R.S. Liptser and A.N. Shiryayev, "Statistics of Random Processes," Springer-Verlag, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The numerator in (2.4) will be denoted by OE k [f ] With this notation, 2.4) becomes f(k) OE k [f ] 3 The Algorithm Assume that the function f satisfies the following growth condition: jf(x)j C(1 jxj ff ) x 2 IR ; 3:1) for some ff; C 0. Then Ejf(X(t) j 1 for all t 0 [7]. To parameterize the function p k , a finite element approximation is used. Let fe 1 ; e n g be a basis in the given finite dimensional set of interpolating functions and fx 1 ; xN g the collection of points in so that e i (x j ) ffi ij . Assume that the support of each ....

R. Sh. Liptser and A. N. Shiryayev. Statistics of Random Processes. Springer, New York, 1977.

....importance in what follows; indeed we shall prove (9) showing that, for any P(A) F ( isequalto1 d( D)and(B ) Before doing this, we need some propedeutic results. P(A) be such that ( a Q ) 1for some Q P(S) Then (B ) 1. Proof. By the hypothesis we can say that (see e.g. [6], Lemma 7.4, page 287) E fQ (a, s)d(a) A fQ (a, s)d(a) #E #A ) 1 where fQ is a jointly measurable function such that fQ (a, s)dQ(s) #X #S ) 1. Hence we have P (T ) 1 and, by Corollary 2, B ) 1. # Lemma 6. For any P(A) there exists a set A # such that F ( A ....

R. S. Liptser, A. N. Shiryiayev, Statistics of Random Processes I, General Theory (Springer Verlag, New York, 1977).

....The same expansion as in Section 4.2. now applies after rede ning T : T ) A : 4.4 General case We consider general local dynamics as introduced in Section 2.3.3. with a Gibbs measure for a nite range interaction as reversible invariant measure. We use Girsanovformula (see [1] or [11]) to go back to case 4.3. Denote P for the path space measure on 12 t] of the interacting process in volume , and P ;0 for the path space measure of the non interacting case. We have ( exp (4.13) and hence expfH ( t )g H ( t ) Z t ....

R. S. Liptser and A. N. Shiryayev. Statistics of Random Processes, part II. SpringerVerlag, New-York, 1977.

.... the conditional mean of the parameter vector under Gaussianity and independence assumptions on fw(t)g and , and (2) The central contribution of [10] is that it shows that square integrability of (t) is not needed for the validity of (7) Thus it extends the results in Liptser and Shiryayev [12] which require such a square integrability; see assumptions (1 4) on page 62 and Theorem 13.4, or Example 1 on page 85, of [12] Since has nite moments of all orders, f (t) Y t g is a convergent martingale, i.e. P a.e. is P a.s. nite, see Doob [13] Thus ....

.... central contribution of [10] is that it shows that square integrability of (t) is not needed for the validity of (7) Thus it extends the results in Liptser and Shiryayev [12] which require such a square integrability; see assumptions (1 4) on page 62 and Theorem 13.4, or Example 1 on page 85, of [12]. Since has nite moments of all orders, f (t) Y t g is a convergent martingale, i.e. P a.e. is P a.s. nite, see Doob [13] Thus there exists a set A with (A) 1, such that ( for all 2 A: Thus Z 1 A ( d P ( 1; where ....

R. S. Liptser and A. N. Shiryayev, Statistics of Random Processes, II: Applications. New York: Springer-Verlag, 1977.

....on [0; t] is absolutely continuous w.r. t the law of independent, homogeneous Poisson processes, and moreover an expression of the corresponding Radon Nikodym derivative in terms of the stochastic intensities of the N i is also known; see for instance Jacod and M emin [15] or Liptser and Shiryayev [21]. Since this result will be required to derive stability results for the dynamics under consideration, we now give sufficient conditions for (30) to hold. Lemma 6 . Assume that there exist non negative coefficients, fl i , m ji such that OE i (x) fl i X j m ji x j ; x 2 (R ) K (31) ....

....and all x 2 [0; A] K , P x (X t 2 C) ae(t) Z C f i (y i e Gammafft x i ) 1iN j dy 1 : dy K (46) where the constant ae(t) is strictly positive. Proof: By Lemma 6, it holds that E x N i (0; t] is finite for all x, i and t. A result due to Jacod and M emin [15] see for instance [21], Theorem 19.7 p. 315) guarantees that, in this case, the law of the N i on (0; t] is absolutely continuous with respect to the law of K independent Poisson processes with intensity 1. Denoting by P the corresponding distribution, an expression for the Radon Nikodym derivative is moreover ....

Liptser, R. S. and A. N. Shiryayev (1978), Statistics of Random Processes, II: Applications, Springer, NY.

....one. This is one of the main reasons for our interest in control functions such as ( Alternatively, we could write the jumps as v 1;l = 1 v 1;l [WL( 1;l ) u( 1;l ) 4:1a) v 2;l = 2 v 2;l u( 2;l ) 4:1b) where u( is a predictable process [15, 23]. All that is important is the nonanticipativity as de ned above, so that, for each t, u(t) is independent of any jump that might occur at t. The cost function. Let c i ( be strictly increasing and continuous real valued function on [0; 1) with c i (0) 0; and satisfying c i (x) Kx K for ....

R. Liptser and A.N. Shiryaev. Statistics of Random Processes. SpringerVerlag, Berlin and New York, 1977.

....and Williams (2000) 3.1 General formulation Since the state variable s is not observed, the decision maker has to infer information about the current state of the system by using the current and the past observations of y. This hidden state model is due to Wonham (1964) and is described in Liptser and Shiryayev (1977) and Elliott, Aggoun, and Moore (1995) It has been used in asset pricing models by David (1997) Veronesi (1999) Veronesi (2000) and David and Veronesi (1999) 5 There are other normalizations that might be adopted. For instance, we could make the jump intensity constant across states, ....

Liptser, R. S. and A. N. Shiryayev (1977). Statistics of Random Processes. New York: Springer-Verlag.

....was characterized by a large number (continuum) of state variables. To exploit the stochastic estimation error of non Gaussian models, David (1997) studied the asset al..location problem in a setting where the unobservable drift jumped between a finite number of states. Using a filtering result in Liptser and Shiryaayev (1977), he showed that learning induces the leverage effect (a negative asymmetric relation between returns and future volatility) on the market portfolio, as agents attempt to hedge the risk of changing estimation error. With a single stock and this same filtering problem characterizing the uncertainty ....

....i i = E t dy y (5) and d f W = Gamma Sigma Sigma 0 Delta Gamma 1 2 dy y Gamma E t dy y = Gamma Sigma Sigma 0 Delta Gamma 1 2 ( t) Gamma )dt Gamma Sigma Sigma 0 Delta Gamma 1 2 SigmadW (6) Moreover, for every t 0, P n i=1 (t) 1. Proof : See Liptser and Shiryaayev (1977). The expressions in (5) are rather intuitive: the drift rate depends only on the prior belief at time t, t) and the transition matrix , and therefore the expression describes how the beliefs are updated between t and t Delta when no other information is available. The diffusion part in (4) ....

Liptser, Robert N. and A. N. Shiryaayev, Statistics of Random Processes I, Springer Verlag, New York, 1977.

....to the basic probability space which supports X( Y ( W ( B( The optimal filter process. We will use the representation of the optimal filter as it was originally developed in [15] This is most convenient for our purposes, and it is completely equivalent to the forms used later, as in [8, 11, 23]. Let X( be a process satisfying (2.1) and which (loosely speaking) is conditionally independent of (X( W ( B( given its initial condition: We formalize this as follows. X( is a process satisfying (2.1) such that there exists a (possibly random) probability measure # # on IR r with ....

R. Liptser and A.N. Shiryaev. Statistics of Random Processes. SpringerVerlag, Berlin and New York, 1977. 39

.... on the data Y 0,t and on #(0) being the initial distribution of X( i.e. #(0) is the current value of what we generically called # # above) This representation is convenient for our purposes, and is equivalent to the forms used subsequently which were based on measure transformations, as in [7, 13, 25]. The Markov property of X( implies that the filter defined by (2.4) satisfies the semigroup relation: ##(t) ## = E #(t s) Y t s,t # #( X(s) R( X 0,s , Y t s,t ) # E #(t s) Y t s,t R( X 0,s , Y t s,t ) 0 s # t. 2.5) In (2.5) #(t s) is the random initial ....

R. Liptser and A.N. Shiryaev. Statistics of Random Processes. SpringerVerlag, Berlin and New York, 1977.

....of the invariant measure. Key Words: nonlinear filtering, invariant measures, asymptotic stability, measure valued processes. AMS Classification:60 G 35, 60 J 05, 60 H 15 2 1 Introduction Stochastic nonlinear filtering is one of the central areas of application of stochastic calculus [16, 20, 22]. The basic object of the study is a pair of stochastic processes (X t , Y t ) t#0 where (X t ) is called the signal process and (Y t ) the observation process. The central problem in nonlinear filtering is the study of the measure valued process (# t ) which is the conditional distribution of X t ....

R.S. Liptser and A.N. Shiryaev. Statistics of Random Processes. Springer-Verlag, New York, 1977.

.... input signals and martingale uncertainty A connection between the disturbance signal uncertainty model and the perturbation martingale uncertainty model is discussed in the paper [19] It is observed in [19] that an arbitrary uncertainty input #( satisfying the conditions of Novikov s theorem [9]) on every finite interval [0,T] can be associated with an uncertainty martingale #( #M. This result is summarized in the following lemma; see [19] Lemma 1 Suppose a random process (#(t) F t ) 0 # t # T satisfies the conditions:P # # T 0 ##(s)# 2 ds # #=1 , E exp #1 2 # ....

....0 #(t)dt, 8) is a Wiener process with respect to the system F t , 0 # t # T and the probability measure Q T defined by equation (2) where #( is defined by equation (7) The above result follows from Novikov s Theorem and Girsanov s Theorem (e.g. see Theorem 6.1 and Theorem 6. 3 of [9]) In particular, it follows from the above results that the martingale #( is given by the equation #(t) exp ## t 0 # # (s)dW (s) 1 2 ##(s)# 2 ds # . 9) Also, on the probability space (#, F T ,Q T ) the system (1) becomes a system of the following form: dx = Ax B 2 #)dt B 2 ....

R. S. Liptser and A. N. Shiryayev. Statistics of Random Processes. I. General Theory. Springer-Verlag, 1977.

....recent date. The Martingale Estimating Functions (MEFs) from the linear family for discretely observed SDEs developed by (Bibby and Srensen, 1995) are inspired by the properties of the pseudo score function, i.e. the score function obtained by discretizing the continuous time likelihood function (Liptser and Shiryayev, 1977) provided that the diffusion function does not depend on the parameter. Requiring that the EF be a martingale implies that the asymptotic properties may be obtained without letting the time between measurements tend to zero. Unfortunately it also implies that the EFs in3 volve conditional ....

Liptser, R. S. and Shiryayev, A. N. (1977), Statistics of Random Processes I, II, Springer-Verlag, New York.

....namely where in every period n, the filter density n (xj y n 0 ) belongs to a same family of densities, parametrized by a finite dimensional parameter. By analogy to Bayesian statistics and generalizing the situation in the classical linear Gaussian models (Kalman filter model, see e.g. [10]) we give the following 2 Definition 1.1 Let the pair ff(yj x) p n (x 0 j x)g be given. Gamma : ffl (x)g 2 Theta is a family of densities filter conjugate to ff(yj x) p n (x 0 j x)g if there exists Phi : Theta Theta J Theta such that, 8 2 Theta; y 2 J; n = 1; 2 Delta Delta ....

.... filtering model (in its time homogeneous form) The corresponding evolution of the filter is then given by ff n = ff n Gamma1 ff 2 n Gamma1 oe 2 C 2 1; z n = C ff n Gamma1 oe 2 C 2 Delta z n Gamma1 y n 8 that corresponds to the familiar Kalman filter, where (see e.g. [10]) m n = C Delta m n Gamma1 i oe 2 C 2 s 2 n Gamma1 j y n 1 oe 2 C 2 s 2 n Gamma1 ; s 2 n = oe 2 C 2 s 2 n Gamma1 1 oe 2 C 2 s 2 n Gamma1 : The above results can be generalized (see [6] to partially observable models of the form X n 1 = a n (Y ....

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Liptser, R.Sh & Shiryaev, A.N. (1977) Statistics of Random processes, SpringerVerlag, New York.

....observations of Y s up to and including t. An important practical aspect of our procedure, especially for high frequency data, will be that this approximation can be computed recursively. Among the two major approaches to nonlinear ltering, namely the so called innovations approach (see [10] [15]) and the reference probability approach (see [3] 13] the latter is better suited for our approximations. In the next subsection we shall present its main features. 3.1 The reference probability approach This approach is based on the fact that, for a probability measure Q on( F ; G t ) ....

R.S. Liptser and A.N.Shiryaev, Statistics of Random Processes I, Springer Verlag, New York, 1977.

.... T i ) log p(t; T i ) for i = 1; n, is Gaussian and thus determined by its conditional mean vector m t = m x t m z t # and covariance matrix V t , i.e. g( t j F t ) N ( t ; m t ; V t ) 39) The evolution of m t and V t is given by the so called Kalman Bucy filter (see e.g. [22], section 10.3) Since Sigma(t) is a square matrix, it is non singular if and only if Sigma(t) Sigma(t) 0 is non singular. Notice also that, without loss of generality, we can restrict ourselves to proving that Sigma(t) is non singular for t T 1 . Lemma 4.1 Let Delta(t; T ) be as in (28) ....

R.S. Liptser, and A.N. Shiryayev, Statistics of Random Processes I, Springer Verlag, New York, 1977.

....on a jump taking place, there is a positive probability of remaining in the same state. The constant intensity speci cation is sometimes used because it simpli es the characterization of the stationary distribution. 7 of y. This hidden state model is due to Wonham (1964) and is described in Liptser and Shiryayev (1977) and in Elliott, Aggoun, and Moore (1995) It has been used in asset pricing models by David (1997) Veronesi (1999) Veronesi (2000) and David and Veronesi (1999) The expected value of the drift of y, s t , given the current information is t = p t . The n dimensional vector p ....

Liptser, R. S. and A. N. Shiryayev (1977). Statistics of Random Processes. New York: Springer-Verlag.

....on a jump taking place, there is a positive probability of remaining in the same state. The constant intensity speci cation is sometimes used because it simpli es the characterization of the stationary distribution. 7 of y. This hidden state model is due to Wonham (1964) and is described in Liptser and Shiryayev (1977) and in Elliott, Aggoun, and Moore (1995) It has been used in asset pricing models by David (1997) Veronesi (1999) Veronesi (2000) and David and Veronesi (1999) The expected value of the drift of y, s t , given the current information is t = p t . The n dimensional vector p ....

Liptser, R. S. and A. N. Shiryayev (1977). Statistics of Random Processes. New York: Springer-Verlag.

....of discrete and continuous time Kalman filtering with uncertain parameters. Key words: minimax estimates, statistically uncertain systems, feedback parameter AMS Subject Classifications: 93E05, 93E10, 93E11, 62C20 1 Introduction In many problems of statistics, filtering, and control [1 5] it is necessary to estimate some function h(x) of the unobservable variable x on the basis of observations of another value y. In this connection, the joint distribution P (dx; dy) of the random variables x and y may depend on an unknown but bounded parameter 2 Theta. One possible approach ....

.... with measurements y t 0 = fy( 0 tg may be found from equation dx = A(t)x v(t) dt PC 0 oe 0 Gamma1 di; x(0) K 0 x 0 ; where di = oe Gamma1 (dy Gamma (Cx w)dt) is the standard Wiener innovation process in R m ; P (t) is a solution of the Riccati type differential equation [2,5]. The process i(t) generates a Wiener probability measure i (di) in the space (C m t ; B m t ) of continuous m dimensional functions i( i(0) 0, given on [0; t] where the Borelian oe algebra B m t is produced by cylindrical sets [5] In view of the fact that the observable process y(t) has ....

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R.Sh. Liptser and A.N. Shiryayev. Statistics of Random Processes 1,2. New York: Springer-Verlag, 1978.

....if the function s 7 Y s is L 1 continuous on (0; 1) and if (X t ) t 0 is a Gaussian process, then (X t ) t 0 is a Brownian motion if and only if E(X s Y t ) 0; 16) for all 0 s t. Proof. 1) The rst part of the rst assertion can be found in Hida Hitsuda [5] or Liptser Shiryaev ([15], Theorem 7.12) For completeness, here is the well known argument: for s t, E X t X s F X s = E Z t s Y u du F X s = E Z t s E Y u F X u du F X s : Hence, B t : X t Z t 0 E Y u F X u du 8 Canonical ....

Liptser, R. Sh., Shiryaev, A. N. (1978) Statistics of random processes, \Nauka", 1974. English transl., Springer, 1977 (Vol I), 1978 (Vol II). Canonical Decomposition 35

....bound on achievable mean square performance for any causal identification algorithm. By causal we mean that the estimate k of the parameter k is Y k measurable, where Y k : oef i ; y i : Gamma1 i kg. Since the optimal estimator k = E[ k j Y k ] can be realized by a Kalman filter [16], such global bounds can be obtained if a suitable bound can be derived for the Kalman filter estimate. For the time varying system model given by (1) and (2) the Kalman filter is given by the system of equations k 1 = F k P k k e k 1 R T k P k k (3) P k 1 = FP k F T Gamma ....

R. S. Liptser and A. N. Shiryayev. Statistics of Random Processes, I Applications. Springer-Verlag, New York, NY, 1977.

....as an input into the solution of the games. When the state variable s is not observed, the decision maker has to infer information about the current state of the system by using the current and the past observations of y. This hidden state model is due to Wonham (1964) and is described in Liptser and Shiryayev (1977) and in Elliott, Aggoun, and Moore (1995) It has been used in asset pricing models by David (1997) David and Veronesi (1999) and Veronesi (2000) The expected value of the drift of y, s t , given the current information is denoted by: t = p t The n dimensional vector p t ....

Liptser, R. S. and A. N. Shiryayev (1977). Statistics of Random Processes. New York: Springer-Verlag.

....studied in [1, 2] see also [3] The desired solution of the filtering problem is an algorithm that provides the best mean square estimate of the given functional of the state process in the form suitable for on line implementation. In the linear case, such a solution is given by the Kalman filter [4, 5]. It is worth mentioning that the exact solution of the continuous discrete time filtering problem is known for a wide class of nonlinear models [6, 4] Specifically, let X = X(t) t0 be the state process and assume that the measurements z(k) are made at moments t k . If f = f(x) is a function ....

R. Sh. Liptser and A. N. Shiryayev, Statistics of Random Processes, Springer, New York, 1992.

....unobserved signal process (x(t) t0 is estimated from the observations y(s) s t. The best mean square estimate is known to be the conditional expectation E[f t (x)jy(s) s t] called the optimal filter. When the observation noise is additive, the Kallianpur Striebel formula (Kallianpur (1980) Liptser and Shiryayev (1992)) provides the representation of the optimal filter as follows: E[f t (x)jy(s) s t] OE t [f ] OE t [1] where OE t [ Delta] is a functional called the unnormalized optimal filter. In the particular case f t (x) f(x(t) there are two approaches to computing OE t [f ] In the first ....

....with respect to the observation process. This approach can be used to obtain representations of general functionals, but these representations are not recursive in time. In fact, there is no closed form differential equation satisfied by OE t [f ] In the second approach (Kallianpur (1980) Liptser and Shiryayev (1992), Rozovskii (1990) it is proved that, under certain regularity assumptions, the functional OE t [f ] can be written as OE t [f ] Z f(x)u(t; x)dx (1.1) y This work was partially supported by ONR Grant #N00014 95 1 0229. z This work was partially supported by ONR Grant #N00014 95 1 0229 ....

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Liptser, R. S. and Shiryayev, A. N. (1992). Statistics of Random Processes, Springer.

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R.S. Liptser,A.N. Shiryaev . Statistics of Random Processes I,II. Springer, New-York, 1977.

....= F (CA) F C while for the second one M = A (AM) A (M#B ) A M)#B = AM)# = M# B . Theorem 1 In the assumptions of Lemma 4 the process (Y t ) t#0 is Markov. Proof. First let CA = # F C. Then according to the properties of pseudoinverse matrices (see [1] or [8]) Lemmas 4 and 3, and C[I MC] CMC = 0 (see (12) we have M = # F C[I M = 0. 6 Now let AM = M#B . Then, by virtue of [I MC]M = M MCM = 0 (see (12) we get MC]M# B = 0. Thus each of the equalities from (12) is su#cient for (10) to be valid. This means that both ....

....from (12) are su#cient conditions for the process (Y t ) t#0 to possess a Markov property. 5 Markov Representation It follows from (2) and (4) that dY t = CAX t dt CB dW t . Let us denote by (F t ) t#0 the filtration generated by the process (Y t ) t#0 and satisfying the standard conditions [8]. Let # t (Y ) F t ) and introduce an innovation process z t = Y t Y 0 CA# s (Y )ds. It is well known that z t is a Gaussian martingale with respect to the filtration (F t ) t#0 with E(z t z CBE(W t W t )B = CBB t : DD t. Therefore, an innovation Wiener process W t ....

[Article contains additional citation context not shown here]

R. S. Liptser and A. N. Shiryayev, Statistics of Random processes, Springer-Verlag, Vol. I, 1977.

....a localization procedure is applied) Assuming that is small enough so that p = 1=2 ) 1, by the H older inequality it holds p : 6.3) Further, by the Doob inequality p 1 p Moreover, similarly to the proof of Lemma 4.12 (Chapter 4, Section 4. 3 in [26]) it is possible to establish E j ds r s 0 s ds: Thus, for U t = E sup t t 0 j we get the integral inequality U t r r U s ds: Now, by the Bellman Gronwall inequality we have U T r and, by the Chebyshev inequality P ....

R.S. Liptser and A.N. Shiryayev (2000) Statistics of Random Processes. I. Springer Verlag.

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R.Sh. Liptser, A.N. Shiryayev, Statistics of Random Processes, Springer-Verlag, Berlin, 1977.

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Liptser R.L., Shiryaev A.N. (1977). Statistics of Random Processes. Springer, New York.

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