STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY GENERALIZED POSITIVE NOISE. Michael Oberguggenberger and Danijela Rajter-Ćirić

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1 PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome , 7 19 STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY GENERALIZED POSITIVE NOISE Michael Oberguggenberger and Danijela Rajter-Ćirić Communicated by Stevan Pilipović Abstract. We consider linear SDEs with the generalized positive noise process standing for the noisy term. Under certain conditions, the solution, a Colombeau generalized stochastic process, is proved to exist. Due to the blowing-up of the variance of the solution, we introduce a new positive noise process, a renormalization of the usual one. When we consider the same equation but now with the renormalized positive noise, we obtain a solution in the space of Colombeau generalized stochastic processes with both, the first and the second moment, converging to a finite limit. 1. Introduction Stochastic differential equations arise in a natural manner in the description of noisy systems appearing mostly in physical and engineering science. They have been studied by many authors and by using different approaches. One of the possible approaches in solving stochastic differential equations uses the Wick product as is done in [2]. Another one, as in [7], uses weighted L 2 -spaces. A possible approach, the one we use in this paper, is considering differential equations in the framework of Colombeau generalized function spaces as is done in papers [5], [6], [8] and in a similar way in [1]. One of the fundamental concepts in stochastic differential equations is the white noise process standing for the noisy term in an equation. Here we are interested in linear SDEs with a nonstandard additive generalized stochastic process. For nonstandard noise we take the positive noise process. The motivation comes from the work of Holden, Øksendal, Ubøe and Zhang. In [2] the analysis of positive noise viewed as a Wick exponential of white noise was developed in detail and a number of equations containing positive noise were considered. Smoothed positive noise process, as discussed in [2], appears to be a good mathematical model for many cases where positive noise occurs. We are interested in considering equations with such a noise but now viewed as a Colombeau generalized process. 2 Mathematics Subject Classification: 46F3, 6G2, 6H1. 7

2 8 OBERGUGGENBERGER AND RAJTER-ĆIRIĆ In this paper we consider the linear Cauchy problem X t = at Xt bt W t, t X = X, where W t is positive noise viewed as a Colombeau generalized stochastic process and X is a Colombeau generalized random variable. We show that, under certain mild conditions on the deterministic functions at and bt, there exists a Colombeau generalized stochastic process Xt which is a solution to the problem above. If, in addition, we suppose that the expectation of a represenatitive of the initial data X converges to a finite limit as ε tends to zero, then we show that the expectation of a representative of the solution Xt converges to a finite limit, too. However, the second moments of the representatives of the solution Xt diverge as ε tends to zero. That was a motivation for introducing a new positive noise which is, in fact, a renormalization of the usual positive noise in the sense of asymptotics. We call that new process a renormalized positive noise process and denote it by W t. Renormalized positive noise has mean value converging to zero and variance converging to infinity, as ε tends to zero. Just as positive noise itself, these properties make it suitable for describing rapid nonnegative fluctuations. When we consider the Cauchy problem above with renormalized positive noise instead of W t, we again obtain a solution in the space of Colombeau generalized stochastic processes but now with both, the first and the second moment, converging to a finite limit. Thus renormalized positive noise is also suitable as a driving term in linear SDEs. 2. Notation and basic definitions Let Ω, Σ, µ be a probability space. A generalized stochastic process on R d is a weakly measurable mapping X : Ω D R d. We denote by D Ω Rd the space of generalized stochastic processes. For each fixed function ϕ DR d, the mapping Ω R defined by ω Xω, ϕ is a random variable. White noise Ẇ on Rd can be constructed as follows. We take as probability space the space of tempered distributions Ω = S R d with Σ the Borel σ-algebra generated by the weak topology. By the Bochner Minlos theorem [2], there is a unique probability measure µ on Ω such that e i ω,ϕ dµω = exp 1 2 ϕ 2 L 2 R d for ϕ SR. The white noise process Ẇ is defined as the identity mapping Ẇ : Ω D R d, Ẇ ω, ϕ = ω, ϕ for ϕ DR d. It is a generalized Gaussian process with mean zero and variance V Ẇ ϕ = EẆ ϕ2 = ϕ 2 L 2 R d, where E denotes expectation. Its covariance is the bilinear functional 1 E Ẇ ϕ Ẇ ψ = ϕyψy dy R d

3 STOCHASTIC DIFFERENTIAL EQUATIONS... 9 represented by the Dirac measure on the diagonal R d R d, showing the singular nature of white noise. A net ϕ ε of mollifiers given by ϕ ε y = 1 y 2 ε d ϕ, ϕ DR d, ϕydy = 1, ϕ, ε is called a nonnegative model delta net. Smoothed white noise process on R d is defined as 3 Ẇ ε x = Ẇ y, ϕ εx y, where Ẇ is white noise on Rd and ϕ ε is a nonnegative model delta net. It follows from 1 that the covariance of smoothed white noise is 4 E Ẇε xẇεy = ϕ ε x zϕ ε y z dz = ϕ ε ˇϕ ε x y R d where ˇϕz = ϕ z. We define the smoothed positive noise process W ε x on R as 5 W ε x = exp Ẇε x 1 2 ϕ ε 2 L, 2 where Ẇε and ϕ ε are as in 3. One can easily show that smoothed positive noise is a family of random stochastic processes, lognormally distributed with mean value 1 and variance V W ε x = e σ2 ε 1 for x R d, where σ 2 ε = ϕ ε 2 L 2. For the remainder of this paper we confine ourselves to the one-dimensional case since that is the case needed for SDEs. We now introduce Colombeau generalized stochastic processes in the one-dimensional case see also [4]. Denote by ER the space of nets X ε ε, ε, 1, of processes X ε with almost surely continuous paths, i.e., the space of nets of processes X ε :, 1 R Ω R such that t, ω X ε t, ω is jointly measurable, for all ε, 1; t X ε t, ω belongs to C R, for all ε, 1 and almost all ω Ω. Definition 1. E Ω M R is the space of nets of processes X ε ε belonging to ER, ε, 1, with the property that for almost all ω Ω, for all T > and α N, there exist constants N, C > and ε, 1 such that sup α X ε t, ω C ε N, ε ε. t [,T ] N Ω R is the space of nets of processes X ε ε ER, ε, 1, with the property that for almost all ω Ω, for all T > and α N and all b R, there exist constants C > and ε, 1 such that sup α X ε t, ω C ε b, ε ε. t [,T ] The differential algebra of Colombeau generalized stochastic processes is the factor algebra G Ω R = E Ω M R/N Ω R.

4 1 OBERGUGGENBERGER AND RAJTER-ĆIRIĆ The elements of G Ω R will be denoted by X = [X ε ], where X ε ε is a representative of the class. White noise can be viewed as a Colombeau generalized stochastic processes having a representative given by 3. This follows from the usual imbedding arguments of Colombeau theory see e.g. [3], since its paths are distributions. What concerns positive noise, a suitably slow scaling in the parameter ε is needed to counterbalance the exponential. Thus taking the scaling ηε = log ε and replacing 5 by 6 W ε x = exp Ẇηε x 1 2 ϕ ηε 2 L 2 produces a family of processes which belongs to EM Ω R and thus defines an element of G Ω R. We refer to this generalized process as the Colombeau positive noise. For evaluation of generalized stochastic process at fixed points of time, we introduce the concept of a Colombeau generalized random variable as follows. Let ER be the space of nets of measurable functions on Ω. Definition 2. ER M is the space of nets X ε ε ER, ε, 1, with the property that for almost all ω Ω there exist constants N, C >, and ε, 1 such that X ε ω Cε N, ε ε. N R is the space of nets X ε ε ER, ε, 1, with the property that for almost all ω Ω and all b R, there exist constants C > and ε, 1 such that X ε ω Cε b, ε ε. The differential algebra GR of Colombeau generalized random variables is the factor algebra GR = ER M /N R. If X G Ω R is a generalized stochastic process and t R, then Xt is a Colombeau generalized random variable, i.e., an element of GR. 3. SDEs with Colombeau generalized positive noise process We consider the Cauchy problem 7 8 X t = at Xt bt W t, t R X = X, as stated in the introduction, where W t G Ω R is Colombeau positive noise and X = [X ε ] GR is a Colombeau generalized random variable. We suppose that at is a deterministic, smooth function on R and denote 9 ãτ = τ at dt. The function bt is supposed to be deterministic and smooth on R. Theorem 1. Under the conditions above, problem 7 8 has an almost surely unique solution X G Ω R.

5 STOCHASTIC DIFFERENTIAL EQUATIONS Proof. Fix ω Ω and ε, 1. representatives reads 1 11 The Cauchy problem 7-8 given by X εt = at X ε t bt W ε t, t R X ε = X ε, where W ε ε EM Ω R is given by 6 and X ε ε ER M R. Problem 1-11 has the solution 12 X ε t = X ε eãt eãt e ãτ bτ W ε τ dτ. Let us show that X ε ε belongs to EM Ω R. First, from 12 we have that sup t [,T ] X ε t X ε exp sup t [,T ] ãt T exp sup t [,T ] ãt inf ãτ τ [,T ] sup bτ τ [,T ] Since W ε ε E Ω M R and X ε ε ER M R we obtain sup X ε t C 1 ε b 1 C 2 T ε b 2, t [,T ] sup W ε τ. τ [,T ] for some b 1, b 2 R and some C 1, C 2 >. Thus, sup t [,T ] X ε t has a moderate bound for any T >. A similar argument applies to subintervals of the negative time axis. We obtain a moderate bound for the first order derivative of X ε from 1: sup X εt C 1 sup X ε t C 2 sup W ε t. t [,T ] t [,T ] t [,T ] Since W ε ε E Ω M R and sup t [,T ] X ε t has a moderate bound, we conclude that sup t [,T ] X εt has a moderate bound, too. By successive derivations, one can estimate higher order derivatives of X ε and obtain their moderate bounds. Thus, X ε ε belongs to E Ω M R and X = [X ε] G Ω R defines a solution to problem 7-8. One can easily show that this solution is almost surely unique in G Ω R by considering the equation X εt = at X ε t N ε t, Xε = N ε, where X ε ε = X 1ε X 2ε ε and X 1ε ε, X 2ε ε EM Ω R are two solutions to equation 1, N ε ε N Ω R and N ε ε N R. After a similar procedure as in the existence part of the proof one obtains that X 1ε X 2ε ε N Ω R. Thus, the solution X to equation 7 is almost surely unique in G Ω R. For fixed ε, 1 denote EX ε = x ε. The expectation of the solution X ε t to problem 1 11 is EX ε t = EX ε eãt eãt e ãτ bτ E W ε τ dτ,

6 12 OBERGUGGENBERGER AND RAJTER-ĆIRIĆ i.e., since E W ε τ = 1, EX ε t = x ε eãt eãt e ãτ bτ dτ. It is obvious that the expectation of the solution X ε to problem 1 11 coincides with the solution to the equation obtained from equation 1 11 by averaging the coefficients: X εt = at X ε t bt, X ε = x ε. If, in addition, we suppose that lim ε x ε = x ±, then EX ε t x eãt eãt e ãτ bτ dτ, as ε. However, the second moment of the solution X ε t may diverge as ε tends to zero. Indeed, assuming that X ε is independent of W ε t, t R, the second moment of X ε t is 13 EXε 2 t = EXε 2 e 2ãt 2EX ε eãt e ãτ bτ dτ 2 e 2ãτ E e ãτ bτ W ε τ dτ. The expectation in the last term in the right-hand side is 2 E e ãτ bτ W ε τ dτ = E = e ãx bx W ε xe ãy by W ε y dx dy e ãx ãy bx by E W ε x W ε y dx dy and we will show that if b is bounded away from zero and the mollifier ϕ is symmetric, then the second moments of X ε t diverge for all t R, t. From now on we assume that the mollifier ϕ satisfies 2 and is symmetric. This entails that 14 ϕ ˇϕr < ϕ ˇϕ for all r R, r, a property which will be essential later. ϕ ˇϕ = ϕ ϕ and 15 ϕ ϕr = Indeed, by the symmetry assumption, ϕzϕz r dz ϕ L 2 R ϕ r L 2 R for all r R by the Cauchy Schwarz inequality. Further, equality in 15 is attained if and only if ϕ and ϕ r are parallel, that is, r =. Thus 14 holds. For technical facilitation we shall also assume that 16 supp ϕ = [ 1/2, 1/2].

7 STOCHASTIC DIFFERENTIAL EQUATIONS We now introduce some notation. By 4, the covariance matrix of smoothed white noise Ẇε at points x, y is σ 2 C ε = ε τε 2 r τε 2, r where r = x y and we use the notation σε 2 = ϕ ηε 2 L = ϕ 2 ηε ϕ ηε, τε 2 r = ϕ ηε ˇϕ ηε r = ϕ ηε ϕ ηε r with the scaling ηε = log ε introduced in 6. Lemma 1. Let [W ε ] G Ω R be Colombeau positive noise. Then its covariance at points x y R equals 17 E W ε x W ε y = e τ 2 ε x y. 18 Proof. When x y, we have that E W ε x W ε y = 1 2π e σ2 ε σ 2 ε 1 e xy exp det Cε 1 2 x, yc 1 ε x, y T dx dy. By 14, det C ε = σε 4 τε 4 r > with r = x y, so the matrix C ε is positive definite. Diagonalization gives that Cε 1 = Q diag σ 2 ε τε 2, σε 2 τε 2 Q T, with Q = The change of variables Cε 1 x, y T x 1, y 1 T gives and turns 18 into E W ε x W ε y = 1 2π Using the relation x y = σ 2 ε τ 2 ε x 1 σ 2 ε τ 2 ε y 1 e σ2 ε exp σ 2 ε τε 2 x 1 y 1 x2 1 2 y2 1 2 we rewrite this as E W ε x W ε y = 1 2π = e τ 2 ε x y, thereby establishing 17. = e τ 2 ε exp 1 2 e σ2 ε exp σ 2 ε τε 2 x 1 y 1 x2 1 2 y2 1 dx 1 dy 1. 2 x 1 σ 2 ε τ 2 ε 2 1 y 1 2 σε 2 2 τε 2 e τ 2 ε exp 1 x 1 2 σε 2 2 τε 2 1 y 1 2 σε 2 2 τε 2 dx 1 dy 1

8 14 OBERGUGGENBERGER AND RAJTER-ĆIRIĆ Proposition 1. Let X G Ω R be the solution to the Cauchy problem 7, 8 constructed in Theorem 1. Assume that bt b for some b > and all t R. Then 19 E X 2 ε t, as ε for all t R, t. Proof. We consider the case t >. As was deduced above, we have to estimate the decisive term 13, which by Lemma 1 equals e ãx ãy bx by e τ 2 ε x y dx dy. Now ã is bounded from above on the interval [, t] and by assumption, b is bounded away from zero. Thus the decisive term above can be estimated from below by c for some constant c >. Recalling that we obtain that e τ 2 ε x y dx dy τ 2 ε x y = ϕ ηε ϕ ηε x y = 1 ηε ϕ ϕ e τ 2 ε x y dx dy = η 2 /η /η x ηε y ηε 1 exp η ϕ ϕx y dx dy with η = ηε. Since ϕ ϕx y c for some c > on a set of positive twodimensional measure, this latter expression tends to infinity as ε. This proves 19. The blow-up of the variance of the solution to equation 1 11 is a motivation for introducing a new positive noise, a renormalization of the usual positive noise in the sense of asymptotics. Renormalized positive noise will depend on the choice of a renormalization interval [, T ] and a free parameter C R, C. We introduce it by means of the representing family 2 W ε t = exp Ẇ ηε t 12 σ2ε 12 C log 2 e τ 2 ε r s dr ds, where t R, ηε = log ε, σ ε, τ ε are defined as before Lemma 1 and Ẇε ε EM Ω R is a representative of smoothed white noise. In fact, 21 W ε t = C W ε t e τ 2 ε r s dr ds t R, where W ε t is a representative of Colombeau positive noise. We shall not display the dependence on T and C in our notation and simply call the class defined by 2 in G Ω R the renormalized positive noise process W.

9 STOCHASTIC DIFFERENTIAL EQUATIONS In the limit, renormalized positive noise W t has mean value zero and infinite variance. More precisely, the following holds. Lemma 2. Let W G Ω R be renormalized positive noise. Then, for any representative and t R, E W ε t, as ε, V W ε t, as ε. Proof. The first assertion follows immediately from the arguments in the proof of Proposition 1: E W E W ε t 1 ε t = = T, as ε. T C e τ 2 ε r s dr ds C e τ 2 ε r s dr ds For the second moment of W ε t we have 2 E W ε t 2 E W ε t = C e τ 2 ε r s dr ds = C 2 e σ2 ε e τ 2 ε r s dr ds = 2 C 2 1 e τ 2 ε r s σ2 ε dr ds. Since τ 2 ε is a symmetric function the denominator of the last term equals By the assumption 16, this in turn equals ε e τ 2 ε r s σ2 ε dr ds = T re τ 2 ε r σ2 ε dr. T re τ 2 ε r σ2 ε dr T re σ2 ε dr using that τε 2 = when r > ε. By 14, the first integrand is bounded; the second integrand goes to zero as ε. Thus the denominator in question converges to zero, and so E W ε t 2 tends to infinity. Finally, the divergence of the second moment of the process W ε t implies divergence of the variance V W ε t Now we consider the equation X t = at Xt bt W t, t R X = X, where W t G Ω R is renormalized positive noise and X = [X ε ] GR is a generalized random variable. Let at be as before, i.e., a deterministic, smooth ε

10 16 OBERGUGGENBERGER AND RAJTER-ĆIRIĆ function on R and let ãτ be given by 9. Also, bt is supposed to be deterministic and smooth on R, bt b > for all t R. Problem in terms of representatives reads X εt = at X ε t bt W ε t, t R X ε = X ε, where W ε ε E Ω M R and X ε ε ER M. The following assertion can be proved similarly as in Theorem 1; we skip the proof. Theorem 2. Under the conditions above, problem has an almost surely unique solution X G Ω R given by 26 X ε t = X ε eãt eãt e ãτ bτ W ε τ dτ. Denote EX ε = x ε and EX 2 ε = x ε and suppose 27 lim ε x ε = x ±, and lim ε x ε = x <. We will show below that the first and the second moment of the solution X ε to problem converge to a finite limit as ε tends to zero, which was exactly what we wanted to achieve by introducing renormalized positive noise W t. Theorem 3. Let X ε be the solution to problem and t R. Then EX ε t x eãt, as ε, EX 2 ε t x e 2ãt e2ãt C 2 T e 2ãy b 2 y dy, as ε. Proof. Note that the expectation of X ε t given by 26 is now EX ε t = EX ε eãt eãt e ãτ bτ E W ε τ dτ. Since E W ε τ, as ε, by using 27 we obtain EX ε t x eãt, as ε. The second moment of the solution X ε t is EX 2 ε t = x ε e 2ãt 2x ε eãt e ãτ bτ E W ε τ dτ 2 e 2ãt E e ãτ bτ W ε τ dτ.

11 STOCHASTIC DIFFERENTIAL EQUATIONS E The expectation in the last term in the right-hand side is 2 E e ãτ bτ W ε τ dτ = E = e ãx bx W ε xe ãy by W ε y dx dy e ãx ãy bx by E W ε x W ε y dx dy. Using the relation 21 one easily obtains E W ε x W ε E W ε xw ε y y = C 2 e τ 2 ε r s dr ds, t [, T ]. We saw in Lemma 1 that E W ε xw ε y = e τ 2 ε x y. That means 2 e ãτ bτ W ε τ dτ = for t [, T ]. We want to evaluate the limit of the term 28 e ãy by dy C 2 dy e ãx e ãy bx by e τ 2 ε x y dx dy, C 2 e τ 2 ε x y dx dy e ãx bx e τ 2 ε x y dx e τ 2 ε x y dx as ε tends to zero. Since τε 2 r = whenever r [ ε, ε], we may rewrite 28 as yε e ãy by dy e ãx bx e τ 2 ε x y dx A ε y y ε 29 yε, C 2 dy e τ 2 ε x y dx B ε y where A ε y = B ε y = y ε y ε y ε e ãx bx dx dx dx. yε yε e ãx bx dx First, note that as ε tends to zero, both the numerator and the denominator in 29 tend to infinity. On the other hand, it is obvious that the quantities A ε and B ε remain finite as ε tends to zero.

12 18 OBERGUGGENBERGER AND RAJTER-ĆIRIĆ Therefore, for evaluating the limit as ε tends to zero of 29 it is enough to consider the limit as ε tends to zero of 3 e ãy by dy C 2 dy Since τ 2 ε is symmetric, we have that yε y ε yε y ε yε y ε e τ 2 ε x y dx = 2 e ãx bx e τ 2 ε x y dx. e τ 2 ε x y dx yε y e τ 2 ε x y dx. Thus 3 equals 31 yε y e ãy by dy e ãx bx e τ 2 ε x y dx y yε 2 C 2 dy e τ 2 ε x y dx y y ε e ãx bx e τ 2 ε x y dx. We shall compute the limit as ε tends to zero of the first summand in 31; due to its structural similarity, the second summand will have the same limit. By the change of variables x y x the first term becomes 32 Introduce ε e ãy by dy e ãxy bx y e τ 2 ε x dx ε. 2 C 2 dy e τ 2 ε x dx R ε = ε e τ 2 ε x dx. The denominator of 32 is then C 2 T R ε. By adding and subtracting the term e ãy by dy the numerator of 32 becomes R ε e 2ãy b 2 y dy e ãy by dy ε ε e ãy by e τ 2 ε x dx, e ãxy bx y e ãy by e τ 2 ε x dx. By supposition, e ãy by is a smooth function and so ε e ãy by dy e ãxy bx y e ãy by e τ 2 ε x dx εr ε,

13 STOCHASTIC DIFFERENTIAL EQUATIONS for small ε. Therefore, we have to compute the limit of 1 2 C 2 R ε e 2ãy b 2 y dy εr ε T R ε as ε tends to zero, which, however, obviously equals 1 2 C 2 e 2ãy b 2 y dy. T Together with the same result for the second summand, this implies that EXε 2 t x e 2ãt e2ãt C 2 T as ε, as claimed. e 2ãy b 2 y dy Acknowledgement. We thank Francesco Russo for valuable discussions. References [1] S. Albeverio, Z. Haba and F. Russo, Trivial solutions for a non-linear two-space-dimensional wave equation perturbed by space-time white noise, Stochastics and Stochastics Reports , [2] H. Holden, B. Øksendal, J. Ubøe and S. Zhang, Stochastic Partial Differential Equations, Birkhäuser, Basel, [3] M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Res. Not. Math. 259, Longman Sci. Techn., Harlow, [4] M. Oberguggenberger 1995, Generalized functions and stochastic processes, In: E. Bolthausen, M. Dozzi, F. Russo Eds., Seminar on Stochastic Analysis, Random Fields and Applications, Prog. Probab 36, Birkhäuser, Basel, [5] M. Oberguggenberger and F. Russo, Nonlinear stochastic wave equations, Integral Transforms and Special Functions , [6] M. Oberguggenberger and F. Russo 1998, Nonlinear SPDEs: Colombeau solutions and pathwise limits, In: L. Decreusefond, J. Gjerde, B. Øksendal, A. S. Üstünel Eds., Stochastic Analysis and Related Topics VI, Prog. Probab. 42, Birkhäuser, Boston, [7] S. Peszat and J. Zabczyk, Nonlinear stochastic wave and heat equations, Probab. Th. Rel. Fields 116 2, [8] F. Russo 1994, Colombeau generalized functions and stochastic analysis, In: A. I. Cardoso, M. de Faria, J. Potthoff, R. Sénéor, L. Streit Eds., Analysis and Applications in Physics, Kluwer, Dordrecht, Institut für Technische Mathematik, Received Geometrie und Bauinformatik, Innsbruck, Austria Institut za matematiku i informatiku Prirodno-matematički fakultet Novi Sad Serbia and Montenegro