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We show that our derivative is an anti Itˆointegral that has the “desired”∈ properties leading to a fundamental theorem of stochastic calculus (Theorem 2.1), a generalized stochastic chain rule (Theorem 3.1) that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolle’s theorems (Lemma 2.1). In addition, it interacts with ba- sic algebraic operations on semimartingales similarly to the way Newton’s deterministic derivative does on deterministic functions (Theorem 3.2), making it natural for computations. Heuristically, our derivative of a semimartingale S with respect to a BM B is the pathwise “velocity” of S relative to B at each point in time (we call it the B-Brownian velocity of S, see the examples at the end of Section 3). It gives a Brownian path view of the changes in a semimartingale’s path S(ω) in time by measuring the rate of change of the covariation of S with the Brownian motion B with respect to the quadratic variation of B (time). The sign of our derivative of S with respect to B tells us, path by path, whether S and B are increasing and decreasing together (positive sign) or whether the value of S changes in the opposite direction of changes in the value of B (negative sign). The magnitude of the B-Brownian velocity of S gives us the B-Brownian speed of S. We believe that this pathwise view, in addition to giving rise to a differentiation theory for Itˆo’s calculus, is a useful tool in the analysis of SDEs and SPDEs; leading to a new smoothness and regularity theory for solutions of these stochastic equations, which in turns leads to new insights into the behavior of solutions to SDEs and SPDEs relative to their driving noise. It also leads to a rich differential stochastic vector calculus as well as to a new stochastic optimization theory (optimization with respect to the noise). We study these different aspects in more details in [1] and in subsequent articles.
Notation 1.1. Throughout this article, let S = St, Ft; t R be a continuous semimartingale, { ∈ +} let M = Mt, Ft; t R+ and V = Vt, Ft; t R+ be the continuous local martingale and the continuous{ process∈ of bounded} variation{ in the∈ decompos} ition of S, respectively, and let B = Bt, Ft; t R be a standard BM on the same usual probability space (Ω, F, Ft , P) (a usual { ∈ +} { } probability space is one where the filtration Ft satisfies the usual conditions: right continuity △ { } and completeness). Also, let D S,B t = d S,B t/dt, which we call the strong derivative of S with h i h i ◦ respect to B. Finally, we denote by λ the Lebesgue measure on R and by R the set R 0 . + + +\{ } Definition 1.1 (Derivative of semimartingale with respect to BM). The stochastic difference and stochastic derivative of S with respect to B at t, respectively, are defined by 3 h 3 r [ S,B t+r S,B t−r] dr; 0
D d S,B t (1.2) t S = h i B t dt A DIFFERENTIATION THEORY FOR ITO’Sˆ CALCULUS 3
(see Theorem 2.1 below), and we call d S,B t/dt the strong derivative of S with respect to B at t. We denote the class of continuous semimartingalesh i whose strong B-derivative exists on S by Ss S Ss S Ss,(k) S B( ). The class of k-times strongly B-differentiable elements of B( ) is denoted by B ( ). If the strong B-derivative of S exists at t and if f C1(R; R) with f ′(x) =0 for all x R and the map x f ′(x) is absolutely continuous, we define∈ the strong derivative6 of S with respect∈ to 7→ △ the semimartingale S(2) = f(B) at t by
D △ d S,B t (1.3) S(2) St = h (2) i . t d S t h i ′ We also have the same definition of D (2) St in the case f is convex and f (x) =0 for all x R, St − ′ 6 ∈ where f−(x) is the left derivative of f at x. The generalized derivative of S with respect to f(B) at t is obtained straightforwardly from (1.1). Remark 1.1. (a) It follows immediately from our Definition 1.1 of the stochastic derivative process that DBV 0 for all processes V of bounded variation on compacts: there changes are “too slow” for the≡ BM to pickup, and they behave like constants in elementary calculus in that their derivative is 0. Additionally, if M and B are independent or orthogonal ( M,B 0); then h i ≡ from the definitions above DBM 0: M is independent of or orthogonal to B and therefore it is “unaffected” by changes in B. ≡
(b) We don’t use the following observation here in this note; but, we formally think of DBt,hSt as the quadratic variation of a generalized stochastic integral which we call the pre-stochastic difference of S with respect to the Brownian motion B: h 3 1 2 √r [ S,B t+r S,B t−r] dBr; 0 1 + − P D D M,B t + D M,B t ; 0 (b) (SMVT and SRT) Let M,B be continuous on the closed interval [a,b] R+ and differ- entiable on the open intervalh i(a,b), a.s. P; then, ⊂ D M,B b(ω) M,B a(ω) (2.3) ( Bt St) t c(ω)= h i − h i , | = b a − for some random variable c(ω) (a,b) a.s. P. In particular, if M,B b(ω)= M,B a(ω) P D ∈ P h i h i a.s. ; then Bt St t c(ω)=0 a.s. . | = A simple application of the SMVT leads to Ss R Corollary 2.1. Suppose that, for some BM B, S B( +) with decomposition St = S0 +Vt +Mt, t 0, a.s. P. ∈ ≥ D P P (i) If Bt St =0 for all t> 0 a.s. . Then, M,B 0 a.s. . D h P i≡ P (ii) If Bt St does not change sign on (a,b) a.s. , then M,B is monotonic over (a,b) a.s. : increasing, nondecreasing, decreasing, or nonincreasingh asi D D D D P Bt St > 0, Bt St 0, Bt St < 0, or Bt St 0; t (a,b) a.s. , ≥ ≤ ∀ ∈ respectively. (iii) If D P Bt St K; a < t < b, a.s. | |≤ then M,B is Lipschitz on (a,b): h i M,B t M,B s K t s ; t,s (a,b) a.s. P. |h i − h i |≤ | − | ∈ We state the Fundamental theorem of stochastic calculus for a class of semimartingales that covers ItˆoSDEs of interest and that can be generalized to cover SPDEs of interest as well: t (2.4) S = St = S + Vt + XsdBs, Ft; 0 t< 0 ≤ ∞ Z0 where V and B are as in Notation 1.1; and the adapted process X = Xt, Ft; t R is in { ∈ +} L2([0,t]; λ) t > 0 a.s. P. We denote by SSIB the class of continuous semimartingales S whose local martingale∀ part M is given by the stochastic integral in (2.4); i.e., t (2.5) Mt = XsdBs, Ft; 0 t< . ≤ ∞ Z0 The elements of SSIB in which the integrand X has a.s. continuous paths form the subclass which we SSIB SSIB SSIB SSIB SSIB SSIB SSIB denote by c . Finally, we denote by 0 and c,0 the subclasses 0 and c,0 c in which V 0 for all of its elements. ⊂ ⊂ We are now≡ ready to present the pathwise fundamental theorem of stochastic calculus (FTSC) Theorem 2.1 (FTSC). Let S SSIB . Then, S Ss (R )—in particular the process M,B is ∈ c ∈ B + h i differentiable for all t R+, a.s. P—and ∈ D D F R D (i) the stochastic derivative process BS = Bt St, t; t is given by Bt St = Xt for all t { ∈ +} ∈ R+, a.s. P. In particular, t D P (2.6) Bt XsdBs = Xt; 0 t< , a.s. . ∀ ≤ ∞ Z0 SIB Moreover, if S, S˜ S with P Mt = M˜ t; t R = 1; then their stochastic derivative ∈ c ∀ ∈ + nP D D ˜ o R processes are indistinguishable: Bt St = Bt St; t =1. ∀ ∈ + n o A DIFFERENTIATION THEORY FOR ITO’Sˆ CALCULUS 5 (ii) t D ˜ ˜ ˜ P (2.7) Bs SsdBs = St S Vt; 0 t< , a.s. − 0 − ∀ ≤ ∞ Z0 ˜ SSIB ˜ for any S c whose local martingale part M is indistinguishable from M. In par- t D∈ R P SSIB ticular; Bs SsdBs = St S Vt for all t , a.s. . Thus, if S ; then, 0 − 0 − ∈ + ∈ c,0 t D R P Bs SsdBs = St S for all t , a.s. . 0 R − 0 ∈ + RemarkR 2.1. In contrast to the fundamental theorem of deterministic calculus, part (ii) of The- orem 2.1 involves the additional term of bounded variation on compacts V˜ , unless S SSIB . ∈ c,0 Remember however that, a.s. P, DBV˜ 0 by Remark 1.1. ≡ In the case the integrand X is not necessarily continuous we state the following Theorem 2.2. Assume that S SSIB ; then, a.s. P, the process M,B is differentiable for all ◦∈ h i t R Z and DBS = Xt; t R Z , for some Z(ω) R with λ(Z)=0. If, additionally, the ∈ +\ ∈ +\ ⊂ + condition (2.1) hold; then we also have t t 1 + − D Xs ds + D Xs ds ; t (0, ) Z, D 2 0 0 ∈ ∞ ∩ (2.8) Bt St = Zt Z + D Xs ds t ; t =0, | =0 Z0 P ˜ SSIB ˜ P D D ˜ a.s. . If S, S with indistinguishable M and M; then, a.s. , Bt St = Bt St for all t ◦ ∈ ∈ R+ O, for some O(ω) R+ with λ(O)=0(we say that DBS and DBS˜ are almost indistinguishable). \ ⊂ ◦ In particular, if M is a B-Brownian martingale; then, a.s. P, DBM = Yt; t R Z for { ∈ +\ } some Z(ω) R+ with λ(Z) =0, where Y is the progressively measurable process such that t 2 ⊂ t EP Y ds < and YsdBs = Mt, for all t R . 0 s ∞ 0 ∈ + R R 3. Pathwise Derivative Rules We now show that our pathwise derivative for Itˆo’s calculus generalizes familiar differentiation rules from deterministic to stochastic calculus, making it useful for computations involving func- tions of semimartingales and algebraic operations on several semimartingales. We start with our chain rule for stochastic calculus. Theorem 3.1 (The chain rule of stochastic calculus). (a) Suppose that f C1(R; R) such that the function x f ′(x) is absolutely continuous. ∈ P 7→ D D R (i) Then, a.s. , the process Bf(S)= Bt f(St); t is given by { ∈ +} t t 1 + ′ − ′ D f (Ss) d M,B s + D f (Ss) d M,B s ; 0 ′ (ii) Suppose further that f (x) = 0 for every x R, d M,B s = XM,B(s)ds and XM,B 6 ∈ △ h i has continuous paths on R a.s. P, and let S(2) = f(B). Then, S,S(2) Ss (R ) and + ∈ B + D D D (2) P (3.4) Bt St = (2) St Bt S (t) 0 t< a.s. . St · ≤ ∞ (b) If f : R R is only assumed to be convex; then (3.1), (3.2), and (3.3) all hold, replacing ′ → ′ f (x) by the left derivative at x, f−(x). Moreover, if in addition the assumptions in part ′ ′ (a)(ii) hold (again replacing f (x) by f−(x)), then (3.4) holds. An interesting question then is when is DBS itself a martingale (or a local martingale)? It is D 2 F for example clear from the above discussion that Bt (Bt t)=2Bt, t; t 0 is a martingale. The next corollary, which follows as an immediate{ consequence− of Theorem≥ 3.1,} gives a sufficient condition. Corollary 3.1. Suppose f C1(R; R) such that the function x f ′(x) is absolutely continuous ∈ 7→ and St = f(Bt)+ Vt, t 0; where B and V are as in Notation 1.1. Then ≥ D ′ Bt St = f (Bt); 0 t< . ≤ ∞ D D F In particular, Bt S = Bt St, t;0 t < is a martingale (local martingale) iff the process ′ { ≤ ∞} f (Bt), Ft;0 t < is a martingale (local martingale). If f : R R is only assumed to be { D≤ ∞}D F → ′ F convex, then Bt S = Bt St, t;0 t< is a martingale (local martingale) iff f−(Bt), t;0 t< is a martingale{ (local martingale≤ )∞}. { ≤ ∞} Another immediate consequence of Theorem 3.1 is the power rule for our pathwise derivative Corollary 3.2 (The power rule of stochastic calculus). Let S SB (R ) with continuous local ∈ + martingale part M. If d M,B s = XM,B(s)ds and XM,B has continuous paths on R+ a.s. P, then S Ss (R ) and h i ∈ B + D p p−1 D (3.5) Bt (St) = p (St) Bt St; 0 t< , p 1. ≤ ∞ ∀ ≥ If in addition St =0 for every t R , a.s. P, then 6 ∈ + D p p−1 D R (3.6) Bt (St) = p (St) Bt St; 0 t< , p . ≤ ∞ ∀ ∈ Theorem 3.2 (The sum, product, and ratio rules ). Let S1,S2 SB (R+) and let a,b R be (1) (2) ∈ ∈ arbitrary but fixed, then aS bS SB(R ) and ± ∈ + D (1) (2) D (1) D (2) P (3.7) Bt aS bS = a Bt S b Bt S ; 0 t< , a.s. . t ± t t ± t ≤ ∞ If in addition the continuous local martingale parts M (1) and M (2) of S(1) and S(2), respectively, satisfy (i) (3.8) d M ,B s = X (i) (s)ds; i =1, 2, h i M ,B (1) (2) (1) (2) s and X (i) has continuous paths on R a.s. P for i =1, 2 then S ,S ,S S S (R ) and M ,B + ∈ B + D (1) (2) (2)D (1) (1)D (2) P (3.9) Bt S S = S Bt S + S Bt S ; 0 t< , a.s. . t t t t t t ≤ ∞ If in addition S(2) =0 for every t R , a.s. P, then S(1)/S(2) Ss (R ) and t 6 ∈ + ∈ B + (1) (2)D (1) (1)D (2) D St St Bt St St Bt St P (3.10) Bt (2) = − 2 ; 0 t< , a.s. . S ! (2) ≤ ∞ t St h i A DIFFERENTIATION THEORY FOR ITO’Sˆ CALCULUS 7 We now briefly look at the Brownian derivatives (Brownian velocity) of several simple semi- martingales obtained as simple applications of Theorem 2.1, Theorem 3.1, and Theorem 3.2. These examples show that our derivative gives intuitive answers: (1) D 1, Bt > 0 Bt Bt = | | 1 Bt 0 (− ≤ I.e., the Brownian speed of Bt is 1 for all t R , and the direction of change of B | | ∈ + | | (increase or decrease) relative to B at t is the same as Bt if Bt > 0 and is opposite to that of Bt if Bt < 0. 2 D R (2) Let St = B Vt, for any process V as in Notation 1.1, then Bt St =2Bt for all t . t − ∈ + So that the Brownian speed of St at any t R is 2 Bt , and the direction of change of St ∈ + | | (increase or decrease) relative to B at t is the same as Bt if Bt > 0 and is opposite to that of Bt if Bt < 0. (3) Let t t X,B 1 2 Ξ = exp XsdBs X ds , t − 2 s Z0 Z0 2 and let the adapted process X = Xt, Ft; t R+ be continuous (thus in L ([0,t]; λ) t> 0 P D X,B X,B{ ∈ } ∀ a.s. ). Then, Bt Ξt = XtΞt . So that the Brownian speed of the exponential local X,B X,B X,B martingale Ξ at any t R is Xt Ξ , and the direction of change of Ξ (increase t ∈ + | | t t or decrease) relative to B at t is the same as Bt if Xt > 0 and is opposite to that of Bt if Xt < 0. More applications of our theory presented here (including SDEs and SPDEs) are dealt with in [1] and subsequent articles. 4. Proofs of results Proof of Lemma 2.1. D (a) Under the assumptions given, the conclusion follows from the definition of Bt St, the facts that S,B t = M,B t and M,B = 0, and the fact that the generalized derivative h i h i h i0 h 3 1 + − lim 3 r [g(t + r) g(t r)] dr = D g(t)+ D g(t) ; 0 (b) Let the assumptions hold on Ω∗ Ω, with P(Ω∗) = 1, and fix ω Ω∗. Then, by the classical mean value theorem a c⊂(ω) (a,b) ∈ ∃ ∈ ∋ M,B b(ω) M,B a(ω) D M,B t t c(ω)= h i − h i . h i | = b a − ∗ D Since M,B is assumed differentiable on (a,b) ω Ω ; then, by part (a), Bt St t=c(ω)= h i ∗ ∀ ∈ | D M,B t t c(ω) for all ω Ω and (2.3) is established. h i | = ∈ 8 HASSAN ALLOUBA The proof is complete. ProofofTheorem2.1. (i) Under the assumptions on S, the stochastic difference and stochastic derivative of S with respect to B are a.s. P given, respectively, by 3 h t+r t−r DBt,hSt = r Xsds Xsds dr; 0 Proof of Theorem 2.2. The proof proceeds exactly as in the proof of part (i) of Theorem 2.1, taking into account the noncontinuity of X and using the fundamental theorem of Lebesgue calculus (e.g., see Theorem 10 on p. 107 in [6]) and (2.2) in Lemma 2.1. Proof of Theorem 3.1. (a) If the function x f ′(x) is absolutely continuous, then f ′′ exists Lebesgue-almost every- 7→ where and we have that the Itˆoformula for f(St) is given by t t ′ 1 ′′ (4.3) f(St)= f(S )+ f (Ss)[dMs + dVs]+ f (Ss)d M s; t R , a.s. P. 0 2 h i ∈ + Z0 Z0 D (i) Using Itˆo’s rule (4.3), Remark 1.1 (a) ( Bt U 0 for any continuous process of bounded variations on compacts) along with the linearity≡ of the cross variation process, and Theorem 2.1 we have a.s. P t t D D ′ 1 ′′ Bt f(St)= Bt f(S )+ f (Ss)[dMs + dVs]+ f (Ss)d M s 0 2 h i 0 0 t Z t Z 1 + ′ − ′ (4.4) D f (Ss) d M,B s + D f (Ss) d M,B s ; 0 D (2) D ′ (ii) From part (i) we have Bt S (t)= Bt f(Bt)= f (Bt). Now, using Itˆo’s formula (2) . . d S t d 1 h i = f(B )+ f ′(B )dB + f ′′(B )ds dt dt 0 s s 2 s (4.5) Z0 Z0 t ′ 2 = [f (Bt)] Also, we have (2) t d S,S t d ′ h i = f (Bs)d M,B s dt dt h i (4.6) Z0 d t = f ′(B )X (s)ds = f ′(B )X (t) dt s M,B t M,B Z0 So that (2) ′ D d S,S t f (Bt)XM,B(t) (4.7) S(2) St = h (2) i = 2 t d S t [f ′(B )] h i t from which it follows that ′ D D (2) f (Bt)XM,B(t) ′ d M,B t D (2) St Bt S (t)= f (Bt)= XM,B(t)= h i = Bt St St ′ 2 · [f (Bt)] · dt as claimed. (b) If f : R R is only assumed to be convex, then from standard results in Itˆo’s calculus we get that→ t ′ 1 f (4.8) f(St)= f(S )+ f (Ss)[dMs + dVs]+ Γ ; t R , a.s. P, 0 − 2 t ∈ + Z0 f ′ where Γ is a continuous increasing process and f−(x) is the left derivative of f at x. The desired results in this convex case then follow by following the same steps in the arguments ′ ′ above, replacing the Itˆoformula (4.3) by (4.8) and replacing f by f− throughout. The proof is complete. Proof of Theorem 3.2. We only need to prove the multiplication and ratio rules (3.9) and (3.10), respectively, as the addition/subtraction rule is clear by the linearity of the cross variation process. We start by proving the multiplication rule (3.9). To this end, let f(x, y) = xy for x, y R. Applying Itˆo’s formula for functions of several continuous semimartingales (since all the ∈ 10 HASSAN ALLOUBA (1) (2) (1) (2) Dif(Ss ,Ss ) and Dij f(Ss ,Ss ) are continuous) we get: D (1) (2) D (1) (2) D (1) (2) Bt St St = Bt f(St ,St )= Bt f(S0 ,S0 ) 2 t t D (1) (2) (i) 1 (1) (2) (i) (j) + Bt Dif(S ,S )dS + Dij f(S ,S )d S ,S s s s s 2 s s h i i=1 0 ≤i,j≤ 0 X Z 0 X 1 Z t t D (2) (1) (1) (2) = Bt Ss dSs + Ss dSs 0 0 (4.9) Z t Z t D (2) (1) (1) (2) = Bt Ss dMs + Ss dMs 0 0 Zt Z t d (2) (1) = S X (1) (s)ds + S X (2) (s)ds dt s M ,B s M ,B Z0 Z0 (2) (1) = St XM (1),B(t)+ St XM (2) ,B(t) (2)D (1) (1)D (2) P = S Bt S + S Bt S ; 0 t< , a.s. , t t t t ≤ ∞ (i) where we have used Remark 1.1 (a); the assumption on d M ,B s for i =1, 2; the fundamental D (hi) i theorem of deterministic calculus; and the fact that Bt St = XM (i) ,B(t) for i =1, 2. Now, for the ratio rule (3.10) we can either use the product rule in conjunction with the power rule to show that under the additional assumption (S(2) = 0 for every t R , a.s. P), we have t 6 ∈ + −1 −1 −2 D (1) (2) (2) D (1) (1) (2) D (2) Bt S S = S Bt S S S Bt S t t t t − t t t h i h i h i (2)D (1) (1)D (2) St Bt St St Bt St P = − 2 ; 0 t< , a.s. . (2) ≤ ∞ St Alternatively, we can let g(x, y)= x/y for everyh x,i y R such that y = 0 and proceed exactly as in (4.9), replacing f(x, y) by g(x, y) to get (3.10). ∈ 6 Acknowledgements. This research is supported in part by NSA grant MDA904-03-1-0089. References [1] Allouba, H., A differentiation theory for Itˆo’s calculus: generalizations and ramifications. In preparation (2004). [2] Bhattacharya, R. and Waymire, E., Stochastic processes with applications. John Wiley & Sons, Inc., 1990. [3] Durrett, R., Stochastic Calculus: A Practical Introduction. New York: CRC Press LLC., 1996. [4] Houdr´e, C. Linear Fourier and stochastic analysis. Probab. Theory Related Fields 87 (1990), no. 2, 167–188. [5] Itˆo, K., Stochastic Integral. Proc. Imperial. Acad. Tokyo, 20 (1944), 519–524. [6] Royden, H. Real Analysis, Third Edition. Macmillan Publishing Company, New York, 1988. Department of Mathematical Sciences, Kent State University, Kent, OH 44242 Phone: (330) 672-9028 email: [email protected]