On the Goodness-of-Fit Tests for Some Continuous Time Processes

Size: px
Start display at page:

Download "On the Goodness-of-Fit Tests for Some Continuous Time Processes"

Transcription

1 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 du Maine Abstract We present a review of several results concerning the construction of the Cramér-von Mises and Kolmogorov-Smirnov type goodnessof-fit tests for continuous time processes. As the models we take a stochastic differential equation with small noise, ergodic diffusion process, Poisson process and self-exciting point processes. For every model we propose the tests which provide the asymptotic size α and discuss the behaviour of the power function under local alternatives. The results of numerical simulations of the tests are presented. Keywords: Hypotheses testing, diffusion process, Poisson process, selfexciting process, goodness-of-fit tests Introduction The goodness-of-fit tests play an important role in the classical mathematical statistics. Particularly, the tests of Cramér-von Mises, Kolmogorov-Smirnov and Chi-Squared are well studied and allow to verify the correspondence of the mathematical models to the observed data see, for example, Durbin 973) or Greenwood and Nikulin 996)). The similar problem, of course, exists for the continuous time stochastic processes. The diffusion and Poisson processes are widely used as mathematical models of many evolution

2 processes in Biology, Medicine, Physics, Financial Mathematics and in many others fields. For example, some theory can propose a diffusion process dx t = S X t ) dt + σ dw t, X, t T as an appropriate model for description of the real data {X t, t T } and we can try to construct an algorithm to verify if this model corresponds well to these data. The model here is totally defined by the trend coefficient S ), which is supposed if the theory is true) to be known. We do not discuss here the problem of verification if the process {W t, t T } is Wiener. This problem is much more complicated and we suppose that the noise is white Gaussian. Therefore we have a basic hypothesis defined by the trend coefficient S ) and we have to test this hypothesis against any other alternative. Any other means that the observations come from stochastic differential equation dx t = S X t ) dt + σ dw t, X, t T, where S ) S ). We propose some tests which are in some sense similar to the Cramér-von Mises and Kolmogorov-Smirnov tests. The advantage of classical tests is that they are distribution-free, i.e., the distribution of the underlying statistics do not depend on the basic model and this property allows to choose the universal thresholds, which can be used for all models. For example, if we observe n independent identically distributed random variables X,..., X n ) = X n with distribution function F x) and the basic hypothesis is simple : F x) F x), then the Cramér-von Mises Wn 2 and Kolmogorov-Smirnov D n statistics are W 2 n = n respectively. Here [ ] 2 ˆFn x) F x) df x), D n = sup ˆF n x) F x) x ˆF n x) = n n j= {Xj <x} is the empirical distribution function. Let us denote by {W s), s } a Brownian bridge, i.e., a continuous Gaussian process with EW s) =, EW s) W t) = t s st. 2

3 Then the limit behaviour of these statistics can be described with the help of this process as follows W 2 n = W s) 2 ds, ndn = sup W s). s Hence the corresponding Cramér-von Mises and Kolmogorov-Smirnov tests ψ n X n ) = {W 2 n >c α }, φ n X n ) = { ndn >d α} with constants c α, d α defined by the equations { } { } P W s) 2 ds > c α = α, P sup W s) > d α = α s are of asymptotic size α. It is easy to see that these tests are distributionfree the limit distributions do not depend of the function F )) and are consistent against any fixed alternative see, for example, Durbin 973)). It is interesting to study these tests for nondegenerate set of alternatives, i.e., for alternatives with limit power function less than. It can be realized on the close nonparametric alternatives of the special form making this problem asymptotically equivalent to the signal in Gaussian noise problem. Let us put F x) = F x) + x h F y)) df y), n where the function h ) describes the alternatives. We suppose that h s) ds =, h s) 2 ds <. Then we have the following convergence under fixed alternative, given by the function h )): [ s 2 Wn 2 = h v) dv + W s)] ds, s ndn = sup h v) dv + W s) s We see that this problem is asymptotically equivalent to the following signal in Gaussian noise problem: dy s = h s) ds + dw s), s. ) 3

4 Indeed, if we use the statistics W 2 = Ys 2 ds, D = sup Y s s then under hypothesis h ) and alternative h ) the distributions of these statistics coincide with the limit distributions of W 2 n and nd n under hypothesis and alternative respectively. Our goal is to see how such kind of tests can be constructed in the case of continuous time models of observation and particularly in the cases of some diffusion and point processes. We consider the diffusion processes with small noise, ergodic diffusion processes and Poisson process with Poisson and self-exciting alternatives. For the first two classes we just show how Cramérvon Mises and Kolmogorov-Smirnov - type tests can be realized using some known results and for the last models we discuss this problem in detail. 2 Diffusion process with small noise Suppose that the observed process is the solution of the stochastic differential equation dx t = S X t ) dt + ε dw t, X = x, t T, 2) where W t, t T is a Wiener process see, for example, Liptser and Shiryayev 2)). We assume that the function S x) is two times continuously differentiable with bounded derivatives. These are not the minimal conditions for the results presented below, but this assumption simplifies the exposition. We are interested in the statistical inference for this model in the asymptotics of small noise : ε. The statistical estimation theory parametric and nonparametric) was developed in Kutoyants 994). Recall that the stochastic process X ε = {X t, t T } converges uniformly in t [, T ] to the deterministic function {x t, t T }, which is a solution of the ordinary differential equation dx t dt = S x t), x, t T. 3) 4

5 Suppose that the function S x) > for x x and consider the following problem of hypotheses testing H : S x) = S x), x x x T H : S x) S x), x x x T where we denoted by x t the solution of the equation 3) under hypothesis H : x t = x + t S x v) dv, t T. Hence, we have a simple hypothesis against the composite alternative. The Cramér-von Mises Wε 2 ) and Kolmogorov-Smirnov D ε ) type statistics for this model of observations can be [ T Wε 2 = [ T D ε = dt S x t ) 2 dt S x t ) 2 ] 2 T ) Xt x 2 t ε S x t ) 2 dt, ] /2 sup X t x t t T S x t ). It can be shown that these two statistics converge as ε ) to the following functionals W 2 ε = W s) 2 ds, ε D ε = sup W s), s where {W s), s } is a Wiener process see Kutoyants 994). Hence the corresponding tests ψ ε X ε ) = {W 2 ε >c α}, φ ε X ε ) = {ε D ε>d α} with the constants c α, d α defined by the equations { } { } P W s) 2 ds > c α = α, P sup W s) > d α = α 4) s are of asymptotic size α. Note that the choice of the thresholds c α and d α does not depend on the hypothesis distribution-free). This situation is quite close to the classical case mentioned above. It is easy to see that if S x) S x), then sup t T x t x t > and, ε D ε. Hence these tests are consistent against any fixed W 2 ε 5

6 alternative. It is possible to study the power function of this test for local contiguous) alternatives of the following form dx t = S X t ) dt + ε h X t) S X t ) dt + ε dw t, t T. We describe the alternatives with the help of the unknown) function h ). The case h ) corresponds to the hypothesis H. One special class of such nonparametric alternatives for this model was studied in Iacus and Kutoyants 2). Let us introduce the composite nonparametric) alternative where H ρ = H : h ) H ρ, { h ) : xt x } h x) 2 µ dx) ρ. To choose alternative we have to precise the natural for this problem distance described by the measure µ ) and the rate of ρ = ρ ε. We show that the choice µ dx) = dx S x) 3 provides for the test statistic the following limit [ s 2 Wε 2 h v) dv + W s)] ds, where we denoted h s) = u /2 T h x u T s), ut = T ds S x s) 2 We see that this problem is asymptotically equivalent to the signal in white Gaussian noise problem: dy s = h s) ds + dw s), s, 5) with the Wiener process W ). It is easy to see that even for fixed ρ > without further restrictions on the smoothness of the function h ) the uniformly good testing is impossible. For example, if we put h n x) = c S x) 3 cos [n x x )] 6

7 then for the power function of the test we have inf β ψ ε, h) β ψ ε, h n ) α. h ) H ρ The details can be found in Kutoyants 26). The construction of the uniformly consistent tests requires a different approach see Ingster and Suslina 23)). Note as well that if the diffusion process is dx t = S X t ) dt + εσ X t ) dw t, X = x, t T, then we can put W 2 ε = [ T σ x t ) S x t ) ) 2 dt] 2 T ) Xt x 2 t ε S x t ) 2 dt and have the same results as above see Kutoyants 26)). 3 Ergodic diffusion processes Suppose that the observed process is one dimensional diffusion process dx t = S X t ) dt + dw t, X, t T, 6) where the trend coefficient S x) satisfies the conditions of the existence and uniqueness of the solution of this equation and this solution has ergodic properties, i.e., there exists an invariant probability distribution F S x), and for any integrable w.r.t. this distribution function g x) the law of large numbers holds T T g X t ) dt g x) df S x). These conditions can be found, for example, in Kutoyants 24). Recall that the invariant density function f S x) is defined by the equality { x } f S x) = G S) exp 2 S y) dy, 7

8 where G S) is the normalising constant. We consider two types of tests. The first one is a direct analogue of the classical Cramér-von Mises and Kolmogorov-Smirnov tests based on empirical distribution and density functions and the second follows the considered above small noise) construction of tests. The invariant distribution function F S x) and this density function can be estimated by the empirical distribution function ˆF T x) and by the local time type estimator ˆf T x) defined by the equalities ˆF T x) = T T {Xt <x} dt, ˆfT x) = 2 T respectively. Note that both of them are unbiased: T {Xt <x} dx t admit the representations E S ˆFT x) = F S x), E S ˆfT x) = f S x), η T x) = 2 T T ζ T x) = 2f S x) T F S X t x) F S X t ) F S x) dw t + o ), f S X t ) {Xt>x} F S X t ) dw t + o ) f S X t ) T and are T asymptotically normal as T ) η T x) = T ˆFT x) F S x)) = N, d F S, x) 2), ζ T x) = T ˆfT x) f S x)) = N, d f S, x) 2). Let us fix a simple basic) hypothesis H : S x) S x). Then to test this hypothesis we can use these estimators for construction of the Cramér-von Mises and Kolmogorov-Smirnov type test statistics W 2 T = T D T = sup x [ ] 2 ˆFT x) F S x) dfs x), ˆF T x) F S x) 8

9 and VT 2 = T d T = sup x [ ] 2 ˆfT x) f S x) dfs x), ˆf T x) f S x) respectively. Unfortunately, all these statistics are not distribution-free even asymptotically and the choice of the corresponding thresholds for the tests is much more complicated. Indeed, it was shown that the random functions η T x), x R) and ζ T x), x R) converge in the space C, B) of continuous functions decreasing to zero at infinity) to the zero mean Gaussian processes η x), x R) and ζ x), x R) respectively with the covariance functions we omit the index S of functions f S x) and F S x) below) R F x, y) = E S [η x) η y)] ) [F ξ x) F ξ) F x)] [F ξ y) F ξ) F y)] = 4E S f ξ) 2 R f x, y) = E S [ζ x) ζ y)] [ {ξ>x} F ξ) ] [ {ξ>y} F ξ) ] ) = 4f x) f y) E S f ξ) 2. Here ξ is a random variable with the distribution function F S x). Of course, d F S, x) 2 = E S [ η x) 2 ], d f S, x) 2 = E S [ ζ x) 2 ]. Using this weak convergence it is shown that these statistics converge in distribution under hypothesis) to the following limits as T ) W 2 T = V 2 T = η x) 2 df S x), ζ x) 2 df S x), T /2 D T = sup η x), x T /2 d T = sup ζ x). x The conditions and the proofs of all these properties can be found in Kutoyants 24), where essentially different statistical problems were studied, but the calculus are quite close to what we need here. Note that the Kolmogorov-Smirnov test for ergodic diffusion was studied in Fournie 992) see as well Fournie and Kutoyants 993) for further details), and the weak convergence of the process η T ) was obtained in Negri 998). 9

10 The Cramér-von Mises and Kolmogorov-Smirnov type tests based on these statistics are Ψ T X T ) = { W 2 T >C α}, ψ T X T ) = { V 2 T >c α}, Φ T φ T X T ) = { T /2 D T >D α}, X T ) = { T /2 d T >d α} with appropriate constants. The contiguous alternatives can be introduced by the following way S x) = S x) + h x) T. Then we obtain for the Cramér-von Mises statistics the limits see, Kutoyants 24)) W 2 T = V 2 T = [2E S [{ξ<x} F S xϕ) ] ξ x 2 [2f S x) E S h s) ds + ζ x)] df S x). ξ ) 2 h s) ds + η x)] df S x), Note that the transformation Y t = F S X t ) simplifies the writing, because the diffusion process Y t satisfies the differential equation dy t = f S X t ) [2S X t ) dt + dw t ], Y = F S X ) with reflecting bounds in and and under hypothesis) has uniform on [, ] invariant distribution. Therefore, W 2 T = V s) 2 ds, T /2 D T = sup V s), s but the covariance structure of the Gaussian process {V s), s } can be quite complicated. To obtain asymptotically distribution-free Cramér-von Mises type test we can use another statistic, which is similar to that of the preceding section. Let us introduce W 2 T = T 2 T [ X t X t S X v ) dv] 2 dt.

11 Then we have immediately under hypothesis) W T 2 = T W 2 T 2 t dt = W s) 2 ds, where we put t = st and W s) = T /2 W st. Under alternative we have W 2 T = = T T T 2 T [ W t + t T [ Wt T + t T The stochastic process X t is ergodic, hence t t h X v ) dv E S h ξ) = t t h X v ) dv] 2 dt h X v ) dv] 2 dt. h x) f S x) dx ρ h as t. It can be shown see section 2.3 in Kutoyants 24), where we have the similar calculus in another problem) that W 2 T = [ρ h s + W s)] 2 ds. Therefore the power function of the test ψ X ) T = { W 2 T >c α} converges to the function ) β ψ ρ h ) = P [ρ h s + W s)] 2 ds > c α. Using standard calculus we can show that for the corresponding Kolmogorov-Smirnov type test the limit will be ) β φ ρ h ) = P sup ρ h s + W s) > c α. s These two limit power functions are the same as in the next section devoted to self-exciting alternatives of the Poisson process. We calculate these functions with the help of simulations in Section 5 below. Note that if the diffusion process is dx t = S X t ) dt + σ X t ) dw t, X, t T,

12 but the functions S ) and σ ) are such that the process is ergodic then we introduce the statistics ŴT 2 T [ t 2 = [ T 2 E S σ ξ) 2 ] X t X S X v ) dv] dt. Here ξ is random variable with the invariant density function { x } f S x) = G S ) σ x) 2 exp S y) 2 σ y) 2 dy. This statistic under hypothesis is equal to Ŵ 2 T = = T 2 E S [ σ ξ) 2 ] T E S [ σ ξ) 2 ] T T [ t ] 2 σ X v ) dw v dt [ t ] 2 σ X v ) dw v dt. T The stochastic integral by the central limit theorem is asymptotically normal η t = te S [ σ ξ) 2 ] t σ X v ) dw v = N, ) and moreover it can be shown that the vector of such integrals converges in distribution to the Wiener process ) η s T,..., η sk T = W s ),..., W s k )) for any finite collection of s < s 2 <... < s k. Therefore, under mild regularity conditions it can be proved that Ŵ 2 T = W s) 2 ds. The power function has the same limit, ) β ψ ρ h ) = P [ρ h s + W s)] 2 ds > c α. 2

13 but with ρ h = E S h ξ) E S [ σ ξ) 2 ]. The similar consideration can be done for the Kolmogorov-Smirnov type test too. We see that both tests can not distinguish the alternatives with h ) such that E S h ξ) =. Note [ that for ergodic processes usually we have E S S ξ) = and E S +h/ T S ξ) + T /2 h ξ) ] = with corresponding random variables ξ, but this does not imply E S h ξ) =. 4 Poisson and self-exciting processes Poisson process is one of the simplest point processes and before taking any other model it is useful first of all to check the hypothesis the observed sequence of events, say, < t,..., t N < T corresponds to a Poisson process. It is natural in many problems to suppose that this Poisson process is periodic of known period. For example, many daily events, signal transmission in optical communication, season variations etc. Another model of point processes as well frequently used is self-exciting stationary point process introduced in Hawkes 972). As any stationary process it can as well describe the periodic changes due to the particular form of its spectral density. Recall that for the Poisson process X t, t of intensity function S t), t we have X t is the counting process) P {X t X s = k} = k!) Λ t) Λ s)) k exp {Λ s) Λ t)}, where we suppose that s < t and put Λ t) = t S v) dv. The self-exciting process X t, t admits the representation X t = t S s, X) ds + π t, where π t, t is local martingale and the intensity function S t, X) = S + t g t s) dx s = S + t i <T 3 g t t i ).

14 It is supposed that ρ = g t) dt <. Under this condition the self-exciting process is a stationary point process with the rate µ = S ρ and the spectral density f λ) = µ 2π G λ) 2, G λ) = e iλt g t) dt see Hawkes 972) or Daley and Vere-Jones 23) for details). We consider two problems: Poisson against another Poisson and Poisson against a close self-exciting point process. The first one is to test the simple basic) hypothesis H : S t) S t), t where S t) is known periodic function of period τ, against the composite alternative H : S t) S t), t, but S t) is always τ-periodic. Let us denote X j t) = X τj )+t X τj ), j =,..., n, suppose that T = nτ and put ˆΛ n t) = n X j t). n The corresponding goodness-of-fit tests of Cramér-von Mises and Kolmogorov-Smirnov type can be based on the statistics τ ] 2 Wn 2 = Λ τ) 2 n [ˆΛn t) Λ t) dλ t), D n = Λ τ) /2 sup ˆΛ n t) Λ t). It can be shown that t τ j= W 2 n = W s) 2 ds, n Dn = sup W s) s 4

15 where {W s), s } is a Wiener process see Kutoyants 998)). Hence these statistics are asymptotically distribution-free and the tests ψ n X T ) = {W 2 n >c α}, φ n X T ) = { ndn>d α} with the constants c α, d α taken from the equations 4), are of asymptotic size α. Let us describe the close contiguous alternatives which reduce asymptotically this problem to signal in white Gaussian noise model 5). We put Λ t) = Λ t) + nλ τ) t h u v)) dλ v), u v) = Λ v) Λ τ). Here h ) is an arbitrary function defining the alternative. satisfies this equality we have the convergence Then if Λ t) W 2 n = [ s 2 h v) dv + W s)] ds. This convergence describes the power function of the Cramér-von Mises type test under these alternatives. The second problem is to test the hypothesis H : S t) = S, t against nonparametric close contiguous) alternative H : S t) = S + t h t s) dx t, t, T We consider the alternatives with the functions h ) having compact support and bounded. We have Λ t) = S t and for some fixed τ > we can construct the same statistics W 2 n = n S τ 2 τ ] 2 [ˆΛn t) S t dt, Dn = S τ) /2 sup t τ Of course, they have the same limits under hypothesis ˆΛ n t) S t. W 2 n = W s) 2 ds, ndn = sup W s). s 5

16 To describe their behaviour under any fixed alternative h ) we have to find the limit distribution of the vector w n = w n t ),..., wn t k ) ), w n t l ) = S τ n n [X j t l ) S t l ], where t l τ. We know that this vector under hypothesis is asymptotically normal L {w n } = N, R) with covariance matrix j= R = R lm ) k k, R lm = τ min t l, t m ). Moreover, it was shown in Dachian and Kutoyants 26) that for such alternatives the likelihood ratio is locally asymptotically normal, i.e., the likelihood ratio admits the representation { Z n h) = exp n h, X n ) } 2 I h) + r n h, X n ) where and n h, X n ) = I h) = S τn τn t h t) 2 dt + S h t s) dx s [dx t S dt], ) 2 h t) dt n h, X n ) = N, I h)), r n h, X n ). 7) To use the Third Le Cam s Lemma we describe the limit behaviour of the vector n h, X n ), w n ). For the covariance Q = Q lm ), l, m =,,..., k of this vector we have E n h, X n ) =, Q = E n h, X n ) 2 = I h) + o )). Further, let us denote dπ t = dx t S dt and H t) = t h t s) dx s, then 6

17 we can write Q l = E [ n h, X n ) w n t l )] n τj n ) τi )+tl = ns 3/2 τ E H t) dπ t dπ t j= τj ) i= τi ) n τj )+tl = nτ E H t) dt = t l S h t) dt + o )), S τ because j= τj ) t E H t) = S h t s) ds = S h s) ds for the large values of t such that [, t] covers the support of h )). Therefore, if we denote h = h s) ds then Q l = Q l = t l S h. τ The proof of the Theorem in Dachian and Kutoyants 26) can be applied to the linear combination of n h, X n ) and w n t ),..., w n t k ) and this yields the asymptotic normality ) L n h, X n ), w n = N, Q). Hence by the Third Lemma of Le Cam we obtain the asymptotic normality of the vector w n ) ) L h w n = L W s ) + s S h,..., W sk ) + s k S h, where we put t l = τ s l. This weak convergence together with the estimates like E h w n t ) w n t 2 ) 2 C t t 2 provides the convergence under alternative) W 2 n = [ S h s + W s) ] 2 ds. 7

18 We see that the limit experiment is of the type dy s = S h ds + dw s), Y =, s. The power βψ n, h) of the Cramer-von Mises type test ψ n X n ) = {W 2 n >c α } is a function of the real parameter ρ h = S h ) β W n, h) = P [ρ h s + W s)] 2 ds > c α + o ) = β ψ ρ h ) + o ). Using the arguments of Lemma 6.2 in Kutoyants 998) it can be shown that for the Kolmogorov-Smirnov type test we have the convergence ndn = sup ρ h s + W s). s The limit power function is ) β φ ρ h ) = P sup ρ h s + W s) > d α. s These two limit power functions will be obtained by simulation in the next section. 5 Simulation First, we present the simulation of the thresholds c α and d α of our Cramérvon Mises and Kolmogorov-Smirnov type tests. Since these thresholds are given by the equations 4), we obtain them by simulating 7 trajectories of a Wiener process on [,] and calculating empirical α quantiles of the statistics W 2 = W s) 2 ds and D = sup W s) s respectively. Note that the distribution of W 2 coincides with the distribution of the quadratic form W 2 = k= ζ 2 k πk) 2, ζ k i.i.d. N, ) 8

19 and both distributions are extensively studied see.9.4)) and.5.4) in Borodin and Salmienen 22)). The analytical expressions are quite complicated and we would like to compare by simulation c α and d α with the real finite time) thresholds giving the tests of exact size α, that is c T α and d T α given by equations P { W 2 n > c T α} = α and P { ndn > d T α} = α respectively. We choose S = and obtain c T α and d T α by simulating 7 trajectories of a Poisson process of intensity on [,T ] and calculating empirical α quantiles of the statistics W 2 n and nd n. The thresholds simulated for T =, T = and for the limiting case are presented in Fig.. The lower curves correspond to the Cramér-von Mises type test, and the upper ones to the Kolmogorov-Smirnov type test. As we can see, for T = the real thresholds are already indistinguishable from the limiting ones, especially in the case of the Cramér-von Mises type test. 3 T= T= Limiting 2 threshold,2,4,6,8 alpha Fig. : Threshold choice It is interesting to compare the asymptotics of the Cramér-von Mises and Kolmogorov-Smirnov type tests with the locally asymptotically uniformly 9

20 most powerful LAUMP) test ˆφ n X n ) = {δt >z α}, δ T = X nτ S nτ S nτ proposed for this problem in Dachian and Kutoyants 26). Here z α is α quantile of the standard Gaussian law, P ζ > z α ) = α, ζ N, ). The limit power function of ˆφ n is β ˆφ ρ h ) = P ρ h + ζ > z α ). In Fig. 2 we compare the limit power functions β ψ ρ), β φ ρ) and β ˆφ ρ). The last one can clearly be calculated directly, and the first two are obtained by simulating 7 trajectories of a Wiener process on [,] and calculating empirical frequencies of the events { } [ρ s + W s)] 2 ds > c α respectively. and { } sup ρ s + W s) > d α s,8 K-S test C-vM test LAUMP test,6 power,4, rho Fig. 2: Limit power functions 2

21 The simulation shows the exact quantitative) comparison of the limit power functions. We see that the power of LAUMP test is higher that the two others and this is of course evident. We see also that the Kolmogorov- Smirnov type test is more powerful that the Cramér-von Mises type test. References. Borodin, A.N. and Salmienen, R. 22). Handbook of Brownian Motion - Facts and Formulae, 2nd ed.), Birkhauser Verlag, Basel. 2. Dachian, S. and Kutoyants, Yu.A. 26). Hypotheses Testing: Poisson versus self-exciting, Scand. J. Statist., 33, Daley, D.J. and Vere-Jones, D. 23). An Introduction to the Theory of Point Processes. vol. I. 2nd ed.), Springer-Verlag, New York. 4. Durbin, J. 973). Distribution Theory for tests Based on the Sample Distribution Function, SIAM, Philadelphia. 5. Fournie, E. 992). Un test de type Kolmogorov-Smirnov pour processus de diffusions ergodic. Rapport de Recherche, 696, INRIA, Sophia-Antipolis. 6. Fournie, E., Kutoyants, Yu. A. 993). Estimateur de la distance minimale pour des processus de diffusion ergodiques. Rapport de Recherche, 952, INRIA, Sophia-Antipolis. 7. Greenwood, P. E. and Nikulin, M. 996). A Guide to Chi-Squared Testing, New-York: John Wiley and Sons. 8. Hawkes, A.G. 972). Spectra for some mutually exciting point processes with associated variable, in Stochastic point processes, P.A.W. Lewis ed., Wiley, New York. 9. Iacus, S. and Kutoyants, Yu.A. 2). Semiparametric hypotheses testing for dynamical systems with small noise, Math. Methods Statist.,,, Ingster, Yu.I. and Suslina, I.A. 23). Nonparametric Goodness-of-Fit Testing Under Gaussian Models, Springer, N.Y. 2

22 . Kutoyants, Yu.A. 994). Identification of Dynamical Systems with Small Noise, Dordrecht: Kluwer. 2. Kutoyants, Yu. A. 998). Statistical Inference for Spatial Poisson Processes. Lect Notes Statist. 34, New York: Springer-Verlag. 3. Kutoyants, Yu.A. 24). Statistical Inference for Ergodic Diffusion Processes, London: Springer-Verlag. 4. Kutoyants, Yu.A. 26). Goodness-of-fit tests for perturbed dynamical systems, in preparation. 5. Liptser, R.S. and Shiryayev, A.N. 2). Statistics of Random Processes. II. Applications, 2nd ed.) Springer, N.Y. 6. Negri, I. 998). Stationary distribution function estimation for ergodic diffusion process, Stat. Inference Stoch. Process.,,,

Goodness of fit test for ergodic diffusion processes

Goodness of fit test for ergodic diffusion processes Ann Inst Stat Math (29) 6:99 928 DOI.7/s463-7-62- Goodness of fit test for ergodic diffusion processes Ilia Negri Yoichi Nishiyama Received: 22 December 26 / Revised: July 27 / Published online: 2 January

More information

On Cramér-von Mises test based on local time of switching diffusion process

On Cramér-von Mises test based on local time of switching diffusion process On Cramér-von Mises test based on local time of switching diffusion process Anis Gassem Laboratoire de Statistique et Processus, Université du Maine, 7285 Le Mans Cedex 9, France e-mail: Anis.Gassem@univ-lemans.fr

More information

!! # % & ( # % () + & ), % &. / # ) ! #! %& & && ( ) %& & +,,

!! # % & ( # % () + & ), % &. / # ) ! #! %& & && ( ) %& & +,, !! # % & ( # % () + & ), % &. / # )! #! %& & && ( ) %& & +,, 0. /! 0 1! 2 /. 3 0 /0 / 4 / / / 2 #5 4 6 1 7 #8 9 :: ; / 4 < / / 4 = 4 > 4 < 4?5 4 4 : / / 4 1 1 4 8. > / 4 / / / /5 5 ; > //. / : / ; 4 ;5

More information

Goodness-of-Fit Tests for Perturbed Dynamical Systems

Goodness-of-Fit Tests for Perturbed Dynamical Systems Goodness-of-Fit Tests for Perturbed Dynamical Systems arxiv:93.461v1 [math.st] 6 Mar 9 Yury A. Kutoyants Laboratoire de Statistique et Processus, Université du Maine 785 Le Mans, Cédex 9, France Abstract

More information

Asymptotical 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 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 information

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS

PROBABILITY: 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 information

ON THE FIRST TIME THAT AN ITO PROCESS HITS A BARRIER

ON THE FIRST TIME THAT AN ITO PROCESS HITS A BARRIER ON THE FIRST TIME THAT AN ITO PROCESS HITS A BARRIER GERARDO HERNANDEZ-DEL-VALLE arxiv:1209.2411v1 [math.pr] 10 Sep 2012 Abstract. This work deals with first hitting time densities of Ito processes whose

More information

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS

PROBABILITY: 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 information

LAN 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 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 information

Investigation of goodness-of-fit test statistic distributions by random censored samples

Investigation of goodness-of-fit test statistic distributions by random censored samples d samples Investigation of goodness-of-fit test statistic distributions by random censored samples Novosibirsk State Technical University November 22, 2010 d samples Outline 1 Nonparametric goodness-of-fit

More information

Large deviations and averaging for systems of slow fast stochastic reaction diffusion equations.

Large deviations and averaging for systems of slow fast stochastic reaction diffusion equations. Large deviations and averaging for systems of slow fast stochastic reaction diffusion equations. Wenqing Hu. 1 (Joint work with Michael Salins 2, Konstantinos Spiliopoulos 3.) 1. Department of Mathematics

More information

(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS

(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS (2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS Svetlana Janković and Miljana Jovanović Faculty of Science, Department of Mathematics, University

More information

ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS

ON 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 information

Brownian Motion. 1 Definition Brownian Motion Wiener measure... 3

Brownian 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 information

Hypoelliptic multiscale Langevin diffusions and Slow fast stochastic reaction diffusion equations.

Hypoelliptic multiscale Langevin diffusions and Slow fast stochastic reaction diffusion equations. Hypoelliptic multiscale Langevin diffusions and Slow fast stochastic reaction diffusion equations. Wenqing Hu. 1 (Joint works with Michael Salins 2 and Konstantinos Spiliopoulos 3.) 1. Department of Mathematics

More information

Parameter Estimation for Stochastic Evolution Equations with Non-commuting Operators

Parameter Estimation for Stochastic Evolution Equations with Non-commuting Operators Parameter Estimation for Stochastic Evolution Equations with Non-commuting Operators Sergey V. Lototsky and Boris L. Rosovskii In: V. Korolyuk, N. Portenko, and H. Syta (editors), Skorokhod s Ideas in

More information

On 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 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 information

Web Appendix for Hierarchical Adaptive Regression Kernels for Regression with Functional Predictors by D. B. Woodard, C. Crainiceanu, and D.

Web Appendix for Hierarchical Adaptive Regression Kernels for Regression with Functional Predictors by D. B. Woodard, C. Crainiceanu, and D. Web Appendix for Hierarchical Adaptive Regression Kernels for Regression with Functional Predictors by D. B. Woodard, C. Crainiceanu, and D. Ruppert A. EMPIRICAL ESTIMATE OF THE KERNEL MIXTURE Here we

More information

Change detection problems in branching processes

Change detection problems in branching processes Change detection problems in branching processes Outline of Ph.D. thesis by Tamás T. Szabó Thesis advisor: Professor Gyula Pap Doctoral School of Mathematics and Computer Science Bolyai Institute, University

More information

Research Article Least Squares Estimators for Unit Root Processes with Locally Stationary Disturbance

Research Article Least Squares Estimators for Unit Root Processes with Locally Stationary Disturbance Advances in Decision Sciences Volume, Article ID 893497, 6 pages doi:.55//893497 Research Article Least Squares Estimators for Unit Root Processes with Locally Stationary Disturbance Junichi Hirukawa and

More information

Reaction-Diffusion Equations In Narrow Tubes and Wave Front P

Reaction-Diffusion Equations In Narrow Tubes and Wave Front P Outlines Reaction-Diffusion Equations In Narrow Tubes and Wave Front Propagation University of Maryland, College Park USA Outline of Part I Outlines Real Life Examples Description of the Problem and Main

More information

Malliavin Calculus in Finance

Malliavin Calculus in Finance Malliavin Calculus in Finance Peter K. Friz 1 Greeks and the logarithmic derivative trick Model an underlying assent by a Markov process with values in R m with dynamics described by the SDE dx t = b(x

More information

Compensating operator of diffusion process with variable diffusion in semi-markov space

Compensating operator of diffusion process with variable diffusion in semi-markov space Compensating operator of diffusion process with variable diffusion in semi-markov space Rosa W., Chabanyuk Y., Khimka U. T. Zakopane, XLV KZM 2016 September 9, 2016 diffusion process September with variable

More information

OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS

OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS APPLICATIONES MATHEMATICAE 29,4 (22), pp. 387 398 Mariusz Michta (Zielona Góra) OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS Abstract. A martingale problem approach is used first to analyze

More information

Introduction to Empirical Processes and Semiparametric Inference Lecture 09: Stochastic Convergence, Continued

Introduction to Empirical Processes and Semiparametric Inference Lecture 09: Stochastic Convergence, Continued Introduction to Empirical Processes and Semiparametric Inference Lecture 09: Stochastic Convergence, Continued Michael R. Kosorok, Ph.D. Professor and Chair of Biostatistics Professor of Statistics and

More information

Some functional (Hölderian) limit theorems and their applications (II)

Some 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 information

UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE

UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (2010), 275 284 UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Iuliana Carmen Bărbăcioru Abstract.

More information

Stochastic Volatility and Correction to the Heat Equation

Stochastic Volatility and Correction to the Heat Equation Stochastic Volatility and Correction to the Heat Equation Jean-Pierre Fouque, George Papanicolaou and Ronnie Sircar Abstract. From a probabilist s point of view the Twentieth Century has been a century

More information

Asymptotic Statistics-VI. Changliang Zou

Asymptotic Statistics-VI. Changliang Zou Asymptotic Statistics-VI Changliang Zou Kolmogorov-Smirnov distance Example (Kolmogorov-Smirnov confidence intervals) We know given α (0, 1), there is a well-defined d = d α,n such that, for any continuous

More information

Brownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539

Brownian 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 information

On the sequential testing problem for some diffusion processes

On the sequential testing problem for some diffusion processes To appear in Stochastics: An International Journal of Probability and Stochastic Processes (17 pp). On the sequential testing problem for some diffusion processes Pavel V. Gapeev Albert N. Shiryaev We

More information

On ADF Goodness-of-Fit Tests for Perturbed Dynamical Systems

On ADF Goodness-of-Fit Tests for Perturbed Dynamical Systems On ADF Goodness-of-Fit Tests for Perturbed Dynamical Systems arxiv:143.7713v1 [math.st] 3 Mar 214 Yu.A. Kutoyants Laboratoire de Statistique et Processus, Université du Maine Le Mans, FRANCE Abstract We

More information

Stat 710: Mathematical Statistics Lecture 31

Stat 710: Mathematical Statistics Lecture 31 Stat 710: Mathematical Statistics Lecture 31 Jun Shao Department of Statistics University of Wisconsin Madison, WI 53706, USA Jun Shao (UW-Madison) Stat 710, Lecture 31 April 13, 2009 1 / 13 Lecture 31:

More information

UDC Anderson-Darling and New Weighted Cramér-von Mises Statistics. G. V. Martynov

UDC Anderson-Darling and New Weighted Cramér-von Mises Statistics. G. V. Martynov UDC 519.2 Anderson-Darling and New Weighted Cramér-von Mises Statistics G. V. Martynov Institute for Information Transmission Problems of the RAS, Bolshoy Karetny per., 19, build.1, Moscow, 12751, Russia

More information

Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )

Mathematical 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 information

Parameter estimation of diffusion models

Parameter estimation of diffusion models 129 Parameter estimation of diffusion models Miljenko Huzak Abstract. Parameter estimation problems of diffusion models are discussed. The problems of maximum likelihood estimation and model selections

More information

Minimax Interpolation Problem for Random Processes with Stationary Increments

Minimax Interpolation Problem for Random Processes with Stationary Increments STATISTICS, OPTIMIZATION AND INFORMATION COMPUTING Stat., Optim. Inf. Comput., Vol. 3, March 215, pp 3 41. Published online in International Academic Press (www.iapress.org Minimax Interpolation Problem

More information

2012 NCTS Workshop on Dynamical Systems

2012 NCTS Workshop on Dynamical Systems Barbara Gentz gentz@math.uni-bielefeld.de http://www.math.uni-bielefeld.de/ gentz 2012 NCTS Workshop on Dynamical Systems National Center for Theoretical Sciences, National Tsing-Hua University Hsinchu,

More information

Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals

Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals Noèlia Viles Cuadros BCAM- Basque Center of Applied Mathematics with Prof. Enrico

More information

Modeling the Goodness-of-Fit Test Based on the Interval Estimation of the Probability Distribution Function

Modeling the Goodness-of-Fit Test Based on the Interval Estimation of the Probability Distribution Function Modeling the Goodness-of-Fit Test Based on the Interval Estimation of the Probability Distribution Function E. L. Kuleshov, K. A. Petrov, T. S. Kirillova Far Eastern Federal University, Sukhanova st.,

More information

THE DVORETZKY KIEFER WOLFOWITZ INEQUALITY WITH SHARP CONSTANT: MASSART S 1990 PROOF SEMINAR, SEPT. 28, R. M. Dudley

THE DVORETZKY KIEFER WOLFOWITZ INEQUALITY WITH SHARP CONSTANT: MASSART S 1990 PROOF SEMINAR, SEPT. 28, R. M. Dudley THE DVORETZKY KIEFER WOLFOWITZ INEQUALITY WITH SHARP CONSTANT: MASSART S 1990 PROOF SEMINAR, SEPT. 28, 2011 R. M. Dudley 1 A. Dvoretzky, J. Kiefer, and J. Wolfowitz 1956 proved the Dvoretzky Kiefer Wolfowitz

More information

Asymptotic inference for a nonstationary double ar(1) model

Asymptotic inference for a nonstationary double ar(1) model Asymptotic inference for a nonstationary double ar() model By SHIQING LING and DONG LI Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong maling@ust.hk malidong@ust.hk

More information

MARKOVIANITY OF A SUBSET OF COMPONENTS OF A MARKOV PROCESS

MARKOVIANITY OF A SUBSET OF COMPONENTS OF A MARKOV PROCESS MARKOVIANITY OF A SUBSET OF COMPONENTS OF A MARKOV PROCESS P. I. Kitsul Department of Mathematics and Statistics Minnesota State University, Mankato, MN, USA R. S. Liptser Department of Electrical Engeneering

More information

Quantum Repeated Measurements, Continuous Time Limit

Quantum Repeated Measurements, Continuous Time Limit Quantum Repeated Measurements, Continuous Time Limit Clément Pellegrini Institut de Mathématiques de Toulouse, Laboratoire de Statistique et Probabilité, Université Paul Sabatier Autrans/ July 2013 Clément

More information

Scaling Limits of Waves in Convex Scalar Conservation Laws under Random Initial Perturbations

Scaling Limits of Waves in Convex Scalar Conservation Laws under Random Initial Perturbations Scaling Limits of Waves in Convex Scalar Conservation Laws under Random Initial Perturbations Jan Wehr and Jack Xin Abstract We study waves in convex scalar conservation laws under noisy initial perturbations.

More information

Understanding Regressions with Observations Collected at High Frequency over Long Span

Understanding Regressions with Observations Collected at High Frequency over Long Span Understanding Regressions with Observations Collected at High Frequency over Long Span Yoosoon Chang Department of Economics, Indiana University Joon Y. Park Department of Economics, Indiana University

More information

On Rescaled Poisson Processes and the Brownian Bridge. Frederic Schoenberg. Department of Statistics. University of California, Los Angeles

On Rescaled Poisson Processes and the Brownian Bridge. Frederic Schoenberg. Department of Statistics. University of California, Los Angeles On Rescaled Poisson Processes and the Brownian Bridge Frederic Schoenberg Department of Statistics University of California, Los Angeles Los Angeles, CA 90095 1554, USA Running head: Rescaled Poisson Processes

More information

Model Specification Testing in Nonparametric and Semiparametric Time Series Econometrics. Jiti Gao

Model 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 information

Spatial Ergodicity of the Harris Flows

Spatial Ergodicity of the Harris Flows Communications on Stochastic Analysis Volume 11 Number 2 Article 6 6-217 Spatial Ergodicity of the Harris Flows E.V. Glinyanaya Institute of Mathematics NAS of Ukraine, glinkate@gmail.com Follow this and

More information

On continuous time contract theory

On 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 information

Studies in Nonlinear Dynamics and Econometrics

Studies in Nonlinear Dynamics and Econometrics Studies in Nonlinear Dynamics and Econometrics Quarterly Journal Volume 4, Number 4 The MIT Press Studies in Nonlinear Dynamics and Econometrics (ISSN 181-1826) is a quarterly journal published electronically

More information

Scaling Limits of Waves in Convex Scalar Conservation Laws Under Random Initial Perturbations

Scaling Limits of Waves in Convex Scalar Conservation Laws Under Random Initial Perturbations Journal of Statistical Physics, Vol. 122, No. 2, January 2006 ( C 2006 ) DOI: 10.1007/s10955-005-8006-x Scaling Limits of Waves in Convex Scalar Conservation Laws Under Random Initial Perturbations Jan

More information

STAT Sample Problem: General Asymptotic Results

STAT Sample Problem: General Asymptotic Results STAT331 1-Sample Problem: General Asymptotic Results In this unit we will consider the 1-sample problem and prove the consistency and asymptotic normality of the Nelson-Aalen estimator of the cumulative

More information

Rice method for the maximum of Gaussian fields

Rice method for the maximum of Gaussian fields Rice method for the maximum of Gaussian fields IWAP, July 8, 2010 Jean-Marc AZAÏS Institut de Mathématiques, Université de Toulouse Jean-Marc AZAÏS ( Institut de Mathématiques, Université de Toulouse )

More information

Asymptotically Efficient Nonparametric Estimation of Nonlinear Spectral Functionals

Asymptotically Efficient Nonparametric Estimation of Nonlinear Spectral Functionals Acta Applicandae Mathematicae 78: 145 154, 2003. 2003 Kluwer Academic Publishers. Printed in the Netherlands. 145 Asymptotically Efficient Nonparametric Estimation of Nonlinear Spectral Functionals M.

More information

L -uniqueness of Schrödinger operators on a Riemannian manifold

L -uniqueness of Schrödinger operators on a Riemannian manifold L -uniqueness of Schrödinger operators on a Riemannian manifold Ludovic Dan Lemle Abstract. The main purpose of this paper is to study L -uniqueness of Schrödinger operators and generalized Schrödinger

More information

On the quantiles of the Brownian motion and their hitting times.

On 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 information

Statistical inference on Lévy processes

Statistical inference on Lévy processes Alberto Coca Cabrero University of Cambridge - CCA Supervisors: Dr. Richard Nickl and Professor L.C.G.Rogers Funded by Fundación Mutua Madrileña and EPSRC MASDOC/CCA student workshop 2013 26th March Outline

More information

On Consistent Hypotheses Testing

On Consistent Hypotheses Testing Mikhail Ermakov St.Petersburg E-mail: erm2512@gmail.com History is rather distinctive. Stein (1954); Hodges (1954); Hoefding and Wolfowitz (1956), Le Cam and Schwartz (1960). special original setups: Shepp,

More information

1. Stochastic Process

1. Stochastic Process HETERGENEITY IN QUANTITATIVE MACROECONOMICS @ TSE OCTOBER 17, 216 STOCHASTIC CALCULUS BASICS SANG YOON (TIM) LEE Very simple notes (need to add references). It is NOT meant to be a substitute for a real

More information

Conditional Full Support for Gaussian Processes with Stationary Increments

Conditional Full Support for Gaussian Processes with Stationary Increments Conditional Full Support for Gaussian Processes with Stationary Increments Tommi Sottinen University of Vaasa Kyiv, September 9, 2010 International Conference Modern Stochastics: Theory and Applications

More information

Introduction to Stochastic SIR Model

Introduction to Stochastic SIR Model Introduction to Stochastic R Model Chiu- Yu Yang (Alex), Yi Yang R model is used to model the infection of diseases. It is short for Susceptible- Infected- Recovered. It is important to address that R

More information

A Lower Bound Theorem. Lin Hu.

A Lower Bound Theorem. Lin Hu. American J. of Mathematics and Sciences Vol. 3, No -1,(January 014) Copyright Mind Reader Publications ISSN No: 50-310 A Lower Bound Theorem Department of Applied Mathematics, Beijing University of Technology,

More information

The 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. 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 information

GAUSSIAN PROCESSES; KOLMOGOROV-CHENTSOV THEOREM

GAUSSIAN 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 information

Convergence at first and second order of some approximations of stochastic integrals

Convergence at first and second order of some approximations of stochastic integrals Convergence at first and second order of some approximations of stochastic integrals Bérard Bergery Blandine, Vallois Pierre IECN, Nancy-Université, CNRS, INRIA, Boulevard des Aiguillettes B.P. 239 F-5456

More information

LANGEVIN THEORY OF BROWNIAN MOTION. Contents. 1 Langevin theory. 1 Langevin theory 1. 2 The Ornstein-Uhlenbeck process 8

LANGEVIN THEORY OF BROWNIAN MOTION. Contents. 1 Langevin theory. 1 Langevin theory 1. 2 The Ornstein-Uhlenbeck process 8 Contents LANGEVIN THEORY OF BROWNIAN MOTION 1 Langevin theory 1 2 The Ornstein-Uhlenbeck process 8 1 Langevin theory Einstein (as well as Smoluchowski) was well aware that the theory of Brownian motion

More information

New ideas in the non-equilibrium statistical physics and the micro approach to transportation flows

New ideas in the non-equilibrium statistical physics and the micro approach to transportation flows New ideas in the non-equilibrium statistical physics and the micro approach to transportation flows Plenary talk on the conference Stochastic and Analytic Methods in Mathematical Physics, Yerevan, Armenia,

More information

Translation Invariant Experiments with Independent Increments

Translation Invariant Experiments with Independent Increments Translation Invariant Statistical Experiments with Independent Increments (joint work with Nino Kordzakhia and Alex Novikov Steklov Mathematical Institute St.Petersburg, June 10, 2013 Outline 1 Introduction

More information

Brownian Motion and lorentzian manifolds

Brownian Motion and lorentzian manifolds The case of Jürgen Angst Institut de Recherche Mathématique Avancée Université Louis Pasteur, Strasbourg École d été de Probabilités de Saint-Flour juillet 2008, Saint-Flour 1 Construction of the diffusion,

More information

Regularity of the density for the stochastic heat equation

Regularity 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 information

OPTIMAL POINTWISE ADAPTIVE METHODS IN NONPARAMETRIC ESTIMATION 1

OPTIMAL POINTWISE ADAPTIVE METHODS IN NONPARAMETRIC ESTIMATION 1 The Annals of Statistics 1997, Vol. 25, No. 6, 2512 2546 OPTIMAL POINTWISE ADAPTIVE METHODS IN NONPARAMETRIC ESTIMATION 1 By O. V. Lepski and V. G. Spokoiny Humboldt University and Weierstrass Institute

More information

Bayesian quickest detection problems for some diffusion processes

Bayesian quickest detection problems for some diffusion processes Bayesian quickest detection problems for some diffusion processes Pavel V. Gapeev Albert N. Shiryaev We study the Bayesian problems of detecting a change in the drift rate of an observable diffusion process

More information

1. Stochastic Processes and filtrations

1. 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 information

3.4 Linear Least-Squares Filter

3.4 Linear Least-Squares Filter X(n) = [x(1), x(2),..., x(n)] T 1 3.4 Linear Least-Squares Filter Two characteristics of linear least-squares filter: 1. The filter is built around a single linear neuron. 2. The cost function is the sum

More information

Pathwise volatility in a long-memory pricing model: estimation and asymptotic behavior

Pathwise 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 information

Evgeny Spodarev WIAS, Berlin. Limit theorems for excursion sets of stationary random fields

Evgeny Spodarev WIAS, Berlin. Limit theorems for excursion sets of stationary random fields Evgeny Spodarev 23.01.2013 WIAS, Berlin Limit theorems for excursion sets of stationary random fields page 2 LT for excursion sets of stationary random fields Overview 23.01.2013 Overview Motivation Excursion

More information

Estimation for the standard and geometric telegraph process. Stefano M. Iacus. (SAPS VI, Le Mans 21-March-2007)

Estimation for the standard and geometric telegraph process. Stefano M. Iacus. (SAPS VI, Le Mans 21-March-2007) Estimation for the standard and geometric telegraph process Stefano M. Iacus University of Milan(Italy) (SAPS VI, Le Mans 21-March-2007) 1 1. Telegraph process Consider a particle moving on the real line

More information

LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC

LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC R. G. DOLGOARSHINNYKH Abstract. We establish law of large numbers for SIRS stochastic epidemic processes: as the population size increases the paths of SIRS epidemic

More information

Man Kyu Im*, Un Cig Ji **, and Jae Hee Kim ***

Man 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 information

Testing Statistical Hypotheses

Testing Statistical Hypotheses E.L. Lehmann Joseph P. Romano Testing Statistical Hypotheses Third Edition 4y Springer Preface vii I Small-Sample Theory 1 1 The General Decision Problem 3 1.1 Statistical Inference and Statistical Decisions

More information

The Poincare map for randomly perturbed periodic mo5on

The Poincare map for randomly perturbed periodic mo5on The Poincare map for randomly perturbed periodic mo5on Georgi Medvedev Drexel University SIAM Conference on Applica1ons of Dynamical Systems May 19, 2013 Pawel Hitczenko and Georgi Medvedev, The Poincare

More information

Tools of stochastic calculus

Tools of stochastic calculus slides for the course Interest rate theory, University of Ljubljana, 212-13/I, part III József Gáll University of Debrecen Nov. 212 Jan. 213, Ljubljana Itô integral, summary of main facts Notations, basic

More information

On pathwise stochastic integration

On pathwise stochastic integration On pathwise stochastic integration Rafa l Marcin Lochowski Afican Institute for Mathematical Sciences, Warsaw School of Economics UWC seminar Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic

More information

Ef ciency of the empirical distribution for ergodic diffusion

Ef ciency of the empirical distribution for ergodic diffusion Bernoulli 3(4), 1997, 445±456 Ef ciency of the empirical distribution for ergodic diffusion YURY A. KUTOYANTS Laboratoire de Statistique et Processus, Universite du Maine, 7217 Le Mans, France e-mail:

More information

Asymptotic Properties of an Approximate Maximum Likelihood Estimator for Stochastic PDEs

Asymptotic 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 information

S. Lototsky and B.L. Rozovskii Center for Applied Mathematical Sciences University of Southern California, Los Angeles, CA

S. Lototsky and B.L. Rozovskii Center for Applied Mathematical Sciences University of Southern California, Los Angeles, CA RECURSIVE MULTIPLE WIENER INTEGRAL EXPANSION FOR NONLINEAR FILTERING OF DIFFUSION PROCESSES Published in: J. A. Goldstein, N. E. Gretsky, and J. J. Uhl (editors), Stochastic Processes and Functional Analysis,

More information

The Codimension of the Zeros of a Stable Process in Random Scenery

The Codimension of the Zeros of a Stable Process in Random Scenery The Codimension of the Zeros of a Stable Process in Random Scenery Davar Khoshnevisan The University of Utah, Department of Mathematics Salt Lake City, UT 84105 0090, U.S.A. davar@math.utah.edu http://www.math.utah.edu/~davar

More information

Issues on quantile autoregression

Issues on quantile autoregression Issues on quantile autoregression Jianqing Fan and Yingying Fan We congratulate Koenker and Xiao on their interesting and important contribution to the quantile autoregression (QAR). The paper provides

More information

SEPARABILITY AND COMPLETENESS FOR THE WASSERSTEIN DISTANCE

SEPARABILITY AND COMPLETENESS FOR THE WASSERSTEIN DISTANCE SEPARABILITY AND COMPLETENESS FOR THE WASSERSTEIN DISTANCE FRANÇOIS BOLLEY Abstract. In this note we prove in an elementary way that the Wasserstein distances, which play a basic role in optimal transportation

More information

Poisson random measure: motivation

Poisson random measure: motivation : motivation The Lévy measure provides the expected number of jumps by time unit, i.e. in a time interval of the form: [t, t + 1], and of a certain size Example: ν([1, )) is the expected number of jumps

More information

Homework # , Spring Due 14 May Convergence of the empirical CDF, uniform samples

Homework # , Spring Due 14 May Convergence of the empirical CDF, uniform samples Homework #3 36-754, Spring 27 Due 14 May 27 1 Convergence of the empirical CDF, uniform samples In this problem and the next, X i are IID samples on the real line, with cumulative distribution function

More information

ASYMPTOTIC EQUIVALENCE OF DENSITY ESTIMATION AND GAUSSIAN WHITE NOISE. By Michael Nussbaum Weierstrass Institute, Berlin

ASYMPTOTIC EQUIVALENCE OF DENSITY ESTIMATION AND GAUSSIAN WHITE NOISE. By Michael Nussbaum Weierstrass Institute, Berlin The Annals of Statistics 1996, Vol. 4, No. 6, 399 430 ASYMPTOTIC EQUIVALENCE OF DENSITY ESTIMATION AND GAUSSIAN WHITE NOISE By Michael Nussbaum Weierstrass Institute, Berlin Signal recovery in Gaussian

More information

Lecture 2: CDF and EDF

Lecture 2: CDF and EDF STAT 425: Introduction to Nonparametric Statistics Winter 2018 Instructor: Yen-Chi Chen Lecture 2: CDF and EDF 2.1 CDF: Cumulative Distribution Function For a random variable X, its CDF F () contains all

More information

Lecture 1: Pragmatic Introduction to Stochastic Differential Equations

Lecture 1: Pragmatic Introduction to Stochastic Differential Equations Lecture 1: Pragmatic Introduction to Stochastic Differential Equations Simo Särkkä Aalto University, Finland (visiting at Oxford University, UK) November 13, 2013 Simo Särkkä (Aalto) Lecture 1: Pragmatic

More information

The Uses of Differential Geometry in Finance

The Uses of Differential Geometry in Finance The Uses of Differential Geometry in Finance Andrew Lesniewski Bloomberg, November 21 2005 The Uses of Differential Geometry in Finance p. 1 Overview Joint with P. Hagan and D. Woodward Motivation: Varadhan

More information

Stochastic differential equation models in biology Susanne Ditlevsen

Stochastic differential equation models in biology Susanne Ditlevsen Stochastic differential equation models in biology Susanne Ditlevsen Introduction This chapter is concerned with continuous time processes, which are often modeled as a system of ordinary differential

More information

Stochastic differential equations in neuroscience

Stochastic differential equations in neuroscience Stochastic differential equations in neuroscience Nils Berglund MAPMO, Orléans (CNRS, UMR 6628) http://www.univ-orleans.fr/mapmo/membres/berglund/ Barbara Gentz, Universität Bielefeld Damien Landon, MAPMO-Orléans

More information

Bayesian Regularization

Bayesian 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 information

n E(X t T n = lim X s Tn = X s

n E(X t T n = lim X s Tn = X s Stochastic Calculus Example sheet - Lent 15 Michael Tehranchi Problem 1. Let X be a local martingale. Prove that X is a uniformly integrable martingale if and only X is of class D. Solution 1. If If direction:

More information