Preliminaries From Calculus

Chpter 1 Preliminries From Clculus Stochstic clculus dels with functions of time t, t T. In this chpter some concepts of the infinitesiml clculus used in the sequel re given. 1.1 Functions in Clculus Continuous nd Differentible Functions A function g is clled continuous t the point t = t if the increment of g over smll intervls is smll, g(t) =g(t) g(t ) s t = t t. If g is continuous t every point of its domin of definition, it is simply clled continuous. g is clled differentible t the point t = t if t tht point g(t) g C t or lim = C, t t this constnt C is denoted by g (t ). If g is differentible t every point of its domin, it is clled differentible. An importnt ppliction of the derivtive is theorem on finite increments. Theorem 1.1 (Men Vlue Theorem): If f is continuous on [, b] nd hs derivtive on (, b), then there is c,<c<b,such tht f(b) f() =f (c)(b ). (1.1) 1

2 CHAPTER 1. PRELIMINARIES FROM CALCULUS Clerly, differentibility implies continuity, but not the other wy round, s continuity sttes tht the increment g converges to zero together with t, wheres differentibility sttes tht this convergence is t the sme rte or fster. Exmple 1.1: The function g(t) = t is not differentible t, s t this point g t t = = 1 t t s t. It is surprisingly difficult to construct n exmple of continuous function which is not differentible t ny point. Exmple 1.2: An exmple of continuous, nowhere differentible function ws given by Weierstrss in 1872: for t 2π cos(3 n t) f(t) =. (1.2) 2 n n=1 We do not give proof of these properties; justifiction for continuity is given by the fct tht if sequence of continuous functions converges uniformly, then the limit is continuous nd justifiction for non-differentibility cn be provided in some sense by differentiting term by term, which results in divergent series. To sve repetition the following nottions re used: continuous function f is sid to be C function; differentible function f with continuous derivtive is sid to be C 1 function; twice differentible function f with continuous second derivtive is sid to be C 2 function; etc. Right- nd Left-Continuous Functions We cn rephrse the definition of continuous function: function g is clled continuous t the point t = t if lim g(t) =g(t ), (1.3) t t it is clled right-continuous (left-continuous) t t if the vlues of the function g(t) pproch g(t ) when t pproches t from the right (left) lim t t g(t) =g(t ), (lim t t g(t) =g(t )). (1.4) If g is continuous it is, clerly, both right- nd left-continuous. The left-continuous version of g, denoted by g(t ), is defined by tking the left limit t ech point, g(t ) = lim s t g(s). (1.5)

1.1. FUNCTIONS IN CALCULUS 3 From the definitions we hve: g is left-continuous if g(t) = g(t ). The concept of g(t+) is defined similrly, g(t+) = lim g(s). (1.6) s t If g is right-continuous function then g(t+) = g(t) for ny t, so tht g + = g. Definition 1.2: A point t is clled discontinuity of the first kind or jump point if both limits g(t+) nd g(t ) exist nd re not equl. The jump t t is defined s g(t) = g(t+) g(t ). Any other discontinuity is sid to be of the second kind. Exmple 1.3: The function sin(1/t) for t nd for t = hs discontinuity of the second kind t zero, becuse the limits from the right or the left do not exist. An importnt result is tht function cn hve t most countbly mny jump discontinuities (see for exmple Hobson (1921)). Theorem 1.3: A function defined on n intervl [, b] cn hve no more thn countbly mny jumps. A function, of course, cn hve more thn countbly mny discontinuities, but then they re not ll jumps, i.e. they would not hve limits from right or left. Another useful result is tht derivtive cnnot hve jump discontinuities t ll. Theorem 1.4: If f is differentible with finite derivtive f (t) in n intervl, then t ll points f (t) is either continuous or hs discontinuity of the second kind. Proof. If t is such tht f (t+) = lim s t f (s) exists (finite or infinite), then by the men vlue theorem the sme vlue is tken by the derivtive from the right f f(t + ) f(t) (t) = lim = lim t f (c) =f (t+).,t<c<t+ Similrly for the derivtive from the left, f (t) =f (t ). Hence f (t) is continuous t t. The result follows. This result explins why functions with continuous derivtives re sought s solutions to ordinry differentil equtions.

4 CHAPTER 1. PRELIMINARIES FROM CALCULUS Functions Considered in Stochstic Clculus Functions considered in stochstic clculus re functions without discontinuities of the second kind, tht is, functions tht hve both right nd left limits t ny point of the domin nd hve one-sided limits t the boundry. These functions re clled regulr functions. It is often greed to identify functions if they hve the sme right nd left limits t ny point. The clss D = D[,T] of right-continuous functions on [,T] with left limits hs specil nme, càdlàg functions (which is the bbrevition of right continuous with left limits in French). Sometimes these processes re clled R.R.C. for regulr right continuous. Note tht this clss of processes includes C, the clss of continuous functions. Let g D be càdlàg function, then by definition ll the discontinuities of g re jumps. According to Theorem 1.3 such functions hve no more thn countbly mny discontinuities. Remrk 1.1: In stochstic clculus g(t) usully stnds for the size of the jump t t. In stndrd clculus g(t) usully stnds for the increment of g over [t, t + ], g(t) =g(t + ) g(t). The mening of g(t) will be cler from the context. 1.2 Vrition of Function If g is function of rel vrible, its vrition over the intervl [, b] is defined s V g ([, b]) = sup g(t n i ) g(t n i 1), (1.7) where the supremum is tken over prtitions: = t n <t n 1 < <t n n = b. (1.8) Clerly, (by the tringle inequlity) the sums in (1.7) increse s new points re dded to the prtitions. Therefore vrition of g is V g ([, b]) = lim g(t n i ) g(t n i 1), (1.9) δ n where δ n = mx 1 i n (t i t i 1 ). If V g ([, b]) is finite then g is sid to be function of finite vrition on [, b]. If g is function of t, then the vrition function of g s function of t is defined by V g (t) =V g ([,t]). Clerly, V g (t) is non-decresing function of t.

1.2. VARIATION OF A FUNCTION 5 Definition 1.5: g is of finite vrition if V g (t) < for ll t. g is of bounded vrition if sup t V g (t) <, in other words, if for ll t, V g (t) <C, constnt independent of t. Exmple 1.4: 1. If g(t) is incresing then for ny i, g(t i) >g(t i 1) resulting in telescoping sum, where ll the terms excluding the first nd the lst cncel out, leving 2. If g(t) is decresing then, similrly, V g(t) =g(t) g(). V g(t) =g() g(t). Exmple 1.5: If g(t) is differentible with continuous derivtive g (t), g(t) = t g (s)ds, nd g (s) ds <, then V g(t) = g (s) ds. This cn be seen by using the definition nd the men vlue theorem. ti g (s)ds = g (ξ t i 1 i)(t i t i 1), for some ξ i (t i 1,t i). Thus i g (s)ds = t i 1 g (ξ i) (t i t i 1), nd i V g(t) = lim g(t i) g(t i 1) = lim g (s)ds t i 1 = sup g (ξ i) (t i t i 1) = g (s) ds. The lst equlity is due to the lst sum being Riemnn sum for the finl integrl. Alterntively, the result cn be seen from the decomposition of the derivtive into the positive nd negtive prts, g(t) = g (s)ds = [g (s)] + ds [g (s)] ds. Notice tht [g (s)] is zero when [g (s)] + is positive, nd the other wy round. Using this one cn see tht the totl vrition of g is given by the sum of the vrition of the bove integrls. However, these integrls re monotone functions with the vlue zero t zero. Hence V g(t) = = [g (s)] + ds + [g (s)] ds ([g (s)] + +[g (s)] )ds = g (s) ds.

6 CHAPTER 1. PRELIMINARIES FROM CALCULUS Exmple 1.6: (Vrition of pure jump function) If g is regulr right-continuous (càdlàg) function or regulr left-continuous (càglàd), nd chnges only by jumps, g(t) = g(s), s t then it is esy to see from the definition tht V g(t) = g(s). s t Exmple 1.7: The function g(t) =t sin(1/t) for t>, nd g() = is continuous on [, 1], differentible t ll points except zero, but hs infinite vrition on ny intervl tht includes zero. Tke the prtition 1/(2πk + π/2), 1/(2πk π/2), k =1, 2,... The following theorem gives necessry nd sufficient conditions for function to hve finite vrition. Theorem 1.6 (Jordn Decomposition): Any function g : [, ) R of finite vrition cn be expressed s the difference of two incresing functions One such decomposition is given by g(t) =(t) b(t). (t) =V g (t) b(t) =V g (t) g(t). (1.1) It is esy to check tht b(t) is incresing nd (t) is obviously incresing. The representtion of function of finite vrition s the difference of two incresing functions is not unique. Another decomposition is g(t) = 1 2 (V g(t)+g(t)) 1 2 (V g(t) g(t)). The sum, the difference, nd the product of functions of finite vrition re lso functions of finite vrition. This is lso true for the rtio of two functions of finite vrition provided the modulus of the denomintor is lrger thn positive constnt. The following result follows by Theorem 1.3 nd its proof is esy. Theorem 1.7: A finite vrition function cn hve no more thn countbly mny discontinuities. Moreover, ll discontinuities re jumps.

1.2. VARIATION OF A FUNCTION 7 Proof. It is enough to estblish the result for monotone functions, since function of finite vrition is difference of two monotone functions. A monotone function hs left nd right limits t ny point, therefore ny discontinuity is jump. The number of jumps of size greter or equl to 1 n is no more thn (g(b) g())n. The set of ll jump points is union of the sets of jump points with the size of the jumps greter thn or equl to 1 n. Since ech such set is finite, the totl number of jumps is t most countble. A sufficient condition for continuous function to be of finite vrition is given by the following theorem, the proof of which is outlined in Exmple 1.5. Theorem 1.8: If g is continuous, g exists nd g (t) dt < then g is of finite vrition. Theorem 1.9 (Bnch): Let g(t) be continuous function on [, 1], nd denote by s() the number of t s with g(t) =. Then the vrition of g is s()d. Continuous nd Discrete Prts of Function Let g(t), t, be right-continuous incresing function. Then it cn hve t most countbly mny jumps, moreover, the sum of the jumps is finite over finite time intervls. Define the discontinuous prt g d of g by g d (t) = ( ) g(s) g(s ) = g(s), (1.11) s t <s t nd the continuous prt g c of g by g c (t) =g(t) g d (t). (1.12) Clerly, g d chnges only by jumps, g c is continuous, nd g(t) =g c (t)+g d (t). Since finite vrition function is the difference of two incresing functions, the decomposition (1.12) holds for functions of finite vrition. Although representtion s the difference of incresing functions is not unique, decomposition (1.12) is essentilly unique, in the sense tht ny two such decompositions differ by constnt. Indeed, if there were nother such decomposition g(t) =h c (t)+h d (t), then h c (t) g c (t) =g d (t) h d (t), implying tht h d g d is continuous. Hence h d nd g d hve the sme set of jump points, nd it follows tht h d (t) g d (t) =c for some constnt c.

8 CHAPTER 1. PRELIMINARIES FROM CALCULUS Qudrtic Vrition If g is function of rel vrible, define its qudrtic vrition over the intervl [,t] s the limit (when it exists) [g](t) = lim (g(t n i ) g(t n i 1)) 2, (1.13) δ n where the limit is tken over prtitions: = t n <t n 1 < <t n n = t, with δ n = mx 1 i n (t n i tn i 1 ). Remrk 1.2: Similrly to the concept of vrition, there is concept of Φ-vrition of function. If Φ(u) is positive function, incresing monotoniclly with u, then the Φ-vrition of g on [,t]is V Φ [g] = sup Φ( g(t n i ) g(t n i 1) ), (1.14) where supremum is tken over ll prtitions. Functions with finite Φ- vrition on [,t] form clss V Φ. With Φ(u) =u one obtins the clss VF of functions of finite vrition, with Φ(u) =u p one obtins the clss of functions of p-th finite vrition, VF p.if1 p<q<, then finite p-vrition implies finite q-vrition. The stochstic clculus definition of qudrtic vrition is different to the clssicl one with p = 2 (unlike for the first vrition p = 1, when they re the sme). In stochstic clculus the limit in (1.13) is tken over shrinking prtitions with δ n = mx 1 i n (t n i tn i 1 ), nd not over ll possible prtitions. We shll use only the stochstic clculus definition. Qudrtic vrition plys mjor role in stochstic clculus, but is hrdly ever met in stndrd clculus due to the fct tht smooth functions hve zero qudrtic vrition. Theorem 1.1: If g is continuous nd of finite vrition then its qudrtic vrition is zero. Proof. n 1 [g](t) = lim (g(t n i+1) g(t n i )) 2 δ n lim i= mx δ n i lim mx δ n i n 1 g(t n i+1) g(t n i ) i= g(t n i+1) g(t n i ) V g (t). g(t n i+1) g(t n i )

1.3. RIEMANN INTEGRAL AND STIELTJES INTEGRAL 9 Since g is continuous, it is uniformly continuous on [,t], hence lim δn mx i g(t n i+1 ) g(tn i ) =, nd the result follows. Note tht there re functions with zero qudrtic vrition nd infinite vrition (clled functions of zero energy). Define the qudrtic covrition (or simply covrition) of f nd g on [,t] by the following limit (when it exists) n 1 ( [f,g](t) = lim f(t n i+1 ) f(t n i ) )( g(t n i+1) g(t n i ) ), (1.15) δ n i= when the limit is tken over prtitions {t n i } of [,t] with δ n = mx i (t n i+1 tn i ). The sme proof s for Theorem 1.1 works for the following result. Theorem 1.11: If f is continuous nd g is of finite vrition, then their covrition is zero, [f,g](t) =. Let f nd g be such tht their qudrtic vrition is defined. By using simple lgebr, one cn see tht covrition stisfies the following result. Theorem 1.12 (Polriztion Identity): [f,g](t) = 1 ([f + g, f + g](t) [f,f](t) [g, g](t)). (1.16) 2 It is obvious tht covrition is symmetric, [f,g](t) =[g, f](t), it follows from (1.16) tht it is liner, tht is, for ny constnts α nd β [αf + βg,h](t) =α [f,h](t)+β[g, h](t). (1.17) Due to symmetry it is biliner, tht is, liner in both rguments. Thus the qudrtic vrition of the sum cn be opened similrly to multipliction of sums (α 1 f +β 1 g)(α 2 h+β 2 k). It follows from the definition of qudrtic vrition tht it is non-decresing function in t nd consequently it is of finite vrition. According to the polriztion identity, covrition is lso of finite vrition. More bout qudrtic vrition is given in Chpters 4, 7, nd 8. 1.3 Riemnn Integrl nd Stieltjes Integrl Riemnn Integrl The Riemnn integrl of f over intervl [, b] is defined s the limit of Riemnn sums f(t)dt = lim f(ξi n )(t n i t n i 1), (1.18) δ

1 CHAPTER 1. PRELIMINARIES FROM CALCULUS where t n i s represent prtitions of the intervl, = t n <t n 1 < <t n n = b, δ = mx 1 i n (tn i t n i 1), nd t n i 1 ξi n t n i. It is possible to show tht the Riemnn integrl is well-defined for continuous functions, nd by splitting up the intervl it cn be extended to functions which re discontinuous t finitely mny points. Clcultion of integrls is often done by using the ntiderivtive, nd is bsed on the following result. Theorem 1.13 (The Fundmentl Theorem of Clculus): If f is differentible on [, b] nd f is Riemnn integrble on [, b] then f(b) f() = f (s)ds. In generl, this result cnnot be pplied to discontinuous functions, see Exmple 1.8 below. For such functions jump term must be dded, see (1.2). Exmple 1.8: Let f(t) = 2 for 1 t 2, f(t) = 1 for t<1. Then f (t) = t ll t 1. f (s)ds = f(t) f(o). f is continuous nd is differentible t ll points but one, the derivtive is integrble, but the function does not equl the integrl of its derivtive. The min tools for clcultions of Riemnn integrls re chnge of vribles nd integrtion by prts. These re reviewed below in the more generl frmework of the Stieltjes integrl. Stieltjes Integrl The Stieltjes integrl is n integrl of the form f(t)dg(t), where g is function of finite vrition. Since function of finite vrition is difference of two incresing functions, it is sufficient to define the integrl with respect to monotone functions. Stieltjes Integrl with Respect to Monotone Functions The Stieltjes integrl of f with respect to monotone function g over n intervl [, b] is defined s fdg = f(t)dg(t) = lim f(ξi n ) ( g(t n i ) g(t n i 1) ), (1.19) δ with the quntities ppering in the definition being the sme s those used bove for the Riemnn integrl. This integrl is generliztion of the Riemnn integrl, which is recovered when we tke g(t) = t. This integrl is lso known s the Riemnn Stieltjes integrl.

1.3. RIEMANN INTEGRAL AND STIELTJES INTEGRAL 11 Prticulr Cses If g (t) exists, nd g(t) =g() + g (s)ds, then it is possible to show tht f(t)dg(t) = f(t)g (t)dt. If g(t) = [t] k= h(k) ( integer, nd [t] stnds for the integer prt of t) then b f(t)dg(t) = f(k)h(k). +1 This property llows us to represent sums s integrls. Exmple 1.9: 1. g(t) =2t 2 b f(t)dg(t) =4 tf(t)dt. t< 2 t<1 2. g(t) = 3 1 t<2 5 2 t f(t)dg(t) = 2f() + f(1) + 2f(2). If, for exmple, f(t) =t then tdg(t) =5.Iff(t) =(t +1)2 then (t + 1) 2 dg(t)=2+4+18=24. Let g be function of finite vrition nd g(t) =(t) b(t) with (t) =V g (t), b(t) =V g (t) g(t), which re non-decresing functions. If f(s) d(s) = f(s) dv g (s) := f(s) dg(s) < then f is Stieltjes integrble with respect to g nd its integrl is defined by f(s)dg(s) = f(s)d(s) f(s)db(s). (,t] (,t] Nottion: f(s)dg(s) = (,b] f(s)dg(s). Note: (,t] dg(s) =g(t) g() nd dg(s) =g(t ) g(). (,t) If f is Stieltjes integrble with respect to function g of finite vrition, then the vrition of the integrl is V (t) = f(s) dg(s) = (,t] f(s) dv g (s).

12 CHAPTER 1. PRELIMINARIES FROM CALCULUS Impossibility of Direct Definition of n Integrl with Respect to Functions of Infinite Vrition In stochstic clculus we need to consider integrls with respect to functions of infinite vrition. Such functions rise, for exmple, s models of stock prices. Integrls with respect to function of infinite vrition cnnot be defined s usul limit of pproximting sums. The following result Theorem 1.14 explins, see for exmple Protter (1992). Theorem 1.14: Let δ n = mx i (t n i tn i 1 ) denote the lrgest intervl in the prtition of [, b]. If lim δ n f(t n i 1)[g(t n i ) g(t n i 1)] exists for ny continuous function f then g must be of finite vrition on [, b]. This shows tht if g hs infinite vrition then the limit of the pproximting sums does not exist for some functions f. Integrtion by Prts Let f nd g be functions of finite vrition. Denote here g(s) =g(s) g(s ), then (with integrls on (, b]) f(b)g(b) f()g() = f(s )dg(s)+ g(s )df (s)+ f(s) g(s) = f(s )dg(s)+ <s b g(s)df (s). (1.2) The lst eqution is obtined by putting together the sum of jumps with one of the integrls. Note tht lthough the sum in (1.2) is written over uncountbly mny vlues <s b, it hs t most countbly mny non-zero terms. This is becuse finite vrition function cn hve t most countble number of jumps. If f is continuous so tht f(s ) = f(s) for ll s then the formul simplifies nd in this cse we hve the fmilir integrtion by prts formul f(b)g(b) f()g() = f(s)dg(s)+ g(s)df (s).

1.3. RIEMANN INTEGRAL AND STIELTJES INTEGRAL 13 Exmple 1.1: Let g(s) be of finite vrition, g() =, nd consider g 2 (s). Using the integrtion by prts with f = g, wehve In other words, g 2 (t) =2 g(s )dg(s) = g2 (t) 2 Now using the formul (1.2) we lso hve g(s)dg(s) =g 2 (t) g(s )dg(s)+ s t( g(s)) 2. 1 ( g(s)) 2. 2 s t g(s )dg(s) = g2 (t) 2 + 1 ( g(s)) 2. 2 s t Thus it follows tht t Chnge of Vribles g(s )dg(s) g2 (t) 2 g(s)dg(s). Let f hve continuous derivtive (f C 1 ) nd g be of finite vrition nd continuous, then f(g(t)) f(g()) = f (g(s))dg(s) = g(t) g() f (u)du. If g is of finite vrition, hs jumps, nd is right-continuous, then f(g(t)) f(g()) = f (g(s ))dg(s) + ( ) f(g(s)) f(g(s )) f (g(s )) g(s), <s t where g(s) =g(s) g(s ) denotes the jump of g t s. This is known in stochstic clculus s Itô s formul. Exmple 1.11: Tke f(x) =x 2, then we obtin g 2 (t) g 2 ()=2 g(s )dg(s)+ s t( g(s)) 2. Remrk 1.3: Note tht for continuous f nd finite vrition g on [,t] the pproximting sums converge s δ = mx i (t n i+1 tn i ), f(g(t n i ))(g(t n i+1) g(t n i )) i f(g(s ))dg(s).

14 CHAPTER 1. PRELIMINARIES FROM CALCULUS Remrk 1.4: One of the shortcomings of Riemnn or Stieltjes integrls is tht they do not preserve the monotone convergence property, tht is, for sequence of functions f n f does not necessrily follow tht their integrls converge. The Lebesgue (or Lebesgue Stieltjes) integrl preserves this property. 1.4 Lebesgue s Method of Integrtion While Riemnn sums re constructed by dividing the domin of integrtion on the x-xis nd the intervl [, b], into smller subintervls, Lebesgue sums re constructed by dividing the rnge of the function on the y-xis, the intervl [c, d], into smller subintervls c = y <y 1 < <y k < y n = d, nd forming sums n 1 y k length({t : y k f(t) <y k+1 }). k= The Lebesgue integrl is the limit of the bove sums s the number of points in the prtition increses. It turns out tht the Lebesgue integrl is more generl thn the Riemnn integrl, nd preserves convergence. This pproch lso llows integrtion of functions in bstrct probbility spces more generl thn R or R n ; it requires dditionl concepts nd is mde more precise in Chpter 2 (see Section 2.3). Remrk 1.5: In folklore the following nlogy is used: imgine tht money is spred out on floor. In the Riemnn method of integrtion, you collect the money s you progress in the room. In the Lebesgue method, first you collect $1 bills everywhere you cn find them, then $5 bills, etc. 1.5 Differentils nd Integrls The differentil df (t) of differentible function f t t is defined s the liner in t prt of the increment t t, f(t + ) f(t). If the differentil of the independent vrible is denoted dt = t, then f(t + dt) f(t) =df (t)+ smller order terms, nd it follows from the existence of the derivtive t t, tht df (t) =f (t)dt. (1.21) If g is lso differentible function of t, then f(g(t)) is differentible, nd df (g(t)) = f (g(t))g (t)dt = f (g(t))dg(t), (1.22) which is known s the chin rule.

1.6. TAYLOR S FORMULA AND OTHER RESULTS 15 Differentil clculus is importnt in pplictions becuse mny physicl problems cn be formulted in terms of differentil equtions. The min reltion between the integrl nd the differentil (or derivtive) is given by the fundmentl theorem of clculus, Theorem 1.13. For differentible functions, differentil equtions of the form df (t) =ϕ(t)dw(t) cn be written in the integrl form f(t) =f() + ϕ(s)dw(s). In stochstic clculus stochstic differentils do not formlly exist nd the rndom functions w(t) re not differentible t ny point. By introducing new (stochstic) integrl, stochstic differentil equtions cn be defined, nd, by definition, solutions to these equtions re given by the solutions to the corresponding stochstic integrl equtions. 1.6 Tylor s Formul nd Other Results This section contins Tylor s formul nd conditions on functions used in results on differentil equtions. It my be treted s n ppendix nd referred to only when needed. Tylor s Formul for Functions of One Vrible If we consider the increment of function f(x) f(x ) over the intervl [x,x], then provided f (x ) exists, the differentil t x is the liner prt in (x x ) of this increment nd it provides the first pproximtion to the increment. Tylor s formul gives better pproximtion by tking higher order terms of powers of (x x ) provided higher derivtives of f t x exist. If f is function of x with derivtives up to order n + 1, then f(x) f(x )=f (x )(x x )+ 1 2 f (x )(x x ) 2 + 1 3! f (3) (x )(x x ) 3 + + 1 n! f (n) (x )(x x ) n + R n (x, x ), where R n is the reminder, nd f (n) is the derivtive of f (n 1). The reminder cn be written in the form 1 R n (x, x )= (n + 1)! f (n+1) (θ n )(x x ) n+1 for some point θ n (x,x).

16 CHAPTER 1. PRELIMINARIES FROM CALCULUS In our pplictions we shll use this formul with two terms. f(x) f(x )=f (x)(x x )+ 1 2 f (θ)(x x ) 2, (1.23) for some point θ (x,x). Tylor s Formul for Functions of Severl Vribles Similrly to the one-dimensionl cse, Tylor s formul gives successive pproximtions to the increment of multivrible function. A function of n rel vribles f(x 1,x 2,...,x n ) is differentible t point x =(x 1,x 2,...,x n ) if the increment t this point cn be pproximted by liner prt, which is the differentil of f t x. f(x) = C i x i + o(ρ), when ρ = n ( x i ) 2 o(ρ) nd lim ρ ρ =. (1.24) If f is differentible t x =(x 1,x 2,...,x n ), then in prticulr it is differentible s function of ny one vrible x i t tht point, when ll the other coordintes re kept fixed. The derivtive with respect to x i is clled the prtil derivtive f/ x i. Unlike in the one-dimensionl cse, the existence of ll prtil derivtives f/ x i t x is necessry but not sufficient for differentibility of f t x. However, if ll prtil derivtives exist nd re continuous t tht point, then f is differentible t tht point, moreover, C i in (1.24) is given by the vlue of f/ x i t x. If we define the differentil of the independent vrible s its increment dx i = x i, then we hve Theorem 1.15: For f to be differentible t point, it is necessry tht f hs prtil derivtives t tht point, nd it is sufficient tht it hs continuous prtil derivtives t tht point. If f is differentible t x, then its differentil t x is given by f df (x 1,x 2,...,x n )= (x 1,x 2,...,x n )dx i. (1.25) x i The first pproximtion of the increment of differentible function is the differentil, f(x) df (x). If f possesses higher order prtil derivtives, then further pproximtion is possible nd it is given by Tylor s formul. In stochstic clculus the second order pproximtion plys n importnt role.

1.6. TAYLOR S FORMULA AND OTHER RESULTS 17 Let f : R n R be C 2,(f(x 1,x 2,...,x n ) hs continuous prtil derivtives up to order two), x =(x 1,x 2,...,x n ), x + x =(x 1 + x 1,x 2 + x 2,...,x n + x n ). Then by considering the function of one vrible g(t) =f(x + t x) for t 1, the following result is obtined. f(x 1,x 2,...,x n ) = f(x + x) f(x) + 1 2 j=1 f x i (x 1,x 2,...,x n )dx i 2 f x i x j (x 1 + θ x 1,...,x n + θ x n )dx i dx j, (1.26) where just like in the cse of one vrible the second derivtives re evluted t some middle point, (x 1 +θ x 1,...,x n +θ x n ) for some θ (, 1), nd dx i = x i. Lipschitz nd Hölder Conditions Lipschitz nd Hölder conditions describe subclsses of continuous functions. They pper s conditions on the coefficients in the results on the existence nd uniqueness of solutions of ordinry nd stochstic differentil equtions. Definition 1.16: f stisfies Hölder condition (Hölder continuous) of order α, <α 1, on [, b] (R) if there is constnt K>, so tht for ll x, y [, b] (R) f(x) f(y) K x y α. (1.27) A Lipschitz condition is Hölder condition with α =1, f(x) f(y) K x y. (1.28) It is esy to see tht Hölder continuous of order α function on [, b] is lso Hölder continuous of ny lesser order. Exmple 1.12: The function f(x) = x on [, ) ishölder continuous with α =1/2 but is not Lipschitz, since its derivtive is unbounded ner zero. To see tht it is Hölder, it is enough to show tht for ll x, y the following rtio is bounded, x y x y K. (1.29) It is n elementry exercise to estblish tht the left-hnd side is bounded by dividing through by y (if y =, then the bound is obviously one), nd pplying l Hôpitl s rule. Similrly x r,<r<1ishölder of order r. A simple sufficient condition for function to be Lipschitz is to be continuous nd piecewise smooth; precise definitions follow.

18 CHAPTER 1. PRELIMINARIES FROM CALCULUS Definition 1.17: f is smooth on [, b] if it possesses continuous derivtive f on (, b) such tht the limits f (+) nd f (b ) exist. Definition 1.18: f is piecewise continuous on [, b] if it is continuous on [, b] except possibly finite number of points t which right-hnd nd left-hnd limits exist. Definition 1.19: f is piecewise smooth on [, b] if it is piecewise continuous on [, b] nd f exists nd is lso piecewise continuous on [, b]. Growth Conditions The liner growth condition lso ppers in the results on existence nd uniqueness of solutions of differentil equtions. f(x) stisfies the liner growth condition if f(x) K(1 + x ). (1.3) This condition describes the growth of function for lrge vlues of x nd sttes tht f is bounded for smll vlues of x. Exmple 1.13: It cn be shown tht if f(,t) is bounded function of t, f(,t) C for ll t, nd f(x, t) stisfies the Lipschitz condition in x uniformly in t, f(x, t) f(y, t) K x y, then f(x, t) stisfies the liner growth condition in x, f(x, t) K 1(1 + x ). The polynomil growth condition on f is the condition of the form f(x) K(1 + x m ), for some K, m >. (1.31) Theorem 1.2 (Gronwll s Inequlity): Let g(t) nd h(t) be regulr non-negtive functions on [,T], then for ny regulr f(t) stisfying the inequlity for ll t T f(t) g(t)+ h(s)f(s)ds (1.32) we hve ( ) f(t) g(t)+ h(s)g(s) exp h(u)du ds. (1.33) s This form is tken from Dieudonné (196). In prticulr, if g is nondecresing, the integrl bove simplifies to give f(t) g(t)e h(s)ds. (1.34) In its simplest form when g = A nd h = B re constnts, f(t) Ae Bt. (1.35)

1.6. TAYLOR S FORMULA AND OTHER RESULTS 19 Solution of First Order Liner Differentil Equtions Liner differentil equtions, by definition, re liner in the unknown function nd its derivtives. A first order liner eqution, in which the coefficient of dx(t) dt does not vnish, cn be written in the form dx(t) + g(t)x(t) =k(t). (1.36) dt These equtions re solved by using the integrting fctor method. The integrting fctor is the function e G(t), where G(t) is chosen by G (t) = g(t). After multiplying both sides of the eqution by e G(t), integrting, nd solving for x(t), we hve ( ) x(t) =e G(t) e G(s) k(s) ds + x()e G() G(t). (1.37) The integrting fctor G(t) is determined up to constnt, but it is cler from (1.37), tht the solution x(t) remins the sme. Further Results on Functions nd Integrtion Results given here re not required to understnd subsequent mteril. Some of these involve the concepts of set of zero Lebesgue mesure. This is given in the next chpter (see Section 2.2); ny countble set hs Lebesgue mesure zero, but there re lso uncountble sets of zero Lebesgue mesure. A prtil converse to Theorem 1.8 lso holds (see for exmple Sks (1964), nd Freedmn (1983), for the following results). Theorem 1.21 (Lebesgue): A finite vrition function g on [, b] is differentible lmost everywhere on [, b]. In wht follows, sufficient conditions for function to be Lipschitz nd not to be Lipschitz re given. 1. If f is continuously differentible on finite intervl [, b], then it is Lipschitz. Indeed, since f is continuous on [, b], it is bounded, f K. Therefore f(x) f(y) = y x f (t)dt y x f (t) dt K x y. (1.38) 2. If f is continuous nd piecewise smooth, then it is Lipschitz, the proof is similr to the bove. 3. A Lipschitz function does not hve to be differentible, for exmple f(x) = x is Lipschitz, but it is not differentible t zero.

2 CHAPTER 1. PRELIMINARIES FROM CALCULUS 4. It follows from the definition of Lipschitz function (1.28) tht if it is differentible, then its derivtive is bounded by K. 5. A Lipschitz function hs finite vrition on finite intervls, since for ny prtition {x i } of finite intervl [, b], f(xi+1 ) f(x i ) K (x i+1 x i )=K(b ). (1.39) 6. As functions of finite vrition hve derivtives lmost everywhere (with respect to Lebesgue mesure), Lipschitz function is differentible lmost everywhere. (Note tht functions of finite vrition hve derivtives which re integrble with respect to Lebesgue mesure, but the function does not hve to be equl to the integrl of the derivtive.) 7. A Lipschitz function multiplied by constnt, nd sum of two Lipschitz functions re Lipschitz functions. The product of two bounded Lipschitz functions is gin Lipschitz function. 8. If f is Lipschitz on [,N] for ny N > but with the constnt K depending on N, then it is clled loclly Lipschitz. For exmple, x 2 is Lipschitz on [,N] for ny finite N, but it is not Lipschitz on [, + ), since its derivtive is unbounded. 9. If f is function of two vribles f(x, t) nd it stisfies the Lipschitz condition in x for ll t, t T, with sme constnt K independent of t, it is sid tht f stisfies the Lipschitz condition in x uniformly in t, t T. A necessry nd sufficient condition for function f to be Riemnn integrble ws given by Lebesgue (see for exmple Sks (1964), Freedmn (1983)). Theorem 1.22 (Lebesgue): A necessry nd sufficient condition for function f to be Riemnn integrble on finite closed intervl [, b] is tht f is bounded on [, b] nd lmost everywhere continuous on [, b], tht is, continuous t ll points except possibly on set of Lebesgue mesure zero. Remrk 1.6: (The following is not used nywhere in the book nd is directed only to reders with knowledge of functionl nlysis). Continuous functions on [, b] with the supremum norm h = sup x [,b] h(x) is Bnch spce, denoted C([, b]). By result in functionl nlysis, ny liner functionl on this spce cn be represented s h(x)dg(x) for some function g of finite vrition. In this wy, (,b] the Bnch spce of functions of finite vrition on [, b] with the norm g = V g ([, b]) cn be identified with the spce of liner functionls on the spce of continuous functions, in other words, the dul spce of C([, b]).