Discretization of Stochastic Differential Systems With Singular Coefficients Part II
|
|
- Mark Williamson
- 6 years ago
- Views:
Transcription
1 Discretization of Stochastic Differential Systems With Singular Coefficients Part II Denis Talay, INRIA Sophia Antipolis joint works with Mireille Bossy, Nicolas Champagnat, Sylvain Maire, Miguel Martinez, Nicolas Perrin ICERM - Brown November 212
2 Outline 1 Introduction 2 The one dimensional case 3 The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics 4 The semi-linear 3D Poisson-Boltzmann PDE in Molecular Dynamics
3 Outline 1 Introduction 2 The one dimensional case 3 The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics 4 The semi-linear 3D Poisson-Boltzmann PDE in Molecular Dynamics
4 On PDEs driven by divergence form operators Consider elliptic or parabolic PDEs driven by the strongly elliptic divergence form operator where L := 1 2 div(a(x) ), < λ ξ 2 (a(x)ξ, ξ) Λ ξ 2 < + for all x, ξ R d.
5 Techniques related to the generation of semigroups in H 1 (R d ): Variational formulations: Aronson, Stroock. Dirichlet form theory applied to forms of the type E(u, u) := 1 u(x) a(x) u(x)q(x)dx, 2 where q is a strictly positive density. Pseudo SDEs (Lyons Zheng decompositions: Fukushima, Roskoz, Slominski): For all function φ in Wp 1,loc (R d ), there exists a pair of local martingales (M φ, N φ ) respectively adapted with respect to the filtration generated by (X t, t T ) and the filtration generated by (X T t, t T ), such that φ(x t ) = φ(x )+ 1 2 M φ t N φ t and M φ t =... t t a φ φ(x θ )dθ and N φ t = a(x θ ) p(θ, x, X θ ) φ(x θ )dθ, p(θ, x, X θ ) t a φ φ(x T θ )dθ.
6 Techniques related to the generation of semigroups in H 1 (R d ): Variational formulations: Aronson, Stroock. Dirichlet form theory applied to forms of the type E(u, u) := 1 u(x) a(x) u(x)q(x)dx, 2 where q is a strictly positive density. Pseudo SDEs (Lyons Zheng decompositions: Fukushima, Roskoz, Slominski): For all function φ in Wp 1,loc (R d ), there exists a pair of local martingales (M φ, N φ ) respectively adapted with respect to the filtration generated by (X t, t T ) and the filtration generated by (X T t, t T ), such that φ(x t ) = φ(x )+ 1 2 M φ t N φ t and M φ t =... t t a φ φ(x θ )dθ and N φ t = a(x θ ) p(θ, x, X θ ) φ(x θ )dθ, p(θ, x, X θ ) t a φ φ(x T θ )dθ.
7 Techniques related to the generation of semigroups in H 1 (R d ): Variational formulations: Aronson, Stroock. Dirichlet form theory applied to forms of the type E(u, u) := 1 u(x) a(x) u(x)q(x)dx, 2 where q is a strictly positive density. Pseudo SDEs (Lyons Zheng decompositions: Fukushima, Roskoz, Slominski): For all function φ in Wp 1,loc (R d ), there exists a pair of local martingales (M φ, N φ ) respectively adapted with respect to the filtration generated by (X t, t T ) and the filtration generated by (X T t, t T ), such that φ(x t ) = φ(x )+ 1 2 M φ t N φ t and M φ t =... t t a φ φ(x θ )dθ and N φ t = a(x θ ) p(θ, x, X θ ) φ(x θ )dθ, p(θ, x, X θ ) t a φ φ(x T θ )dθ.
8 Techniques related to the generation of semigroups in H 1 (R d ): Variational formulations: Aronson, Stroock. Dirichlet form theory applied to forms of the type E(u, u) := 1 u(x) a(x) u(x)q(x)dx, 2 where q is a strictly positive density. Pseudo SDEs (Lyons Zheng decompositions: Fukushima, Roskoz, Slominski): For all function φ in Wp 1,loc (R d ), there exists a pair of local martingales (M φ, N φ ) respectively adapted with respect to the filtration generated by (X t, t T ) and the filtration generated by (X T t, t T ), such that φ(x t ) = φ(x )+ 1 2 M φ t N φ t and M φ t =... t t a φ φ(x θ )dθ and N φ t = a(x θ ) p(θ, x, X θ ) φ(x θ )dθ, p(θ, x, X θ ) t a φ φ(x T θ )dθ.
9 Remark. Pardoux Williams have exhibited a Lyons Zheng decomposition for Dirichlet forms with degenerate Neumann boundary conditions. The Lyons Zheng decompositions cannot lead to algorithms since one should first compute the transition density p(t, x, y) of the Markov process, that is, the fundamental solution.
10 Remark. Pardoux Williams have exhibited a Lyons Zheng decomposition for Dirichlet forms with degenerate Neumann boundary conditions. The Lyons Zheng decompositions cannot lead to algorithms since one should first compute the transition density p(t, x, y) of the Markov process, that is, the fundamental solution.
11 Parabolic diffraction problems Given a finite time horizon T and a positive matrix-valued function a(x) which is smooth except at the interface surfaces between subdomains of R d, consider the parabolic diffraction problem t u(t, x) 1 2 div(a(x) )u(t, x) = for all (t, x) (, T ] Rd, u(, x) = f (x) for all x R d, Compatibility transmission conditions along the interfaces surfaces. Suppose that 1 2 div(a(x) ) is a strongly elliptic operator. Existence and uniqueness of continuous solutions with possibly discontinuous derivatives along the surfaces hold true: see, e.g. Ladyzenskaya et al. Motivations: Neurosciences (3D brains!), Molecular Dynamics, Geophysics, etc.
12 Outline 1 Introduction 2 The one dimensional case 3 The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics 4 The semi-linear 3D Poisson-Boltzmann PDE in Molecular Dynamics
13 We consider the one dimensional parabolic problem ( ) t u(t, x) 1 2 x (a(x) x u(t, x)) =, (t, x) (, T ] (R {}), u(t, +) = u(t, ), t [, T ], u(, x) = f (x), x R, a(+) x u(t, +) = a( ) x u(t, ), t [, T ]. ( ) Suppose λ >, Λ >, < λ a(x) = (σ(x)) 2 Λ < + for all x R. Suppose also that σ is of class C 3 b (R {}) and is left and right continuous at point. Suppose finally that the first derivative of the function σ has finite left and right limits at.
14 The key SDE with weighted local time The one dimensional case allows specific analytical and numerical tools: Portenko (1979), Le Gall (1985), Lejay-Martinez (23)... Consider the one-dimensional stochastic differential equation with local time dx t = σ(x t )db t + σ(x t )σ (X t )dt + σ2 (+) σ 2 ( ) 2σ 2 dl t (X ). (+) Here L t (X ) is the right-sided local time corresponding to the sign function defined as sgn(x) := 1 for x > and sgn(x) := 1 for x and σ is the left derivative of σ. Under mild hypotheses on σ this SDE has a unique weak solution which is a strong Markov process : see, e.g., Le Gall (1984).
15 Probabilistic interpretation Theorem 1 Let the bounded function f be in the set { W 2 := g Cb 2 (R {}), g (i) L 2 (R) L 1 (R) for i = 1, 2, Then the function a(+)g (+) = a( )g ( )}. u(t, x) := E x f (X t ), (t, x) [, T ] R, is the unique function in C 1,2 b ([, T ] (R {})) and continuous on [, T ] R which satisfies the diffraction PDE.
16 Difficulties: One cannot apply Itô-Tanaka s formula to u(t, X t ) because u is time dependent. Astonishingly, proving that u(t, x) satisfies the transmission condition is not simple.
17 Difficulties: One cannot apply Itô-Tanaka s formula to u(t, X t ) because u is time dependent. Astonishingly, proving that u(t, x) satisfies the transmission condition is not simple.
18 Proof of Theorem 1 Key observation: for all function g of class Cb 2 (R {}) having a second derivative in the sense of the distributions which is a Radon measure and satisfying the transmission condition a(+)g (+) = a( )g ( ), the Itô Tanaka formula applied to g(x t ) leads to where t x R, t >, E x g(x t ) = g(x) + E x Lg(X s )ds Lg(x) := σ(x)σ (x) x g (x) a(x) 2 xx g(x)i x.
19 First step: smoothness and boundedness. Let σ + (x) be an arbitrary C 3 b (R) extension of the function σ(x)i x> which satisfies, for a + (x) := (σ + (x)) 2, < λ a + (x) Λ < + for all x R. Denote by (X t + ) the unique strong solution to dx t + = σ + (X t + )db t + σ + (X t + )(σ + ) (X t + )dt. Let τ (X ) be the first passage time of the process (X t ) at point : τ (X ) := inf{s > : X s = }. Notice that τ (X ) = τ (X + ). Let r x (s) be the density under P x of τ (X ) T. For all function φ such that E φ(x t ) is finite, for all x >, t t E x φ(x t ) = E x φ(x t + ) E φ(x s + )r x (t s)ds+ E φ(x s )r x (t s)ds. It remains to prove estimates on the derivatives of r x (s).
20 Second step: the differential part of the diffraction PDE In view of the preceding key observation, for all < t < T, < ɛ < T t and x in R, u(t + ɛ, x) u(t, x) = E x f (X t+ɛ ) E x f (X t ) = In addition, by the strong Markov property, t+ɛ u(t + ɛ, x) u(t, x) = E x u(t, X ɛ ) u(t, x). Then easy calculations lead to t u(t, x) = Lu(t, x). t E x Lf (X s )ds.
21 Third step: u(t, x) satisfies the transmission condition. In view of of the preceding first step, for all fixed t the second partial derivative w.r.t. x of u(t, x) is a Radon measure. Thus the Itô-Tanaka formula applied to u(t, X s ) for s ɛ and fixed time t leads to (a(+) x u(t, +) a( ) x u(t, )) E L ɛ(x ) = 2a(+) It then remains to prove ( t+ɛ t lim inf ɛ E Lf (X s )ds E L ɛ(x ) ɛ = +. ɛ ) E Lu(t, X s )ds.
22 Last step: uniqueness. As, for all real number x, x = 1 2 (x + x ) and x = 1 2 (x x ), Itô Tanaka s formula implies d(x t ) = 1 2 dx t sgn(x t)dx t dl t (X ) = I Xt>dX t dl t (X ), d(x t ) = 1 2 dx t 1 2 sgn(x t)dx t 1 2 dl t (X ) = I Xt<dX t a( ) 2a(+) dl t (X ). Now, let U (t, x) be an arbitrary solution. For all fixed t in [, T ] the function U (t s, x) is of class C 1,2 b ([, t] R {}) and its partial derivatives have left and right limits when x tends to. Thus we may apply the classical Itô s formula (no need of Itô-Tanaka s formula!) to this function and the semimartingales (X s ) and (X s ) and to use that, by hypothesis, U (t, x) satisfies the transmission condition.
23 Smoothness properties in L 1 (R) of the transition semigroup of (X t ) Theorem 2 The probability distribution of X t under P x has a density q X (x, t, y) which satisfies: and C >, x R, t >, Leb-a-e. y R {}, q X (x, t, y) C t C >, x R, t (, T ], f L 1 (R), u(t, x) = E x f (X t ) C t f 1.
24 Suppose in addition that the function σ is of class Cb 4 (R {}) and that its three first derivatives have finite left and right limits at. Set { W 4 := g Cb 4 (R {}), g (i) L 2 (R) L 1 (R) for i = 1,..., 4 a(+)g (+) = a( )g ( ) and a(+)(lg) (+) = a( )(Lg) ( Then, for all j =, 1, 2 and i = 1,..., 4 such that 2j + i 4, C >, x R, t (, T ], f W 4, j t i x u(t, x) C t f γ,1, where γ = 1 if 2j + i = 1 or 2, and γ = 3 if 2j + i = 3 or 4, and g l,p := l x i g p. i=
25 Proof of Theorem 2 A key argument: we can closely follow a part of the proof of Aronson s estimate (see, e.g., Bass and Stroock), starting with observing that, owing to the condition transmision satisfied by fonctions in W 2, integrating by parts leads to φ Cb 1 (R), φ(x) Lf (x) dx = φ (x) a(x) f (x) dx; similarly, as P t f (x) satisfies the transmission condition, two successive integrations by parts lead to t >, φ W 2, φ(x) L(P t f )(x) dx = Lφ(x) (P t f )(x) dx.
26 A SDE with discontinuous coefficients without local time Set Set also β + := β := 2a( ) a(+)+a( ), 2a(+) a(+)+a( ), β(x) := x ( β I x< + β + I x> ), β 1 (x) = x β I x< + x β + I x>. σ(x) := σ β 1 (x) ( β I x + β + I x> ), b(x) := σ β 1 (x)σ β 1 (x) ( β I x + β + I x> ). Adapt a calculation in Le Gall and apply Itô Tanaka s formula to β(x t ). The process Y := β(x ) satisfies the SDE with discontinuous coefficients : t t Y t = β(x ) + σ(y s )db s + b(y s )ds.
27 Our Euler type discretization scheme We now present a Euler type scheme. For other methods: see A. Lejay. Let Approximate (Y t ) by h n := T n and t n k := k h n. Y n t = Y n t n k + σ(y n t n k )I Y n t n k (B t B t n k ) + b(y n t n k )I Y n t n k (t t n k ). Then set X n t = β 1 (Y n t ), t T. Remark. The Euler scheme (X t ) converges weakly to (X t ) since (Y t ) converges weakly to (Y t ) (see Yan). However, as the coefficients b and σ are discontinuous, no classical convergence rate estimate applies.
28 Convergence rate estimates Theorem 3 Under the above hypotheses on the function σ, there exists a positive number C such that, for all initial condition f in W 4, all parameter < ɛ < 1 2, all n large enough, and all x in R, E x f (X T ) E x f (X n T ) C f 1,1 h (1 ɛ)/2 +C f 1,1 hn +C f 3,1 h 1 ɛ n n. Theorem 4 Let f : R R be in the space W := { g C 4 b (R {}), g (i) L 2 (R) L 1 (R) for i = 1,..., 4, There exists a positive number C (depending on f ) such that, for all < ɛ < 1 2, all n large enough, and all x in R, E x f (X T ) E x f (X n T ) Ch 1/2 ɛ n. }.
29 A discretization error decomposition The discretization error satisfies ɛ T := E(f β 1 (Y T )) E(f β 1 (Y n T )) n 1 = (E(u(T tk n, β 1 (Y n t ))) Eu(T t n k k+1, β 1 (Y n n t ))), k+1 n k= from which n 2 { ɛ T E u(θk n, β 1 (Y n t )) k n u(θn k+1, β 1 (Y n t )) k n k= } +u(θk+1, n β 1 (Y n t )) k n u(θn k+1, β 1 (Y n t )) k+1 n + Eu(θ1 n, β 1 (Y n t )) Eu(, n 1 n β 1 (Y n T )) n 2 =: E{T k S k } + ER n 1. k=
30 Methodology We distinguish several cases. When Y n t and Y n k n t are simultaneously positive or negative, we k+1 n use a Taylor expansion of u(tk+1 n, ) around (t k n, Y n t ) and then k n apply accurate estimates of the derivatives of u(t, x) for t in (, T ] and x in R {}. We prove that Y n t and Y n k n t have opposite signs with small k+1 n probability when Y n t is large enough. k n When Y n t is small, we explicit the expansion of u(t n k n k+1, ) around and use the fact that u(t, x) solves the transmission problem, which allows us to cancel the lower order term in the expansion.
31 Methodology We distinguish several cases. When Y n t and Y n k n t are simultaneously positive or negative, we k+1 n use a Taylor expansion of u(tk+1 n, ) around (t k n, Y n t ) and then k n apply accurate estimates of the derivatives of u(t, x) for t in (, T ] and x in R {}. We prove that Y n t and Y n k n t have opposite signs with small k+1 n probability when Y n t is large enough. k n When Y n t is small, we explicit the expansion of u(t n k n k+1, ) around and use the fact that u(t, x) solves the transmission problem, which allows us to cancel the lower order term in the expansion.
32 Methodology We distinguish several cases. When Y n t and Y n k n t are simultaneously positive or negative, we k+1 n use a Taylor expansion of u(tk+1 n, ) around (t k n, Y n t ) and then k n apply accurate estimates of the derivatives of u(t, x) for t in (, T ] and x in R {}. We prove that Y n t and Y n k n t have opposite signs with small k+1 n probability when Y n t is large enough. k n When Y n t is small, we explicit the expansion of u(t n k n k+1, ) around and use the fact that u(t, x) solves the transmission problem, which allows us to cancel the lower order term in the expansion.
33 A key estimates A discrete version of Krylov s inequality: There exists C > such that, for all ξ R d and < ε < 1/2, there exists h > satisfying N h h h, h f (kh)p( X ph ξ h 1/2 ε ) Ch 1/2 ε, k= where N h := T /h 1.
34 Outline 1 Introduction 2 The one dimensional case 3 The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics 4 The semi-linear 3D Poisson-Boltzmann PDE in Molecular Dynamics
35 The Poisson-Boltzmann PDE The Poisson-Boltzmann (PB) PDE in Molecular Dynamics describes the electrostatic potential around a biomolecular assembly, and is used to compute global characteristics of the system such as the solvatation free energy, the electrostatic forces exerted by the solvent on the molecule. The implicit solvent equation, which means that the solvent is considered as a continuum, reads where (ε(x) u(x)) + κ 2 (x)u(x) = f (x), x R 3, ε(x) is the permittivity of the medium, κ 2 (x) is called the ion accessibility parameter.
36
37 The geometry of the problem The atomic structure of the molecule modelled as N atoms at positions x 1,..., x N in Ω i with radii r 1,..., r N and charge q i, Ω i = N i=1 B(x i, r i ).
38 Other difficulties The source term is singular f := N q i δ xi. i=1 This difficulty can be removed by considering the solution G of ε i G = f, that is, G(x) = 1 4πε int N l=1 q j x x l x Ω int. Then v := u χg solves the smoothened PB equation with a smooth source term g provided that χ has compact support in Ω i and χ 1 in the neighborhood of {x 1,..., x N }. The function κ is discontinuous. We must deal with it. The operator has divergence form with discontinuous coefficient ε. We must deal with it.
39 The general case Assume that Γ is a smooth (C 3 ) manifold in R d. Notation: π(x) for the orthogonal projection of x on Γ, n(y) as the outward normal to Γ for y Γ, ρ(x) as the signed distance between x and Γ. ρ(x) := (x π(x)) n(π(x)).
40 A martingale problem We say that (P x ) x R d on (C, B) solves the martingale problem (MP) for L if, for all x R d, one has and, for all ϕ satisfying one has M ϕ t (w) := ϕ(w(t)) ϕ(w()) P x {w C : w() = x} = 1, ϕ C b (R d ) C 2 b (R d \ Γ), ε ϕ (n π) C b (N ), t Lϕ(w(s))ds is a P x martingale. Remark: the test functions satisfy the transmission property ε int int ϕ(x) n(x) = ε ext ext ϕ(x) n(x).
41 Our main result Theorem The martingale problem for L is well-posed. In addition, there is weak existence and uniqueness in law for the SDE dx t = 2ε(X t )db t + ε ext ε int n(x t )dl t (Y ), 2ε ext Y t = ρ(x t ), and the probability law of X solves the martingale problem for L.
42 Sketch of the proof We construct a smooth local straightening ψ of Γ defined on a neighborhood U of x, s.t. ψ 1 = ρ and Z t := ψ(x t ) satisfies dz 1 t = 2ε(X t )db t + ε ext ε int dl t (Z 1 ) + (drift)(z t )dt, 2ε ext and there is no local time term in the SDE solved by Zt 2,..., Zt d. Girsanov s formula allows one to remove the drift term, so that Z 1 t solves a one-dimensional SDE. Conditionnally to Z 1, Z 2,..., Z d solves a classical SDE with time dependent coefficients. Only weak existence.
43 Lemma (Generalized Itô-Meyer formula) If X is a continuous semimartingale, Y := ρ(x ), and if φ is a test function for the MP for L, then t φ(x t ) = φ(x )+ int φ(x s ) dx s t 3 i,j =1 t + 1 g(x s )dl 2 s(y ), ( ) εint where g(x) := 1 int φ(π(x)) n(π(x)). ε ext 2 u x i x j (X s )d X i, X j s t a.s., The formula would be easily obtained from Itô s and Itô-Tanaka s formulas if the functions φ(x) g(x)[ρ(x)] + and g(x) were C 2. If (X t ) solves the preceding SDE, the local time terms cancel.
44 Feynman-Kac formulas Proposition (First Feynman-Kac representation) Let v be the solution of (ε v) + κ 2 v = g, where g is a smooth function. Then, for all x R 3, [ + ( t ) ] v(x) = E x g(x t ) exp κ 2 (X s )ds dt. This representation does not allow one to develop an efficient numerical scheme because One needs to precisely discretize X everywhere where g is nonzero. In general, the computation of g is costly. Since X has (scaled) Brownian paths aaway from Γ, it is better to have formulas only involving informations on the entrance time and position in small neighborhoods of Γ.
45 A second Feynman-Kac formula Fix h >. We define the stopping times τ k = inf{t τ k 1 : ρ(x t ) = h} τ k = inf{t τ k : X t Γ} Since (u G) = in Ω i, for all x s.t. ρ(x) h, u(x) = E x [u(x τ 1 ) G(X τ 1 )] + G(x). For all x Ω e, ( τ1 )] u(x) = E x [u(x τ1 ) exp κ 2 (X t )dt. Applying these two formulas recursively yields:
46 Application Theorem One has u(x) = E x [ + k=1 ( ( τk G(Xτk ) G(X τ )) exp κ 2 (X k t )dt) ]. Application: Analyze the convergence rates of (improved) Walk on Spheres algorithms introduced in this context by Mascagni and Simonov.
47 Outline 1 Introduction 2 The one dimensional case 3 The linear 3D Poisson-Boltzmann PDE in Molecular Dynamics 4 The semi-linear 3D Poisson-Boltzmann PDE in Molecular Dynamics
48 The semi-linear Poisson-Boltzmann PDE The semi-linear PB equation reads (ε(x) v(x)) + κ 2 (x) sinh(v(x)) = g(x), x R 3.
49 Interpretation in terms of Backward Stochastic Differential Equations Consider the Backward Stochastic Differential Equation T >, t T, Y x t = Y x T + T t T t Z x s db s. (g(x x s ) κ 2 (X x s ) sinh(y x s ))ds Theorem (N. Perrin: Ph.D. thesis) 1. There exists a unique solution (Y x, Z x ) (in an appropriate space of processes). 2. There exists a unique weak solution to the smoothened PB equation in the space M := {v H 1 (R 3 ) ; cosh(v) 2 1 L 2 (R 3 )}. This solution belongs to C b (R3 ) C 2 (R 3 Γ). In addition, v(x) = Y x + χ(x) G(x).
Simulation of diffusion. processes with discontinuous coefficients. Antoine Lejay Projet TOSCA, INRIA Nancy Grand-Est, Institut Élie Cartan
Simulation of diffusion. processes with discontinuous coefficients Antoine Lejay Projet TOSCA, INRIA Nancy Grand-Est, Institut Élie Cartan From collaborations with Pierre Étoré and Miguel Martinez . Divergence
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 informationIntroduction to Random Diffusions
Introduction to Random Diffusions The main reason to study random diffusions is that this class of processes combines two key features of modern probability theory. On the one hand they are semi-martingales
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 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 informationSome SDEs with distributional drift Part I : General calculus. Flandoli, Franco; Russo, Francesco; Wolf, Jochen
Title Author(s) Some SDEs with distributional drift Part I : General calculus Flandoli, Franco; Russo, Francesco; Wolf, Jochen Citation Osaka Journal of Mathematics. 4() P.493-P.54 Issue Date 3-6 Text
More informationBackward Stochastic Differential Equations with Infinite Time Horizon
Backward Stochastic Differential Equations with Infinite Time Horizon Holger Metzler PhD advisor: Prof. G. Tessitore Università di Milano-Bicocca Spring School Stochastic Control in Finance Roscoff, March
More informationBSDEs and PDEs with discontinuous coecients Applications to homogenization K. Bahlali, A. Elouain, E. Pardoux. Jena, March 2009
BSDEs and PDEs with discontinuous coecients Applications to homogenization K. Bahlali, A. Elouain, E. Pardoux. Jena, 16-20 March 2009 1 1) L p viscosity solution to 2nd order semilinear parabolic PDEs
More informationApplied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.
Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R
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 informationStrong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term
1 Strong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term Enrico Priola Torino (Italy) Joint work with G. Da Prato, F. Flandoli and M. Röckner Stochastic Processes
More informationProbabilistic Interpretation for the Nonlinear Poisson-Boltzmann Equation in Molecular Dynamics
Probabilistic Interpretation for the Nonlinear Poisson-Boltzmann quation in Molecular Dynamics Nicolas Perrin To cite this version: Nicolas Perrin. Probabilistic Interpretation for the Nonlinear Poisson-Boltzmann
More informationOn countably skewed Brownian motion
On countably skewed Brownian motion Gerald Trutnau (Seoul National University) Joint work with Y. Ouknine (Cadi Ayyad) and F. Russo (ENSTA ParisTech) Electron. J. Probab. 20 (2015), no. 82, 1-27 [ORT 2015]
More informationOn semilinear elliptic equations with measure data
On semilinear elliptic equations with measure data Andrzej Rozkosz (joint work with T. Klimsiak) Nicolaus Copernicus University (Toruń, Poland) Controlled Deterministic and Stochastic Systems Iasi, July
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationDiscretization of SDEs: Euler Methods and Beyond
Discretization of SDEs: Euler Methods and Beyond 09-26-2006 / PRisMa 2006 Workshop Outline Introduction 1 Introduction Motivation Stochastic Differential Equations 2 The Time Discretization of SDEs Monte-Carlo
More informationHarmonic Functions and Brownian motion
Harmonic Functions and Brownian motion Steven P. Lalley April 25, 211 1 Dynkin s Formula Denote by W t = (W 1 t, W 2 t,..., W d t ) a standard d dimensional Wiener process on (Ω, F, P ), and let F = (F
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 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 informationBrownian Motion on Manifold
Brownian Motion on Manifold QI FENG Purdue University feng71@purdue.edu August 31, 2014 QI FENG (Purdue University) Brownian Motion on Manifold August 31, 2014 1 / 26 Overview 1 Extrinsic construction
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 informationReflected Brownian Motion
Chapter 6 Reflected Brownian Motion Often we encounter Diffusions in regions with boundary. If the process can reach the boundary from the interior in finite time with positive probability we need to decide
More informationElectrodynamics PHY712. Lecture 3 Electrostatic potentials and fields. Reference: Chap. 1 in J. D. Jackson s textbook.
Electrodynamics PHY712 Lecture 3 Electrostatic potentials and fields Reference: Chap. 1 in J. D. Jackson s textbook. 1. Poisson and Laplace Equations 2. Green s Theorem 3. One-dimensional examples 1 Poisson
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 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 informationu xx + u yy = 0. (5.1)
Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function
More informationMarkov Chain Approximation of Pure Jump Processes. Nikola Sandrić (joint work with Ante Mimica and René L. Schilling) University of Zagreb
Markov Chain Approximation of Pure Jump Processes Nikola Sandrić (joint work with Ante Mimica and René L. Schilling) University of Zagreb 8 th International Conference on Lévy processes Angers, July 25-29,
More informationGeometric projection of stochastic differential equations
Geometric projection of stochastic differential equations John Armstrong (King s College London) Damiano Brigo (Imperial) August 9, 2018 Idea: Projection Idea: Projection Projection gives a method of systematically
More informationExact Simulation of Diffusions and Jump Diffusions
Exact Simulation of Diffusions and Jump Diffusions A work by: Prof. Gareth O. Roberts Dr. Alexandros Beskos Dr. Omiros Papaspiliopoulos Dr. Bruno Casella 28 th May, 2008 Content 1 Exact Algorithm Construction
More informationSTATISTICS 385: STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 4 DUE NOVEMBER 23, = (2n 1)(2n 3) 3 1.
STATISTICS 385: STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 4 DUE NOVEMBER 23, 26 Problem Normal Moments (A) Use the Itô formula and Brownian scaling to check that the even moments of the normal distribution
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationObliquely Reflected Brownian motions (ORBMs) in Non-Smooth Domains
Obliquely Reflected Brownian motions (ORBMs) in Non-Smooth Domains Kavita Ramanan, Brown University Frontier Probability Days, U. of Arizona, May 2014 Why Study Obliquely Reflected Diffusions? Applications
More informationThe Pedestrian s Guide to Local Time
The Pedestrian s Guide to Local Time Tomas Björk, Department of Finance, Stockholm School of Economics, Box 651, SE-113 83 Stockholm, SWEDEN tomas.bjork@hhs.se November 19, 213 Preliminary version Comments
More informationHomework #6 : final examination Due on March 22nd : individual work
Université de ennes Année 28-29 Master 2ème Mathématiques Modèles stochastiques continus ou à sauts Homework #6 : final examination Due on March 22nd : individual work Exercise Warm-up : behaviour of characteristic
More informationADAPTIVE WEAK APPROXIMATION OF REFLECTED AND STOPPED DIFFUSIONS
ADAPTIVE WEAK APPROXIMATION OF REFLECTED AND STOPPED DIFFUSIONS CHRISTIAN BAYER, ANDERS SZEPESSY, AND RAÚL TEMPONE ABSTRACT. We study the weak approximation problem of diffusions, which are reflected at
More informationStochastic Calculus February 11, / 33
Martingale Transform M n martingale with respect to F n, n =, 1, 2,... σ n F n (σ M) n = n 1 i= σ i(m i+1 M i ) is a Martingale E[(σ M) n F n 1 ] n 1 = E[ σ i (M i+1 M i ) F n 1 ] i= n 2 = σ i (M i+1 M
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 informationUniformly Uniformly-ergodic Markov chains and BSDEs
Uniformly Uniformly-ergodic Markov chains and BSDEs Samuel N. Cohen Mathematical Institute, University of Oxford (Based on joint work with Ying Hu, Robert Elliott, Lukas Szpruch) Centre Henri Lebesgue,
More informationFinite element approximation of the stochastic heat equation with additive noise
p. 1/32 Finite element approximation of the stochastic heat equation with additive noise Stig Larsson p. 2/32 Outline Stochastic heat equation with additive noise du u dt = dw, x D, t > u =, x D, t > u()
More informationOn some weighted fractional porous media equations
On some weighted fractional porous media equations Gabriele Grillo Politecnico di Milano September 16 th, 2015 Anacapri Joint works with M. Muratori and F. Punzo Gabriele Grillo Weighted Fractional PME
More informationINVARIANT MANIFOLDS WITH BOUNDARY FOR JUMP-DIFFUSIONS
INVARIANT MANIFOLDS WITH BOUNDARY FOR JUMP-DIFFUSIONS DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN Abstract. We provide necessary and sufficient conditions for stochastic invariance of finite dimensional
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 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 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 informationFrequency functions, monotonicity formulas, and the thin obstacle problem
Frequency functions, monotonicity formulas, and the thin obstacle problem IMA - University of Minnesota March 4, 2013 Thank you for the invitation! In this talk we will present an overview of the parabolic
More informationarxiv: v3 [math.na] 26 Jan 2015
Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation Mireille Bossy Nicolas Champagnat Hélène Leman Sylvain Maire Laurent Violeau January 27, 215 Mariette Yvinec arxiv:1411.234v3 [math.na]
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Numerical Methods for Partial Differential Equations Finite Difference Methods
More informationIntegral Representation Formula, Boundary Integral Operators and Calderón projection
Integral Representation Formula, Boundary Integral Operators and Calderón projection Seminar BEM on Wave Scattering Franziska Weber ETH Zürich October 22, 2010 Outline Integral Representation Formula Newton
More informationIntroduction to numerical schemes
236861 Numerical Geometry of Images Tutorial 2 Introduction to numerical schemes Heat equation The simple parabolic PDE with the initial values u t = K 2 u 2 x u(0, x) = u 0 (x) and some boundary conditions
More informationLecture 17 Brownian motion as a Markov process
Lecture 17: Brownian motion as a Markov process 1 of 14 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 17 Brownian motion as a Markov process Brownian motion is
More informationStochastic integration. P.J.C. Spreij
Stochastic integration P.J.C. Spreij this version: April 22, 29 Contents 1 Stochastic processes 1 1.1 General theory............................... 1 1.2 Stopping times...............................
More informationON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME
ON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME SAUL D. JACKA AND ALEKSANDAR MIJATOVIĆ Abstract. We develop a general approach to the Policy Improvement Algorithm (PIA) for stochastic control problems
More informationObstacle problems for nonlocal operators
Obstacle problems for nonlocal operators Camelia Pop School of Mathematics, University of Minnesota Fractional PDEs: Theory, Algorithms and Applications ICERM June 19, 2018 Outline Motivation Optimal regularity
More informationFinal: Solutions Math 118A, Fall 2013
Final: Solutions Math 118A, Fall 2013 1. [20 pts] For each of the following PDEs for u(x, y), give their order and say if they are nonlinear or linear. If they are linear, say if they are homogeneous or
More informationIn this chapter we study elliptical PDEs. That is, PDEs of the form. 2 u = lots,
Chapter 8 Elliptic PDEs In this chapter we study elliptical PDEs. That is, PDEs of the form 2 u = lots, where lots means lower-order terms (u x, u y,..., u, f). Here are some ways to think about the physical
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 informationWEYL S LEMMA, ONE OF MANY. Daniel W. Stroock
WEYL S LEMMA, ONE OF MANY Daniel W Stroock Abstract This note is a brief, and somewhat biased, account of the evolution of what people working in PDE s call Weyl s Lemma about the regularity of solutions
More informationThe Azéma-Yor Embedding in Non-Singular Diffusions
Stochastic Process. Appl. Vol. 96, No. 2, 2001, 305-312 Research Report No. 406, 1999, Dept. Theoret. Statist. Aarhus The Azéma-Yor Embedding in Non-Singular Diffusions J. L. Pedersen and G. Peskir Let
More informationWalsh Diffusions. Andrey Sarantsev. March 27, University of California, Santa Barbara. Andrey Sarantsev University of Washington, Seattle 1 / 1
Walsh Diffusions Andrey Sarantsev University of California, Santa Barbara March 27, 2017 Andrey Sarantsev University of Washington, Seattle 1 / 1 Walsh Brownian Motion on R d Spinning measure µ: probability
More informationSkew Brownian Motion and Applications in Fluid Dispersion
Skew Brownian Motion and Applications in Fluid Dispersion Ed Waymire Department of Mathematics Oregon State University Corvallis, OR 97331 *Based on joint work with Thilanka Appuhamillage, Vrushali Bokil,
More informationChapter 1. Introduction to Electrostatics
Chapter. Introduction to Electrostatics. Electric charge, Coulomb s Law, and Electric field Electric charge Fundamental and characteristic property of the elementary particles There are two and only two
More informationLAN 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 informationThe heat kernel meets Approximation theory. theory in Dirichlet spaces
The heat kernel meets Approximation theory in Dirichlet spaces University of South Carolina with Thierry Coulhon and Gerard Kerkyacharian Paris - June, 2012 Outline 1. Motivation and objectives 2. The
More informationTwo viewpoints on measure valued processes
Two viewpoints on measure valued processes Olivier Hénard Université Paris-Est, Cermics Contents 1 The classical framework : from no particle to one particle 2 The lookdown framework : many particles.
More informationElliptic Operators with Unbounded Coefficients
Elliptic Operators with Unbounded Coefficients Federica Gregorio Universitá degli Studi di Salerno 8th June 2018 joint work with S.E. Boutiah, A. Rhandi, C. Tacelli Motivation Consider the Stochastic Differential
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 informationThe Dirichlet boundary problems for second order parabolic operators satisfying a Carleson condition
The Dirichlet boundary problems for second order parabolic operators satisfying a Carleson condition Sukjung Hwang CMAC, Yonsei University Collaboration with M. Dindos and M. Mitrea The 1st Meeting of
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 information1 Brownian Local Time
1 Brownian Local Time We first begin by defining the space and variables for Brownian local time. Let W t be a standard 1-D Wiener process. We know that for the set, {t : W t = } P (µ{t : W t = } = ) =
More informationRigorous Functional Integration with Applications to Nelson s and the Pauli-Fierz Model
Rigorous Functional Integration with Applications to Nelson s and the Pauli-Fierz Model József Lőrinczi Zentrum Mathematik, Technische Universität München and School of Mathematics, Loughborough University
More informationRandom G -Expectations
Random G -Expectations Marcel Nutz ETH Zurich New advances in Backward SDEs for nancial engineering applications Tamerza, Tunisia, 28.10.2010 Marcel Nutz (ETH) Random G-Expectations 1 / 17 Outline 1 Random
More informationWeak convergence and large deviation theory
First Prev Next Go To Go Back Full Screen Close Quit 1 Weak convergence and large deviation theory Large deviation principle Convergence in distribution The Bryc-Varadhan theorem Tightness and Prohorov
More informationREMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID
REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID DRAGOŞ IFTIMIE AND JAMES P. KELLIHER Abstract. In [Math. Ann. 336 (2006), 449-489] the authors consider the two dimensional
More informationModel Specification Testing in Nonparametric and Semiparametric Time Series Econometrics. Jiti Gao
Model Specification Testing in Nonparametric and Semiparametric Time Series Econometrics Jiti Gao Department of Statistics School of Mathematics and Statistics The University of Western Australia Crawley
More informationBIHARMONIC WAVE MAPS INTO SPHERES
BIHARMONIC WAVE MAPS INTO SPHERES SEBASTIAN HERR, TOBIAS LAMM, AND ROLAND SCHNAUBELT Abstract. A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed.
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 PARABOLIC HARNACK INEQUALITY
ON PARABOLIC HARNACK INEQUALITY JIAXIN HU Abstract. We show that the parabolic Harnack inequality is equivalent to the near-diagonal lower bound of the Dirichlet heat kernel on any ball in a metric measure-energy
More informationLecture Notes on PDEs
Lecture Notes on PDEs Alberto Bressan February 26, 2012 1 Elliptic equations Let IR n be a bounded open set Given measurable functions a ij, b i, c : IR, consider the linear, second order differential
More informationMartingale Problems. Abhay G. Bhatt Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Delhi
s Abhay G. Bhatt Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Delhi Lectures on Probability and Stochastic Processes III Indian Statistical Institute, Kolkata 20 24 November
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 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 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 informationSébastien Chaumont a a Institut Élie Cartan, Université Henri Poincaré Nancy I, B. P. 239, Vandoeuvre-lès-Nancy Cedex, France. 1.
A strong comparison result for viscosity solutions to Hamilton-Jacobi-Bellman equations with Dirichlet condition on a non-smooth boundary and application to parabolic problems Sébastien Chaumont a a Institut
More informationLecture 12. F o s, (1.1) F t := s>t
Lecture 12 1 Brownian motion: the Markov property Let C := C(0, ), R) be the space of continuous functions mapping from 0, ) to R, in which a Brownian motion (B t ) t 0 almost surely takes its value. Let
More informationMulti-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form
Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form Yipeng Yang * Under the supervision of Dr. Michael Taksar Department of Mathematics University of Missouri-Columbia Oct
More informationAhmed Mohammed. Harnack Inequality for Non-divergence Structure Semi-linear Elliptic Equations
Harnack Inequality for Non-divergence Structure Semi-linear Elliptic Equations International Conference on PDE, Complex Analysis, and Related Topics Miami, Florida January 4-7, 2016 An Outline 1 The Krylov-Safonov
More informationA probabilistic interpretation of the transmission conditions using the Skew Brownian motion
A probabilistic interpretation of the transmission conditions using the Skew Brownian motion Antoine Lejay 1, Projet OMEGA, INRIA & Institut Élie Cartan, Nancy Abstract: In order to solve with a Monte
More informationStrong Markov property of determinantal processes associated with extended kernels
Strong Markov property of determinantal processes associated with extended kernels Hideki Tanemura Chiba university (Chiba, Japan) (November 22, 2013) Hideki Tanemura (Chiba univ.) () Markov process (November
More informationStochastic Lagrangian Transport and Generalized Relative Entropies
Stochastic Lagrangian Transport and Generalized Relative Entropies Peter Constantin Department of Mathematics, The University of Chicago 5734 S. University Avenue, Chicago, Illinois 6637 Gautam Iyer Department
More informationEXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS
EXISTECE AD REGULARITY RESULTS FOR SOME OLIEAR PARABOLIC EUATIOS Lucio BOCCARDO 1 Andrea DALL AGLIO 2 Thierry GALLOUËT3 Luigi ORSIA 1 Abstract We prove summability results for the solutions of nonlinear
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 informationSplitting methods with boundary corrections
Splitting methods with boundary corrections Alexander Ostermann University of Innsbruck, Austria Joint work with Lukas Einkemmer Verona, April/May 2017 Strang s paper, SIAM J. Numer. Anal., 1968 S (5)
More informationInverse Scattering with Partial data on Asymptotically Hyperbolic Manifolds
Inverse Scattering with Partial data on Asymptotically Hyperbolic Manifolds Raphael Hora UFSC rhora@mtm.ufsc.br 29/04/2014 Raphael Hora (UFSC) Inverse Scattering with Partial data on AH Manifolds 29/04/2014
More informationChap. 1. Some Differential Geometric Tools
Chap. 1. Some Differential Geometric Tools 1. Manifold, Diffeomorphism 1.1. The Implicit Function Theorem ϕ : U R n R n p (0 p < n), of class C k (k 1) x 0 U such that ϕ(x 0 ) = 0 rank Dϕ(x) = n p x U
More informationLAN property for sde s with additive fractional noise and continuous time observation
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,
More informationSTOCHASTIC PERRON S METHOD AND VERIFICATION WITHOUT SMOOTHNESS USING VISCOSITY COMPARISON: OBSTACLE PROBLEMS AND DYNKIN GAMES
STOCHASTIC PERRON S METHOD AND VERIFICATION WITHOUT SMOOTHNESS USING VISCOSITY COMPARISON: OBSTACLE PROBLEMS AND DYNKIN GAMES ERHAN BAYRAKTAR AND MIHAI SÎRBU Abstract. We adapt the Stochastic Perron s
More informationSingular Perturbations of Stochastic Control Problems with Unbounded Fast Variables
Singular Perturbations of Stochastic Control Problems with Unbounded Fast Variables Joao Meireles joint work with Martino Bardi and Guy Barles University of Padua, Italy Workshop "New Perspectives in Optimal
More informationHJB equations. Seminar in Stochastic Modelling in Economics and Finance January 10, 2011
Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague petrasek@karlin.mff.cuni.cz Seminar in Stochastic Modelling in Economics and Finance
More informationA Feynman-Kac Path-Integral Implementation for Poisson s Equation
for Poisson s Equation Chi-Ok Hwang and Michael Mascagni Department of Computer Science, Florida State University, 203 Love Building Tallahassee, FL 32306-4530 Abstract. This study presents a Feynman-Kac
More informationElectrodynamics PHY712. Lecture 4 Electrostatic potentials and fields. Reference: Chap. 1 & 2 in J. D. Jackson s textbook.
Electrodynamics PHY712 Lecture 4 Electrostatic potentials and fields Reference: Chap. 1 & 2 in J. D. Jackson s textbook. 1. Complete proof of Green s Theorem 2. Proof of mean value theorem for electrostatic
More information