 22 Aug 2009
On the structure of Gaussian random variables arxiv:97.25v2 [math.pr] 22 Aug 29 Ciprian A. Tudor SAMOS/MATISSE, Centre d Economie de La Sorbonne, Université de Panthéon-Sorbonne Paris, 9, rue de Tolbiac,
More informationBranching Processes II: Convergence of critical branching to Feller s CSB
Chapter 4 Branching Processes II: Convergence of critical branching to Feller s CSB Figure 4.1: Feller 4.1 Birth and Death Processes 4.1.1 Linear birth and death processes Branching processes can be studied
More informationMultivariate Generalized Ornstein-Uhlenbeck Processes
Multivariate Generalized Ornstein-Uhlenbeck Processes Anita Behme TU München Alexander Lindner TU Braunschweig 7th International Conference on Lévy Processes: Theory and Applications Wroclaw, July 15 19,
More informationStochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno
Stochastic Processes M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno 1 Outline Stochastic (random) processes. Autocorrelation. Crosscorrelation. Spectral density function.
More informationRough Burgers-like equations with multiplicative noise
Rough Burgers-like equations with multiplicative noise Martin Hairer Hendrik Weber Mathematics Institute University of Warwick Bielefeld, 3.11.21 Burgers-like equation Aim: Existence/Uniqueness for du
More informationStochastic Viral Dynamics with Beddington-DeAngelis Functional Response
Stochastic Viral Dynamics with Beddington-DeAngelis Functional Response Junyi Tu, Yuncheng You University of South Florida, USA you@mail.usf.edu IMA Workshop in Memory of George R. Sell June 016 Outline
More informationCOVARIANCE IDENTITIES AND MIXING OF RANDOM TRANSFORMATIONS ON THE WIENER SPACE
Communications on Stochastic Analysis Vol. 4, No. 3 (21) 299-39 Serials Publications www.serialspublications.com COVARIANCE IDENTITIES AND MIXING OF RANDOM TRANSFORMATIONS ON THE WIENER SPACE NICOLAS PRIVAULT
More informationThe multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications
The multidimensional Ito Integral and the multidimensional Ito Formula Eric Mu ller June 1, 215 Seminar on Stochastic Geometry and its applications page 2 Seminar on Stochastic Geometry and its applications
More informationarxiv: v1 [math.pr] 23 Jan 2018
TRANSFER PRINCIPLE FOR nt ORDER FRACTIONAL BROWNIAN MOTION WIT APPLICATIONS TO PREDICTION AND EQUIVALENCE IN LAW TOMMI SOTTINEN arxiv:181.7574v1 [math.pr 3 Jan 18 Department of Mathematics and Statistics,
More informationA Fourier analysis based approach of rough integration
A Fourier analysis based approach of rough integration Massimiliano Gubinelli Peter Imkeller Nicolas Perkowski Université Paris-Dauphine Humboldt-Universität zu Berlin Le Mans, October 7, 215 Conference
More informationRegularity of the density for the stochastic heat equation
Regularity of the density for the stochastic heat equation Carl Mueller 1 Department of Mathematics University of Rochester Rochester, NY 15627 USA email: cmlr@math.rochester.edu David Nualart 2 Department
More informationCBMS Lecture Series. Recent Advances in. the Numerical Approximation of Stochastic Partial Differential Equations
CBMS Lecture Series Recent Advances in the Numerical Approximation of Stochastic Partial Differential Equations or more accurately Taylor Approximations of Stochastic Partial Differential Equations A.
More informationOn the quantiles of the Brownian motion and their hitting times.
On the quantiles of the Brownian motion and their hitting times. Angelos Dassios London School of Economics May 23 Abstract The distribution of the α-quantile of a Brownian motion on an interval [, t]
More informationStochastic optimal control with rough paths
Stochastic optimal control with rough paths Paul Gassiat TU Berlin Stochastic processes and their statistics in Finance, Okinawa, October 28, 2013 Joint work with Joscha Diehl and Peter Friz Introduction
More information13 The martingale problem
19-3-2012 Notations Ω complete metric space of all continuous functions from [0, + ) to R d endowed with the distance d(ω 1, ω 2 ) = k=1 ω 1 ω 2 C([0,k];H) 2 k (1 + ω 1 ω 2 C([0,k];H) ), ω 1, ω 2 Ω. F
More informationStability of Stochastic Differential Equations
Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010
More informationMan Kyu Im*, Un Cig Ji **, and Jae Hee Kim ***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 19, No. 4, December 26 GIRSANOV THEOREM FOR GAUSSIAN PROCESS WITH INDEPENDENT INCREMENTS Man Kyu Im*, Un Cig Ji **, and Jae Hee Kim *** Abstract.
More informationHomogenization with stochastic differential equations
Homogenization with stochastic differential equations Scott Hottovy shottovy@math.arizona.edu University of Arizona Program in Applied Mathematics October 12, 2011 Modeling with SDE Use SDE to model system
More informationAsymptotic properties of maximum likelihood estimator for the growth rate for a jump-type CIR process
Asymptotic properties of maximum likelihood estimator for the growth rate for a jump-type CIR process Mátyás Barczy University of Debrecen, Hungary The 3 rd Workshop on Branching Processes and Related
More informationRandom attractors and the preservation of synchronization in the presence of noise
Random attractors and the preservation of synchronization in the presence of noise Peter Kloeden Institut für Mathematik Johann Wolfgang Goethe Universität Frankfurt am Main Deterministic case Consider
More informationShort-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility
Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility José Enrique Figueroa-López 1 1 Department of Statistics Purdue University Statistics, Jump Processes,
More informationSmoothness of the distribution of the supremum of a multi-dimensional diffusion process
Smoothness of the distribution of the supremum of a multi-dimensional diffusion process Masafumi Hayashi Arturo Kohatsu-Higa October 17, 211 Abstract In this article we deal with a multi-dimensional diffusion
More informationLarge Deviations Principles for McKean-Vlasov SDEs, Skeletons, Supports and the law of iterated logarithm
Large Deviations Principles for McKean-Vlasov SDEs, Skeletons, Supports and the law of iterated logarithm Gonçalo dos Reis University of Edinburgh (UK) & CMA/FCT/UNL (PT) jointly with: W. Salkeld, U. of
More informationLarge Deviations for Perturbed Reflected Diffusion Processes
Large Deviations for Perturbed Reflected Diffusion Processes Lijun Bo & Tusheng Zhang First version: 31 January 28 Research Report No. 4, 28, Probability and Statistics Group School of Mathematics, The
More informationErgodicity of Stochastic Differential Equations Driven by Fractional Brownian Motion
Ergodicity of Stochastic Differential Equations Driven by Fractional Brownian Motion April 15, 23 Martin Hairer Mathematics Research Centre, University of Warwick Email: hairer@maths.warwick.ac.uk Abstract
More informationLower Tail Probabilities and Normal Comparison Inequalities. In Memory of Wenbo V. Li s Contributions
Lower Tail Probabilities and Normal Comparison Inequalities In Memory of Wenbo V. Li s Contributions Qi-Man Shao The Chinese University of Hong Kong Lower Tail Probabilities and Normal Comparison Inequalities
More informationFast-slow systems with chaotic noise
Fast-slow systems with chaotic noise David Kelly Ian Melbourne Courant Institute New York University New York NY www.dtbkelly.com May 1, 216 Statistical properties of dynamical systems, ESI Vienna. David
More informationApproximation of BSDEs using least-squares regression and Malliavin weights
Approximation of BSDEs using least-squares regression and Malliavin weights Plamen Turkedjiev (turkedji@math.hu-berlin.de) 3rd July, 2012 Joint work with Prof. Emmanuel Gobet (E cole Polytechnique) Plamen
More informationRegularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions
Nils Berglund nils.berglund@univ-orleans.fr http://www.univ-orleans.fr/mapmo/membres/berglund/ EPFL, Seminar of Probability and Stochastic Processes Regularity structures and renormalisation of FitzHugh-Nagumo
More informationSequential maximum likelihood estimation for reflected Ornstein-Uhlenbeck processes
Sequential maximum likelihood estimation for reflected Ornstein-Uhlenbeck processes Chihoon Lee Department of Statistics Colorado State University Fort Collins, CO 8523 Jaya P. N. Bishwal Department of
More informationASYMPTOTIC PROPERTIES OF MAXIMUM LIKELIHOOD ES- TIMATION: PARAMETERISED DIFFUSION IN A MANIFOLD
ASYMPOIC PROPERIES OF MAXIMUM LIKELIHOOD ES- IMAION: PARAMEERISED DIFFUSION IN A MANIFOLD S. SAID, he University of Melbourne J. H. MANON, he University of Melbourne Abstract his paper studies maximum
More informationIntertwinings for Markov processes
Intertwinings for Markov processes Aldéric Joulin - University of Toulouse Joint work with : Michel Bonnefont - Univ. Bordeaux Workshop 2 Piecewise Deterministic Markov Processes ennes - May 15-17, 2013
More informationThe Cameron-Martin-Girsanov (CMG) Theorem
The Cameron-Martin-Girsanov (CMG) Theorem There are many versions of the CMG Theorem. In some sense, there are many CMG Theorems. The first version appeared in ] in 944. Here we present a standard version,
More informationOn the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem
On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem Koichiro TAKAOKA Dept of Applied Physics, Tokyo Institute of Technology Abstract M Yor constructed a family
More informationRisk-Sensitive and Robust Mean Field Games
Risk-Sensitive and Robust Mean Field Games Tamer Başar Coordinated Science Laboratory Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, IL - 6181 IPAM
More information