An Itō formula in S via Itō s Regularization

Transcription

1 isibang/ms/214/3 January 31st, statmath/eprints An Itō formula in S via Itō s Regularization Suprio Bhar Indian Statistical Institute, Bangalore Centre 8th Mile Mysore Road, Bangalore, 5659 India

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3 AN ITŌ FORMULA IN S VIA ITŌ S REGULARIZATION SUPRIO BHAR Abstract. Following the work of Itaru Mitoma in [8], we give an alternative construction of Stochastic Integrals of Hermite Sobolev space valued predictable processes {G t } with respect to real semimartingales {X t }. We consider Gs, ϕ dxs for ϕ in the Schwartz space as random linear functionals. Our main technique involves obtaining regularized versions of these random linear functionals via Itō s Regularization Theorem. These results allow us to prove an Itō formula in S. 1. Introduction In various situations we come across random evaluation functionals. Given a real valued random variable X and a space E of real valued test functions, f f(x) for f E is an example of such a functional. Formally writing f(x) = δ X, f (where δ X takes values in the dual of E, some suitable space of distributions) gives a much clearer picture of this setup. More generally, we have random linear functionals which may take values in some space of distributions. Note that δ X actually takes values in some Hilbert space (see [11, Lemma 2.1]). Likewise if we can identify some regularity properties of a random linear functional (such as it takes values in a Hilbert space), analysis of such a functional becomes easier. First we use the setup of random linear functionals on Schwartz space to define Stochastic Integrals of Hermite-Sobolev space valued processes with respect to real semimartingales. Our main tool is Itō s Regularization Theorem which relates random linear functionals on the Schwartz space to processes taking values in some Hermite-Sobolev space. Using this identification, we then go on to prove an Itō formula. In section 2, we recall the countably Hilbertian topology defined on S(R d ) which gives rise to the Hermite-Sobolev spaces S p (R d ). We also recall a version of Itō s Regularization Theorem, which we use thoughout this paper. In section 3, we give a construction of the stochastic integral of appropriate predictable processes (taking values in the Hermite-Sobolev spaces) with respect to real square integrable martingales, finite variation processes and real semimartingales. We start with a Hermite-Sobolev space valued predictable processes {G t } and a real square integrable martingale {M t } and consider the random linear functional defined on the Schwartz space, viz. ϕ G s, ϕ [ dm s. In [8, section 3, Application ] t IV] the author remarked upon the condition E G s, ξ 2 d M s <, ξ E 21 Mathematics Subject Classification. Primary: 6H5; Secondary: 6G2. Key words and phrases. Hermite-Sobolev spaces, Tempered distributions, S valued processes, Random Linear functionals, Itō formula, Local times, Stochastic Integral, Tensor Quadratic Variation. 1

4 2 SUPRIO BHAR (E is some Nuclear Fréchet space) to obtain a E valued [ version of the stochastic integral. We consider a stronger assumption that E ] t G s 2 p d M s <, where G s takes values in some Hermite Sobolev space S p. This allows us to construct the version in S p, which we also connect to the theory of Hilbert space valued stochastic integrals as developed in [7]. We also compute the Tensor Quadratic Variation of the stochastic integral. Next we consider a norm bounded S p valued predictable process {G t } and produce a regularized version of the random linear functional ϕ G s, ϕ da s, where {A t } is some real valued process of locally finite variation. We show that the regularized version is a S p valued process of locally finite variation. We then extend this construction by considering a real valued semimartingale {X t } as the integrator. We use the decomposition of X into a martingale part M and a finite variation part A, to decompose the random linear functional ϕ G s, ϕ dx s as ϕ G s, ϕ dm s + G s, ϕ da s. We show that the regularized version of this random linear functional does not depend upon the decomposition M + A. We have followed the procedure for general stochastic integration as set out here [5, Chapter 23]. The results are extendable to multi dimensions. In section 4, we prove an Itō formula (see Theorem 4.2) which generalises [11, Theorem 2.3]. This result has been used in [12] to show existence of solution of some stochastic differential equations in S, the space of tempered distributions. A version of this Itō formula was also proved in [19, Theorem III.1] with equality in S. In [6, Theorem 3], the author has proved this formula for twice continuously (Fréchet) differentiable function while dealing with a single Hilbert space. Note that derivatives of tempered distributions may not not be in the same Hermite Sobolev space as the original one. Using regularization, the result [6, Theorem 3] was also proved in [8, Theorem 8] in the case of a E valued continuous martingale. These results are also connected to [11, Proposition 1.3], where the random linear functional nature of the stochastic integral has been pointed out. We also mention some connections with the Local Time process for a continuous real valued semimartingale (also see [11, Lemma 2.1]). The author would like to inform the interested reader that the general case of regularizing S valued semimartingales has been considered in [1]. 2. Preliminaries 2.1. Topologies on S and S. Let Z d + := {n = (n 1,, n d ) : n i integers}. If n = (n 1,, n d ), we define n := n n d. Let {h n : n Z d +} be the orthonormal basis (ONB) for L 2 (R d ) given by the Hermite functions (see [4, Chapter 1.3], [17, Chapter 1]). Let, represent the L 2 (R d ) inner product. For any fixed p R, consider the following formal sums { f, g p := k= n =k (2.1) (2k + d)2p f, h n g, h n, f 2 p := k= n =k (2k + d)2p f, h n 2 Let S(R d ) L 2 (R d ) be the space of smooth rapidly decreasing R-valued functions on R d with the topology given by L. Schwartz (see [18]). Then (S(R d ), p ) are pre-hilbert spaces and completing them one obtains the separable Hilbert spaces

5 AN ITŌ FORMULA IN S VIA ITŌ S REGULARIZATION 3 (S p (R d ), p ) known as the Hermite-Sobolev spaces (see [4, Chapter 1.3]). Let S (R d ) denote the space of tempered distributions. In [4], it was shown that (S p (R d ), p ) are dual to (S p (R d ), p ). Furthermore, the following are also known: L 2 (R d ) = (S (R d ), ), for p < q, (S q (R d ), q ) (S p (R d ), p ), S(R d ) = p R (S p(r d ), p ), S (R d ) = p R (S p(r d ), p ) The following notations will be used thoughout this paper: S, S, S p will stand for S(R), S (R), S p (R) respectively. Given ψ S(R d ) (or S p (R d )) and ϕ S (R d ) (or S p (R d )), the action of ϕ on ψ will be denoted by ϕ, ψ Regularization of random linear functionals. Definition 2.1. Let E be a countably Hilbertian Nuclear space. A stochastic process {X(x), x E} is a random linear functional if for each x, y E and a, b R we have X(ax + by) = ax(x) + bx(y) a.s. We shall take S(R d ) as our Nuclear space. The main tool for our analysis is Itō s Regularization Theorem for S(R d ). The result is as follows: Theorem 2.2 ([4, Theorem 2.3.2]). Let τ denote the topology on S(R d ). If Y is a τ-continuous random linear functional on S(R d ), then there exists a τ-regular version of Y say Ỹ, i.e. (1) Y (ϕ) = Ỹ, ϕ a.s. for all ϕ S(R d ). (2) There exists a countably Hilbertian topology θ on S(R d ) weaker than τ such that Ỹ is almost surely (S(Rd ), θ) valued. Furthermore such a τ-regular version is unique in the sense that if Ỹ and Y are two τ-regular version of Y, then Ỹ = Y a.s. For our purpose we need a specialized version of the above Theorem. Theorem 2.3 ([2, Theorem 4.1]). Let X be a random linear functional on S(R d ) which is continuous in probability in p for some p R. If p HS q, for some q R, then X has a version which is in S q (R d ) a.s. Remark 2.4. Note that in dimension 1, h q n := (2n + 1) q h n is an orthonormal basis for S q. Then, h q n 2 p = (2n + 1) 2(p q). n= n= Now for p HS q we need the above sum to be finite which happens if q > p For dimension d, the appropriate condition is q > p + d 2. Here is an outline of the proof. We have k= n =k hq n 2 p k= (2k +d)2(p q) #{n Z d + : n = k}. By [2, Chapter II, section 5], #{n Z d + : n = k} = ( ) k+d 1 d 1. We can find a constant C > such that ( ) k+d 1 d 1 C.(2k + d) d 1. Hence for q > p + d 2, the series is finite.

6 4 SUPRIO BHAR Remark 2.5. Continuity in probability can be metrized by the well-known Ky Fan metric. (2.2) d(y 1, Y 2 ) := E{ Y 1 Y 2 1} where Y 1, Y 2 are two real-valued random variables. Now, given a random linear functional X on S, to check the continuity property, we only need to check it at the point. Continuity in probability in some norm p means the following: given any ϵ >, δ > such that Since X() = we need to check x p < δ = d(x(x), X()) < ϵ, (2.3) x p < δ = E ( X(x) 1) < ϵ. Observe that (2.4) E ( X(x) 1) E ( X(x) ) ( E (X(x)) 2) 1 2 Mostly we shall estimate this second moment to establish the required continuity. 3. Regularized Version of the Stochastic Integral Unless stated otherwise the we shall use the following notations thoughout the section. To avoid cumbersome notations we mostly present the version of the results in dimension d = 1. We also follow the procedure for general stochastic integration as set out here [5, Chapter 23]. Fix p, q R with p > and q > p Note that this implies p HS q. Let T be a positive real number or. Our time interval will be [, T ]. For T =, [, T ] will mean [, ). Let (Ω, F, {F t } t [,T ], P ) be a probability space, with the filtration satisfying the usual conditions. For any real valued martingale M or a process of finite variation A or a semimartingale X, we shall assume M, A, X. Furthermore, any such real valued process (martingales, process of finite variation or semimartingales) will be assumed to have rcll (right continuous with left limits) paths. Unless stated otherwise stopping times or processes will be assumed to be adapted to the filtration {F t } Stochastic Integral with respect to a real square integrable Martingale. Let {M t } t [,T ] be a real-valued square integrable martingale adapted to the given filtration. Let {G t } t [,T ] be a S p valued predictable process such that (3.1) E T G s 2 p d M s <, Here M is the predictable increasing process such that M 2 M is a F t martingale. Note that for any ϕ S p, the real valued process G s, ϕ is a predictable process. Furthermore, we have for any t [, T ] (3.2) E G s, ϕ 2 d M s ϕ 2 p E G s 2 p d M s < Then for any t [, T ] we may consider the linear map I t,g on S p defined by (3.3) I t,g (ϕ) := G s, ϕ dm s.

7 AN ITŌ FORMULA IN S VIA ITŌ S REGULARIZATION 5 We can restrict this map to S and obtain a random linear functional on S. Theorem 3.1. For each t [, T ], the random linear functional I t,g has a regularized version Ĩt,G in S p. Proof. For any t [, T ], we have E(I t,g (ϕ)) 2 = E ϕ 2 p E ϕ 2 p E G s, ϕ 2 d M s, (by Itō s isometry) T G s 2 pd M s G s 2 pd M s Now using (2.4) we have ϵ ϕ p < 1 + E = E( I T t,g(ϕ) 1) < ϵ. G s 2 pd M s Hence I t,g is continuous in probability in p at ϕ =, which by linearity of I t,g is true globally. Hence by Theorem 2.3, we get a S q valued regularized version, say Ĩt,G, such that Ĩt,G, ϕ = I t,g (ϕ), a.s., ϕ S. Now another application of Itō s isometry gives ( ) 2 > E G s 2 p d M s = (2n + 1) 2p E G s, h n dm s = E n= 2 (2n + 1) 2p Ĩ t,g, h n = E Ĩ t,g 2 p. n= Hence Ĩt,G is S p valued a.s. The next result is an application of the well-known Pettis Theorem (see [15, Theorem 1.1]) Lemma 3.2. Let t [, T ] and p R. Let Y : Ω S p be a function. Then Y is measurable with respect to F t and B(S p ) (the Borel σ algebra on S p ) iff for all ϕ S, Y, ϕ is measurable with respect to F t and B(R) (the Borel σ algebra on R). Proof. For ϕ S, the map ψ ψ, ϕ is continuous on S p. If Y is measurable with respect to F t and B(S p ), then Y, ϕ is measurable as in the statement. For the converse, note that S p is a separable Hilbert space and its dual is S p. Furthermore, S is dense in S p in the norm p. If for all ϕ S, Y, ϕ is measurable with respect to F t and B(R), then by above density, we have Y, ϕ is measurable with respect to F t and B(R) for all ϕ S p. Applying Pettis Theorem we get the converse implication. We list properties of Ĩt,G in the next result. Theorem 3.3. Ĩt,G satisfies the following conditions:

8 6 SUPRIO BHAR (a) E Ĩt,G 2 p <. (b) Ĩt,G is measurable with respect to F t and the Borel σ-algebra of S p. (c) For any s, t [, T ] with s t and A F s, ] ] E [Ĩt,G 1 A = E [Ĩs,G 1 A. Hence Ĩt,G is a S p valued square integrable martingale. Proof. Part (a) is included in the proof of the previous Theorem. Note that Ĩt,G, ϕ = I t,g (ϕ), a.s., ϕ S. But Ĩt,G is S p valued and I t,g (ϕ) makes sense for all ϕ S p. Since S p is dual to S p and S is dense in S p we have Ĩt,G, ϕ = I t,g (ϕ), a.s., ϕ S p. Now I t,g (ϕ) is F t /B(R) measurable for all ϕ S and hence for all ϕ S p. Therefore Ĩt,G, ϕ is F t /B(R) measurable for all ϕ S p. Applying Lemma 3.2 we get part (b). For part (c), it is enough to prove (see discussion in [16, pp ]) ] ] E [Ĩs,G 1 A, ϕ = E [Ĩt,G 1 A, ϕ, ϕ S p. But, {I t,g (ϕ)} t is a martingale for any ϕ S. Hence, we have the following implications: E [I s,g (ϕ)1 A ] = E [I t,g (ϕ)1 A ], ϕ S. ] ] = E [ Ĩs,G, ϕ 1 A = E [ Ĩt,G, ϕ 1 A, ϕ S. = E [ Ĩs,G 1 A, ϕ ] = E [ Ĩt,G 1 A, ϕ ], ϕ S. ] ] = E [Ĩs,G 1 A, ϕ = E [Ĩt,G 1 A, ϕ, ϕ S. ] ] = E [Ĩs,G 1 A, ϕ = E [Ĩt,G 1 A, ϕ, ϕ S p. This proves part (c). Proposition 3.4. {Ĩt,G} has a version with rcll (right continuous with left limits) paths in q for any q > p Proof. Note that E(I t,g (ϕ)) 2 ϕ 2 p E T G s 2 pd M s. From [9, Theorem 1] we get the required version. Remark 3.5. Observe that for predictable rectangles G t = 1 (s,u] F (t, ω)ϕ, where s, u [, T ] with < s < u and ϕ S p, we have Ĩt,G = 1 F (M t u M t s )ϕ, which agrees with the usual stochastic integral defined on S p (as a Hilbert space, see [7]). Therefore the construction above actually gives the stochastic integral G s dm s for predictable G satisfying the integrability condition (3.1). Remark 3.6. We list some general observations.

9 AN ITŌ FORMULA IN S VIA ITŌ S REGULARIZATION 7 (1) If the martingale is continuous, then the argument works for integrands which are progressively measurable. In this case, we can replace M (in the integrability conditions) by [M], the quadratic variation process. (2) If M t = (Mt 1,, Mt n ) is a F t -adapted R n -valued square integrable martingale and {G t } is a S p (R d ) valued predictable process then under appropriate integrability conditions we may define the random linear functionals on S(R d ) I t,g,i (ϕ) := G s, ϕ dm i s, ϕ S(R d ); i = 1,, n. We can obtain regularized versions of the integrals. (3) we can do this procedure with locally square integrable martingales. The linear functionals need to be appropriately defined. The predictable process {G t } can also be taken to satisfy integrability condition (3.1) locally. Let B be a real Banach space and H be a real Hilbert space. Let B denote the norm in B. Let H and, H denote the norm and the inner product in H respectively. (see [7, Chapter 4, Section 21]) Definition 3.7. Let u B B. We define a norm on B B by { n } n u 1 := inf x i B y i B : u = x i y i, with x i, y i B. The projective tensor product B 1 B is the completion of B B with respect to 1. Given elements u, v of H H with u = n x i y i, v = m j=1 x j y j where x i, x j, y i, y j H, we define an inner product n m u, v 2 := xi, x j yi, y j j=1 The Hilbert-Schmidt tensor product denoted by H 2 H is the completion of H H with respect to this inner product. We also recall the definition of the tensor quadratic variation of a Hilbert space valued martingale (see [7, Theorem 21.6 and 22.8]). Definition 3.8 (Tensor Quadratic Variation). Given a H valued square integrable martingale, its tensor quadratic variation denoted by {[[M]] t } is the process [[M]] t = lim prob. P, P={=t <t 1 < <t n =t} H H. n 1 (M ti+1 M ti ) 2, where the limit (in probability) is of H 2 H valued random variables and P is the maximum length of the sub-intervals in P. We denote by { M t } the dual predictable projection of the tensor quadratic variation. Proposition 3.9. The tensor quadratic variation of Ĩt,G is given by ]] [[Ĩ,G = t i= G s G s d [M] s, a.s.

10 8 SUPRIO BHAR and its dual predictable projection is Ĩ,G t = G s G s d M s, a.s. Proof. Given any partition P = { = t < t 1 < < t n = t} and ϕ, ψ S, we have n 1 ( Iti+1,G(ϕ) I ) ( ti,g(ϕ) I ti+1,g(ψ) I ) ti,g(ψ) i= n 1 = (Ĩt i+1,g Ĩt i,g) 2, ϕ ψ, a.s. = i= n 1 i= ]] [[Ĩ,G (Ĩt i+1,g Ĩt i,g) 2, ϕ ψ, a.s. Since both and [I,G(ϕ), I,G (ψ)] t exist as appropriate limits (in probability) over shrinking partitions, we have t ]] (3.4) [I,G (ϕ), I,G (ψ)] t = [[Ĩ,G, ϕ ψ, a.s. Now [I,G (ϕ), I,G (ψ)] t = G s, ϕ G s, ψ d [M] s = t G 2 s, ϕ ψ d [M] s, a.s. Note that E G s 2 S p d [M] s = E G s 2 S p d M s is finite and G s 2 S p = G s G s 2. Hence we can write (using the theory of Bochner-Stieltjes integration) [I,G (ϕ), I,G (ψ)] t = G 2 s d [M] s, ϕ ψ, a.s. From (3.4) we now can identify process. ]] [[Ĩ,G t as the indicated S p 2 S p valued Since G s G s 1 = G s 2 S p = G s G s 2, then from (3.1) we get T E G s G s 1 d M s <. Now G s G s is a S p 1 S p valued predictable process. Therefore G s G s d M s is a S p 1 S p valued predictable process (using the theory of Bochner-Stieltjes integration). Note that H 1 H H 2 H. Claim: {Ĩt,G Ĩt,G } t G s G s d M s is a S p 1 S p valued martingale. ]] } Assuming the claim, since {Ĩt,G Ĩt,G [[Ĩ,G is a S p 1 S p valued martingale (see [7, Theorem 22.8(2 )]), then taking the difference we get { [[Ĩ,G]] } G s G s d M s t t

11 AN ITŌ FORMULA IN S VIA ITŌ S REGULARIZATION 9 is a S p 1 S p valued martingale. Since { G s G s d M s } is a predictable process, this proves that (also see [7, Proposition 22.7, equation (22.7.5)]) Ĩ,G = t G s G s d M s, a.s. Now we are going to prove the claim. For ϕ S we have by the Itō isometry ( ) 2 E Ĩt,G, ϕ = E(I t,g (ϕ)) 2 = E G s, ϕ 2 d M s or ) E ( Ĩt,G Ĩt,G, ϕ ϕ = E G s G s, ϕ ϕ d M s Given ϕ, ψ S, we can apply the previous equation to ϕ+ψ which on simplification gives ) (3.5) E ( Ĩt,G Ĩt,G, ϕ ψ = E G s G s, ϕ ψ d M s Again from Theorem 3.3(a) we get E Ĩt,G 2 S p < which implies E Ĩt,G Ĩt,G 1 <. Rewriting equation (3.5) we get ) E (Ĩt,G Ĩt,G, ϕ ψ = E G s G s d M s, ϕ ψ which implies (3.6) E (Ĩt,G Ĩt,G ) == E S p S p G s G s d M s Now fix u, t [, T ] with u t. Let F F u. Since I t,g (ϕ) = G s, ϕ dm s is a martingale, we have E [ [ (I t,g (ϕ)) 2 ] t ] 1 F = E 1 F G s, ϕ 2 d M s. Following the same derivation as in (3.6) we get ) ] ] (3.7) E [(Ĩt,G Ĩt,G 1 F == E [1 F G s G s d M S s p S p This proves the claim Stochastic Integral with respect to a real finite variation process. Let {A t } t [,T ] be a real valued F t -adapted process of finite variation with right continuous paths such that its total variation on the time interval [, T ] is bounded by a constant, i.e. C > such that (3.8) V [,T ] (A ) C, a.s. Note that if T =, then by (3.8) we mean that for any t >, V [,t] (A ) C a.s. Let {G t } t [,T ] be a S p valued norm bounded (i.e. there exists a constant R > such that G s p R a.s. for all t) predictable process. In particular we have for any t [, T ] and ϕ S p, (3.9) G s, ϕ G s p ϕ p R. ϕ p

12 1 SUPRIO BHAR and (3.1) G s p da s R.V [,t] (A ) R.C <. This also implies G s, ϕ da s R.C. ϕ p. Now consider the linear map J t,g on S p defined by (3.11) J t,g (ϕ) := G s, ϕ da s. We can restrict this map to S and obtain a random linear functional on S. Theorem 3.1. For each t [, T ], the random linear functional J t,g has a regularized version J t,g in S p. Proof. Note that Then, E (J t,g (ϕ) R.C ϕ p. So J t,g (ϕ) ϕ p G s p da s RC ϕ p. ϕ p < ϵ R.C = E( J t,g(ϕ) 1) < ϵ. Hence J t,g is continuous in probability in p at ϕ =, which by linearity of J t,g is true globally. Hence by Theorem 2.3, we get a S q valued regularized version, say J t,g, such that Jt,G, ϕ = J t,g (ϕ), a.s., ϕ S. Note that if for some ω Ω, V [,t] A (ω) =, then (J t,g (ϕ))(ω) =, which does not contribute to E J t,g (ϕ). For further calculations we shall ignore these ω. Now, E J t,g 2 p = E = = 2 (2n + 1) 2p Jt,G, h n n= (2n + 1) 2p E (J t,g (h n )) 2 n= ( (2n + 1) 2p E n= ) 2 G s, h n da s ( (2n + 1) 2p da s E G s, h n V [,t] (A ) V [,t] (A ) n= using Jensen s inequality, ( (2n + 1) 2p E n= ) 2 ) G s, h n 2 (V [,t] (A )) 2 da s V [,t] (A )

13 AN ITŌ FORMULA IN S VIA ITŌ S REGULARIZATION 11 using (3.8), C.E ( = C. E <. ( ) (2n + 1) 2p G s, h n 2 da s n= ) G s 2 p da s C.R 2 ( E V [,t] (A ) ) Hence J t,g is S p valued a.s. Proposition Jt,G has the following properties: (a) J t,g is measurable with respect to F t and the Borel σ-algebra of S p. (b) J t,g is a S p valued finite variation process. Proof. Proof of part (a) is similar to Theorem (3.3) part (b). To prove part (b), it is enough to show the variation exists for time intervals of the form [, t] where t [, T ]. Fix such a t. Let P := { = t < t 1 < t n = t} be a partition of [, t]. Fix a ω Ω. Take any arbitrary ϵ >. Then there exist ϕ, ϕ 1,, ϕ n 1 S (all of which depends on ω) such that ϕ i p 1 for i =, 1,, n 1 and J ti+1,g J ti,g p (ω) Then, n 1 J ti+1,g J n 1 ti,g p (ω) i= Using (3.1), we have Jti+1,G J ti,g, ϕ i (ω) + i= Jti+1,G J n 1 ti,g, ϕ i (ω) + ϵ, i =, 1,, n 1. 2i+1 i= n 1 < ( J ti+1,g(ϕ i ) J ti,g(ϕ i ) ) (ω) + ϵ i= n 1 i= i+1 t i G s, ϕ i da s (ω) + ϵ ϵ 2 i+1 n 1 (i+1 ) ϕ i p G s p da s (ω) + ϵ i= t i n 1 (i+1 ) G s p da s (ω) + ϵ, ( ϕ i p 1) i= t i ( ) = G s p da s (ω) + ϵ n 1 J ti+1,g J ti,g p < RC + ϵ, a.s. i= But P is an arbitrary partition of [, t] and the constant RC does not depend on such a partition. Hence taking supremum over all such partitions we conclude that the variation is finite a.s.

14 12 SUPRIO BHAR Proposition Jt,G has rcll (right continuous with left limits) paths in q for any q > p Proof. Note that sup s [,t] J s,g (ϕ) ϕ p G s p da s RC ϕ p. The arguments are same as in [9, Theorem 1]. Remark Observe that for predictable rectangles G t = 1 (s,u] F (t, ω)ϕ, where s, u [, T ] with < s < u and ϕ S p, we have J t,g = 1 F (A t u A t s )ϕ, which agrees with the Bochner integral defined on S p (in the Stieltjes sense). Therefore the construction above actually gives the stochastic integral G s da s for normbounded and predictable G. The construction given above can be extended to any finite variation process. Proposition Let {A t } t [,T ] be a F t -adapted process of finite variation with right continuous paths. Let {G t } t [,T ] be a S p valued norm bounded [i.e. there exists a constant R > such that G s p R a.s. for all t] predictable process. Then there exist a sequence of F t stopping times τ n with τ n and a sequence of (n) S p valued regularized versions J t,g of J t,g (as defined in (3.11)) such that J (n) (a) t,g is measurable with respect to F t and the Borel σ-algebra of S p. (n) (b) J t,g is a S p valued finite variation process. (n) (c) For t [, τ n ) and for all ψ S, J t,g, ψ = J t,g (ψ) a.s.. Proof. Note that V [,t] (A ) is a non-decreasing process adapted to the filtration. Consider the stopping times τ n defined by τ n := inf{t : V [,t] (A ) n}. Note that there might be jumps of A (and hence of V [,t] (A )) at the time points t = τ n (ω). But these are prelocally bounded. In particular, the processes A τ n t := A t 1 (t<τn) + A τn 1 (τn t) are F t adapted and bounded. Furthermore we actually have V [,t] (A τ n ) n. Now consider random linear functionals J (n) t,g (ϕ) := G s, ϕ da τ n s. Note that J (n) t,g (ϕ) = J t,g(ϕ) a.s. for t < τ n. We can obtain regularized versions of J (n) t,g (ϕ) (as in Theorem 3.1 and Proposition 3.11), which gives the appropriate versions mentioned in the statement. Remark If the paths of the finite variation process are continuous, then so are the paths of the total variation. In this situation we may take t [, τ n ] in part (c) of Proposition 3.14 (since there are no jumps at t = τ n ). Remark We can relax the hypothesis using the following: (1) Given a process of locally finite variation, by stopping we get a process of finite variation and hence the procedure given above works for processes of locally finite variation. The predictable process {G t } can also be taken to be locally norm bounded.

15 AN ITŌ FORMULA IN S VIA ITŌ S REGULARIZATION 13 (2) We can take {G t } to be S p (R d ) valued. In which case the random linear functional J t,g (ϕ) needs to be defined for ϕ S(R d ). All the results mentioned above can be proved in this general setup. (3) If A t = (A 1 t,, A n t ) is a F t -adapted R n -valued process of finite variation with right continuous paths and {G t } a S p (R d ) valued predictable process then under appropriate integrability conditions we may define the random linear functionals on S(R d ) J t,g,i (ϕ) := G s, ϕ da i s, ϕ S(R d ); i = 1,, n. We can obtain regularized versions of the integrals Stochastic Integral with respect to a real semimartingale. A important step in defining stochastic integrals of real valued bounded predictable processes with respect to a real semimartingale is the following: Lemma Let {V t } t be a real valued bounded predictable processes. Let {M t } t be a square integrable martingale and {A t } t is a process of finite variation such that M t = A t, a.s. t. Then V s dm s = V s da s, a.s. t. Proof. This result is included in the proof of Theorem 23.4 in [5]. A classical result on local martingales goes like this: (see [5, Lemma 23.5] or [1, Theorem 25]) Theorem 3.18 (Decomposition of Local Martingales). Given any local martingale {M t } t, two local martingales {M t} t, {M t } t one of which has bounded jumps and the other is of locally integrable variation and M t = M t + M t, a.s. Thus any semimartingale has a decomposition (not necessarily unique) X t = M }{{} t + A }{{} t, a.s. locally square integrable locally finite variation Suitably stopping, we can assume them to be a square integrable martingale and a process of finite variation respectively. Again using further stopping, the process of finite variation can be made to satisfy (3.8) prelocally (see Proposition 3.14). Theorem Let {X t } be a real semimartingale such that it has a decomposition X t = M t + A t, a.s. t, where {M t } t is a square integrable martingale and {A t } t is a process of finite variation satisfying (3.8). Let {G t } t be a norm bounded predictable S p valued process (i.e. G t p is bounded). Then G s, ϕ dx s has a regularized version in S p. Furthermore, this version is independent of the decomposition X = M + A.

16 14 SUPRIO BHAR Proof. Under the hypothesis the integrability conditions (3.1) and (3.1) are satisfied on any finite time interval [, T ]. Then, for any t [, T ] and ϕ S, we have Hence (Ĩt,G + J ) t,g G s, ϕ dx s = G s, ϕ dm s + G s, ϕ da s }{{}}{{} I t,g (ϕ) J t,g (ϕ) = Ĩt,G, ϕ + Jt,G, ϕ, a.s. gives a regularized version of the integral G s, ϕ dx s. Apriori, it seems that this version will depend on the decomposition of X. Now consider the real bounded predictable process V s = G s, ϕ. Hence Lemma 3.17 gives, G s, ϕ dm s = G s, ϕ da s, a.s. if M = A with M a square integrable martingale and A a process of finite variation. This says the stochastic integral G s, ϕ dx s does not depend on the decomposition of X. We apply the same result here to conclude that the regularized version is indeed independent of the decomposition. Remark 3.2. As mentioned before in the remarks in previous subsections, we may also consider the case where {G t } is a S p (R d ) valued locally norm-bounded predictable process and X t = (X 1 t,, X n t ) is a R n valued semimartingale, to obtain regularized versions of stochastic integrals under appropriate integrability conditions. 4. Itō Formula Consider the derivative maps denoted by i : S(R d ) S(R d ) for i = 1,, d. We can extend these maps by duality to i : S (R d ) S (R d ) as follows: for ψ S (R d ), i ψ, ϕ := ψ, i ϕ, ϕ S(R d ). Let {e i : i = 1,, d} be the standard basis vectors in R d. Let n = (n 1,, n d ) Z d + be a multi-index. Then (see [3, Appendix A.5]) ni e i ni + e i i h n = h n ei h n+ei, 2 2 with the convention that for a multi-index n = (n 1,, n d ), if n i < for some i, then h n. Above recurrence implies that i : S p (R d ) S p 1 (Rd ) is a 2 bounded linear operator. First we recall some useful facts about translation operators on S (R d ). For x R d, let τ x denote the operators on S(R d ) defined by (τ x ϕ)(y) := ϕ(y x), y R d. This operators can be extended to τ x : S (R d ) S (R d ) by τ x ϕ, ψ := ϕ, τ x ψ, ψ S(R d ). Proposition 4.1. The translation operators defined have the following properties:

17 AN ITŌ FORMULA IN S VIA ITŌ S REGULARIZATION 15 (a) For x R d and any p R, τ x : S p (R d ) S p (R d ) is a bounded linear map. In particular, there exists a real polynomial P k of degree k = 2( p + 1) such that τ x ϕ p P k ( x ) ϕ p, ϕ S p (R d ). (b) For x R d and any i = 1,, d we have τ x i = i τ x. Proof. See [13, Theorem 2.1] for the proof of part (a). Fix an element ψ S(R d ). Then for y R d, (τ x i ψ)(y) = ( i ψ)(y + x) = i (ψ(y + x)) = ( i τ x ψ)(y) i.e. τ x i ψ = i τ x ψ. Now for ϕ S (R d ) and any ψ S(R d ), Hence we have part (b). i τ x ϕ, ψ = τ x ϕ, i ψ = ϕ, τ x i ψ = ϕ, i τ x ψ = i ϕ, τ x ψ = τ x i ϕ, ψ. Given a ϕ S (R d ), there exists a p > such that ϕ S p (R d ). Let {X t } be a (R d ) valued semimartingale. By part (a) of the Proposition 4.1, we have τ Xt ϕ is S p (R d ) valued for any t. By stopping, if required, we may assume that {X t } is prelocally bounded. But i : S p (R d ) S p 1 (Rd ) is a bounded linear operator. 2 Hence, by part (a) of Proposition 4.1 we get i τ Xs ϕ p 1 2 C. τ X s ϕ p C.P k ( X s ) ϕ p C, s [, T ] where C, C > are appropriate constants. Similarly, there exists a constant C > such that ij τ Xs ϕ p 1 C, s [, T ]. By stopping, if required, we may assume that the total variations of [X i, X j ] c t are also bounded (see Proposition 3.14). Hence as per the results obtained in the previous section, the random linear functionals d It 1 (ψ) := i τ Xs ϕ, ψ dxs i and I 2 t (ψ) := d i,j=1 2 ij τ Xs ϕ, ψ d[x i, X j ] c s both have regularized versions which live in S p 1 (Rd ) and S 2 p 1 (R d ) respectively and are given by d iτ Xs ϕ dxs i and d i,j=1 2 ij τ X s ϕ d[x i, X j ] c s respectively. We now prove an Itō formula which generalises [11, Theorem 2.3]. This result has been used in [12] to show existence of solution of some stochastic differential

18 16 SUPRIO BHAR equations in S, the space of tempered distributions. A version of this Itō formula was also proved in [19, Theorem III.1] with equality in S. In [6, Theorem 3], the author has proved this formula for twice continuously (Fréchet) differentiable function while dealing with a single Hilbert space. Note that derivatives of tempered distributions may not not be in the same Hermite Sobolev space as the original one. Using regularization, the result [6, Theorem 3] was also proved in [8, Theorem 8] in the case of a E valued continuous martingale. Theorem 4.2. Let p > and ϕ S p (R d ). Let X = (X 1,, X d ) be a R d valued semimartingale. Let Xs i denote the jump of Xs. i Then [ ] d τ Xs ϕ τ Xs ϕ + ( Xs i i τ Xs ϕ) s t is a S p 1 (R d ) valued process and we have the following equality in S p 1 (R d ), (4.1) d τ Xt ϕ = τ X ϕ i τ Xs ϕ dxs i + 1 d 2 ijτ 2 Xs ϕ d[x i, X j ] c s i,j=1 + [ ] d τ Xs ϕ τ Xs ϕ + ( Xs i i τ Xs ϕ), a.s. s t Proof. By [11, Proposition 1.4], there exists some positive integer n such that the map x τ x ϕ S n is a C 2 map. For any fixed ψ S(R d ) we have x τ x ϕ, ψ is a C 2 map and i τ x ϕ, ψ = i ϕ, ψ( + x) = ϕ, i ψ( + x) = ϕ, τ x i ψ = i τ x ϕ, ψ etc. We follow the proof of [5, Theorem 23.7] and define B t := {x R d : x sup s t X s }. Then there exist constants C, C > such that d τ Xs ϕ τ Xs ϕ + ( Xs i i τ Xs ϕ), ψ s t = τ X s ϕ, ψ τ Xs ϕ, ψ d + i τ Xs ϕ, ψ Xs i s t = τ X s ϕ, ψ τ Xs ϕ, ψ d i τxs ϕ, ψ Xs i s t C. d X s 2 sup ijτ 2 y ϕ, ψ s t i,j=1 y B t C. d X s 2 sup ijτ 2 y ϕ p 1 ψ p+1 s t i,j=1 y B t C. ( ) X s 2 sup τ y ϕ p ψ p+1. y B s t t

19 AN ITŌ FORMULA IN S VIA ITŌ S REGULARIZATION 17 Since for each s [, t], the random variable τ Xs ϕ τ Xs ϕ + d ( Xi s i τ Xs ϕ) is S p 1 valued, in view of the above estimate, [ ] d τ Xs ϕ τ Xs ϕ + ( Xs i i τ Xs ϕ) s t is a S p 1 valued random variable. To complete the proof we need to verify the following equality in S p 1, [ ] d d τ Xs ϕ τ Xs ϕ + ( Xs i i τ Xs ϕ) = τ Xt ϕ τ X ϕ + i τ Xs ϕ dxs i s t 1 2 d i,j=1 2 ijτ Xs ϕ d[x i, X j ] c s It is enough to show that the action of both the sides on ψ S agree. Now applying the Itō formula (see [5, Theorem 23.7]) to this C 2 map x τ x ϕ, ψ we have d τ Xt ϕ, ψ = τ X ϕ, ψ i τ Xs ϕ, ψ dxs i (4.2) } {{ } =It 1(ψ) + 1 d 2 2 ij τ Xs ϕ, ψ d[x i, X j ] c s i,j=1 }{{} + s t [ =I 2 t (ψ) τ Xs ϕ, ψ τ Xs ϕ, ψ + where X i s denotes the jump of X i s. This completes the proof. a.s. d i τ Xs ϕ, ψ ] Xs i, a.s., Given a real valued semimartingale {X t }, consider the local time process denoted by {L t (x)} t [, ),x R. Note that this process is jointly measurable in (x, t, ω) and for each x R, {L t (x)} t is a continuous adapted process. By [1, p. 216, Corollary 1], we have for any ϕ S, (4.3) L t (x)ϕ(x) dx = ϕ(x s )d [X, X] c s. As an application of Proposition 3.14 we have the following result. Proposition 4.3. The random tempered distribution δ X s d [X, X] c s is S p valued for any p > 1 4 and is given by the function x L t(x). Proof. Note that for any fixed x R, the distribution δ x is in S p for any p > 1 4 (see [14, Theorem 4.1 part (a)]). Also τ x δ = δ x. The map x τ x δ is continuous (see [14, proof of Proposition 3.1]). By stopping if required, we may assume {X t } is bounded. Hence {δ Xt } is a norm bounded predictable process. Hence the random linear functional ϕ ϕ(x s )d [X, X] c s = δxs, ϕ d [X, X] c s has a regularized version (see Proposition 3.14) given by

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