On pathwise stochastic integration

Transcription

1 On pathwise stochastic integration Rafa l Marcin Lochowski Afican Institute for Mathematical Sciences, Warsaw School of Economics UWC seminar Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 1 / 29

2 The Riemann-Stieltjes integral The Riemann-Stieltjes integral of a deterministic function f : [a, b] R (integrand) with respect to another deterministic function g : [a, b] R (integrator) is defined as the limit of sums n f (s i ) {g (t i ) g (t i 1 )}, i=1 where s i [t i 1 ; t i ], π = {a = t < t 1 <... < t n = b}, as the mesh of the partition π, mesh(π) := max i=1,2,...,n t i t i 1, goes to. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 2 / 29

3 The Riemann-Stieltjes integral The Riemann-Stieltjes integral of a deterministic function f : [a, b] R (integrand) with respect to another deterministic function g : [a, b] R (integrator) is defined as the limit of sums n f (s i ) {g (t i ) g (t i 1 )}, i=1 where s i [t i 1 ; t i ], π = {a = t < t 1 <... < t n = b}, as the mesh of the partition π, mesh(π) := max i=1,2,...,n t i t i 1, goes to. The Riemann-Stieltjes integral, denoted by (RS) f dg [a;b] may not exist for a given pair (f, g). The simplest situation when it happens is when the total variation of g is infinite. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 2 / 29

4 The total variation and the existence of the R-S integral Recall that the total variation of a deterministic function g : [a, b] R is defined as TV (g, [a; b]) := sup n sup a t <t 1 <...<t n b i=1 n g (t i ) g (t i 1 ), Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 3 / 29

5 The total variation and the existence of the R-S integral Recall that the total variation of a deterministic function g : [a, b] R is defined as TV (g, [a; b]) := sup n sup a t <t 1 <...<t n b i=1 n g (t i ) g (t i 1 ), The finiteness of TV (g, [a; b]) together with the continuity of f guarantee the existence of (RS) [a;b] f dg. However, still, for a bounded, Borel-measurable integrand f and finite (total) variation integrator g, (RS) [a;b] f dg may not exist. This may happen e.g. when the jumps of the function f coincide with the jumps of g (the invention of an appropriate example might be a good, instructive exercise for students). Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 3 / 29

6 The refinement of the R-S integral - the Lebesgue-Stieltjes integral A fine refinement of the R-S integral is the Lebesgue-Stieltjes integral. The construction is made by the introduction of the measure space ([a; b], B ([a; b]), µ g ), where B ([a; b]) denotes the σ-field of Borel subsets of [a; b] and µ g is a signed, σ-finite measure on [a; b]. To define µ g we consier the càdlàg version of g (right-continuous with left limits) which we will also denote by g and for a c d b define µ g (c; d] := g(d) g(c). Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 4 / 29

7 The refinement of the R-S integral - the Lebesgue-Stieltjes integral A fine refinement of the R-S integral is the Lebesgue-Stieltjes integral. The construction is made by the introduction of the measure space ([a; b], B ([a; b]), µ g ), where B ([a; b]) denotes the σ-field of Borel subsets of [a; b] and µ g is a signed, σ-finite measure on [a; b]. To define µ g we consier the càdlàg version of g (right-continuous with left limits) which we will also denote by g and for a c d b define µ g (c; d] := g(d) g(c). Now we extend the measure µ g to all Borel subsets of [a; b] (Caratheodory s extension) and define the Lebesgue-Stieltjes integral as the usual Lebesgue integral of f with respect to the measure µ g : (LS) f dg := f dµ g. [a;b] [a;b] Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 4 / 29

8 The Lebesgue-Stieltjes integral - properties Now, finiteness of TV (g, [a; b]) together with boundedness and Borel-measurability of the integrand f guarantee the existence of (LS) [a;b] f dg. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 5 / 29

9 The Lebesgue-Stieltjes integral - properties Now, finiteness of TV (g, [a; b]) together with boundedness and Borel-measurability of the integrand f guarantee the existence of (LS) [a;b] f dg. If both, f and g are càdlàg and have finite total variation we have important integration by parts formula (LS) f (t )dg(t) =f (b)g(b) f (a)g(a) (1) (a;b] (LS) g(t )df (t) f (s) g(s). (a;b] a<s b where f (s) = f (s) f (s ), g(s) = g(s) g(s ). Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 5 / 29

10 The Lebesgue-Stieltjes integral - properties Now, finiteness of TV (g, [a; b]) together with boundedness and Borel-measurability of the integrand f guarantee the existence of (LS) [a;b] f dg. If both, f and g are càdlàg and have finite total variation we have important integration by parts formula (LS) f (t )dg(t) =f (b)g(b) f (a)g(a) (1) (a;b] (LS) g(t )df (t) f (s) g(s). (a;b] a<s b where f (s) = f (s) f (s ), g(s) = g(s) g(s ). Notice, that using this formula one may propose a reasonable value for (LS) (a;b] f (t )dg(t) whenever f and g are càdlàg, TV (f, [a; b]) < + and g is bounded! (The same observation may be used to differentiate distributions but it is another story...) Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 5 / 29

11 The Lebesgue-Stieltjes integral still insufficient (A standard) Brownian motion B is one of the simplest continuous-time process with continuous trajectories, widely used in stochastic modelling and optimisation. Unfortunately, its paths have a.s. infinite total variation. When one wants to integrate locally finite variation deterministic function or locally finite variation stochastic process with respect to a Brownian trajectory B t, t [; T ], one may use the relation (1) (this idea is due to Paley, Wiener and Zygmund). Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 6 / 29

12 The Lebesgue-Stieltjes integral still insufficient (A standard) Brownian motion B is one of the simplest continuous-time process with continuous trajectories, widely used in stochastic modelling and optimisation. Unfortunately, its paths have a.s. infinite total variation. When one wants to integrate locally finite variation deterministic function or locally finite variation stochastic process with respect to a Brownian trajectory B t, t [; T ], one may use the relation (1) (this idea is due to Paley, Wiener and Zygmund). Unfortunately, this is still not sufficient to calculate (or at least to give a reasonable meaning if the calculations were too hard) e.g. (LS) B t db t. (a;b] Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 6 / 29

13 Functions with finite and infinite total variation - examples t Figure : A typical path of a Brownian motion (green) and a function with finite total variation (blue) Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 7 / 29

14 A standard Brownian motion - an axiomatic definition A standard Brownian motion is a stochastic process B t, t defined on some (rich enough) probability space (Ω, F, P) with the following properties B ; B has continuous paths, i.e. the probability P that for ω Ω, the path [; + ) t B t (ω) R is continuous equals 1; for any s < t < u increments B u B t and B t B s are independent; for any s < t the increment B t B s has normal distribution with mean and variance t s, N (, t s). Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 8 / 29

15 Finiteness of the quadratic variation of a Brownian motion Cruicial observation which is sometimes utilised to define a stochastic integral with respect to the Brownian motion or even with respect to much more general family of processes - semimartingales - is the finiteness of their quadratic variation. But even with the quadratic variation we must be careful. When we define V 2 (B, [a; b]) := sup n then V 2 (B, [a; b]) is a.s. infinite. sup a t <t 1 <...<t n b i=1 n ( ) 2 Bti B ti 1, Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 9 / 29

16 Finiteness of the quadratic variation of a Brownian motion Cruicial observation which is sometimes utilised to define a stochastic integral with respect to the Brownian motion or even with respect to much more general family of processes - semimartingales - is the finiteness of their quadratic variation. But even with the quadratic variation we must be careful. When we define V 2 (B, [a; b]) := sup n sup a t <t 1 <...<t n b i=1 n ( ) 2 Bti B ti 1, then V 2 (B, [a; b]) is a.s. infinite. However, one may relatively easily prove that for } any sequence of partitions π k = {a = t k < tk 1 <... < tk n(k) = b, with mesh(π k ) as k + we have n(k) i=1 ( B t k i B t k i 1 ) 2 P [B] b [B] a := b a, where P denotes the convergence in probability. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 9 / 29

17 ψ- variation of a Brownian motion For x > define x 2 ψ(x) := ln(ln(1/x) 2) and ψ() := then, by the result of S. J. Taylor (Exact asymptotic estimates of Brownian path variation, Duke Math. J. 39), almost surely: V ψ (B, [a; b]) := sup n sup t <t 1 <...<t n T i=1 n ψ ( ) B ti B ti 1 < +, (2) Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 1 / 29

18 ψ- variation of a Brownian motion For x > define x 2 ψ(x) := ln(ln(1/x) 2) and ψ() := then, by the result of S. J. Taylor (Exact asymptotic estimates of Brownian path variation, Duke Math. J. 39), almost surely: V ψ (B, [a; b]) := sup n sup t <t 1 <...<t n T i=1 n ψ ( ) B ti B ti 1 < +, (2) moreover, ψ is a function with the greatest possible order at, for which (2) holds. Remark Moreover, when mesh ( π (n)), then lim sup n n sup n t <t 1 <...<t n T i=1 ( ) ψ B (n) t B (n) i t = T a.s. i 1 Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 1 / 29

19 A small detour - why the quadratic variation [B] of the Brownian motion on the interval [a; b] equals b a? To give a hint why the quadratic variation of the Brownian motion on the interval [a; b] equals b a let us recall the simplest construction of a standard Brownian motion (due to Donsker we know that this construction works, though the convergence is relatively slow). Pick (very) large integer n; set t =, 1 n, 2 n,..., n n, n+1 n,...; dt = 1 n ; set B = and for t =, 1 n, 2 n,..., n n, n+1 n,... Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 11 / 29

20 A small detour - why the quadratic variation [B] of the Brownian motion on the interval [a; b] equals b a? To give a hint why the quadratic variation of the Brownian motion on the interval [a; b] equals b a let us recall the simplest construction of a standard Brownian motion (due to Donsker we know that this construction works, though the convergence is relatively slow). Pick (very) large integer n; set t =, 1 n, 2 n,..., n n, n+1 n,...; dt = 1 n ; set B = and for t =, 1 n, 2 n,..., n n, n+1 n,... { n 1 with probability 1/2; B t+dt = B t+1/n = B t + 1 n with probability 1/2. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 11 / 29

21 A small detour - why the quadratic variation [B] of the Brownian motion on the interval [a; b] equals b a? To give a hint why the quadratic variation of the Brownian motion on the interval [a; b] equals b a let us recall the simplest construction of a standard Brownian motion (due to Donsker we know that this construction works, though the convergence is relatively slow). Pick (very) large integer n; set t =, 1 n, 2 n,..., n n, n+1 n,...; dt = 1 n ; set B = and for t =, 1 n, 2 n,..., n n, n+1 n,... { n 1 with probability 1/2; B t+dt = B t+1/n = B t + 1 n with probability 1/2. Now notice that (db t ) 2 = dt (the simplest version of Itô s formula) and thus (db t ) 2 = dt = [B] T = T. t=1/n,2/n,...,t t=1/n,2/n,...,t Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 11 / 29

22 Another small detour - Young s integral When the integrand has finite p-variation, V p the integrator has finite q-variation, V q, p > 1, q > 1 and 1/p + 1/q > 1, then one still may define (RYS) f dg. where (RYS) denotes some refinement of the Riemann-Stieltjes integral (the refinement is made to avoid problems with discontinuities) and when g is continuous, it coincides with the Riemann-Stieltjes integral. [a;b] Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 12 / 29

23 Another small detour - Young s integral When the integrand has finite p-variation, V p the integrator has finite q-variation, V q, p > 1, q > 1 and 1/p + 1/q > 1, then one still may define (RYS) f dg. where (RYS) denotes some refinement of the Riemann-Stieltjes integral (the refinement is made to avoid problems with discontinuities) and when g is continuous, it coincides with the Riemann-Stieltjes integral. This is proved with the following Love-Young inequality: n f (s i ) {g(t i ) g(t i 1 )} f (s i ) {g(b) g(a)} ζ ( p 1 + q 1), i=1 [a;b] for any partition π = {a = t < t 1 <... < t n = b}, s i [t i 1 ; t i ], i, i {1, 2,..., n}. Here ζ(r) = k r. k=1 Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 12 / 29

24 The Young integral is still insufficient Since the ψ- variation of a Brownian motion is finite then any p- variation with p > 2 of the Brownian motion is locally finite. (Just to make it precise let us write a formula for p-variation: ) V p (B, [a; b]) := sup n sup t <t 1 <...<t n T i=1 n B ti B ti 1 p. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 13 / 29

25 The Young integral is still insufficient Since the ψ- variation of a Brownian motion is finite then any p- variation with p > 2 of the Brownian motion is locally finite. (Just to make it precise let us write a formula for p-variation: V p (B, [a; b]) := sup n sup t <t 1 <...<t n T i=1 n B ti B ti 1 p. ) Thus, the Young integral may be used for the pathwise integration (RYS) X s db s whenever X is a stochastic process with locally finite q-variation, where 1 q < 2. Unfortunately, this is still unsufficient for calcuation of the integral [a;b] B sdb s. [a;b] Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 13 / 29

26 Stochastic integral - the quadratic variation approach For simplicity we will fix on integration with respect to the Brownian motion B. We consider a family of stochastic processes X with càglàd (left continuous with right limits) paths [; + ) t X t R such that for every process X X the following hold 1 for every u > t the increment B u B t is independent of the values of X s, s t; 2 X H := E + Xs 2 d[b] s = E + Xs 2 ds = + EXs 2 ds < +. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 14 / 29

27 Stochastic integral - the quadratic variation approach For simplicity we will fix on integration with respect to the Brownian motion B. We consider a family of stochastic processes X with càglàd (left continuous with right limits) paths [; + ) t X t R such that for every process X X the following hold 1 for every u > t the increment B u B t is independent of the values of X s, s t; 2 X H := E + Xs 2 d[b] s = E + Xs 2 ds = + EXs 2 ds < +. It appears that H = (X, H ) is a Hilbert space and the family of simple processes of the form K = K i= K i 1 (ti,t i+1 ], where = t < t 1 < t 2 <... with t i + as i + and B u B ti, u t i being independent from K i is dense in H. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 14 / 29

28 Stochastic integral - the quadratic variation approach cont. Now, for any t > and every simple process K we define t n 1 KdB := K 1 B + K i (B ) ti+1 B ti + Kn (B t B tn ), (3) whenever t n t < t n+1. i= Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 15 / 29

29 Stochastic integral - the quadratic variation approach cont. Now, for any t > and every simple process K we define t n 1 KdB := K 1 B + K i (B ) ti+1 B ti + Kn (B t B tn ), (3) i= whenever t n t < t n+1. By this definition and the independence of the increments B ti+1 B ti from K i we easily calculate ( t 2 ( + ) 2 E KdB) E KdB = K 2 H. (4) Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 15 / 29

30 Stochastic integral - the quadratic variation approach cont. Now, for any t > and every simple process K we define t n 1 KdB := K 1 B + K i (B ) ti+1 B ti + Kn (B t B tn ), (3) i= whenever t n t < t n+1. By this definition and the independence of the increments B ti+1 B ti from K i we easily calculate ( t 2 ( + ) 2 E KdB) E KdB = K 2 H. (4) The equality (4) is called Itô s isometry. Now, for any X X and ε > we find a simple process K ε such that X K ε H < ε and define t X db = lim ε By (4) this limit exists and is unique. t K ε db. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 15 / 29

31 Stochastic integral - the quadratic variation approach cont. The construction presented on two previous slides may be extended to martingales, local martingales and semimartingales. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 16 / 29

32 Stochastic integral - the quadratic variation approach cont. The construction presented on two previous slides may be extended to martingales, local martingales and semimartingales. A continuous-time martingale M t, t, is a special (càdlàg) process, conditional increments of which, M t M s, s < t, with respect to the available information - filtration till moment s, F s, are centered E [M t M s F s ] =. Every such a process has finite quadratic variation [M]. Moreover, the process M 2 [M] is also a martingale. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 16 / 29

33 Stochastic integral - the quadratic variation approach cont. The construction presented on two previous slides may be extended to martingales, local martingales and semimartingales. A continuous-time martingale M t, t, is a special (càdlàg) process, conditional increments of which, M t M s, s < t, with respect to the available information - filtration till moment s, F s, are centered E [M t M s F s ] =. Every such a process has finite quadratic variation [M]. Moreover, the process M 2 [M] is also a martingale. A local martingale is a (càdlàg) process, which stopped at the appropriate stopping times is a martingale. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 16 / 29

34 Stochastic integral - the quadratic variation approach cont. The construction presented on two previous slides may be extended to martingales, local martingales and semimartingales. A continuous-time martingale M t, t, is a special (càdlàg) process, conditional increments of which, M t M s, s < t, with respect to the available information - filtration till moment s, F s, are centered E [M t M s F s ] =. Every such a process has finite quadratic variation [M]. Moreover, the process M 2 [M] is also a martingale. A local martingale is a (càdlàg) process, which stopped at the appropriate stopping times is a martingale. A semimartingale is a (càdlàg) process which is a sum of a local martingale and a locally finite variation process. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 16 / 29

35 Semimartingales The family of semimartingales is rich enough to encompass majority of processes used in stochastic modelling (except maybe fractional Brownian motions). Moreover, the deep result of Bichteler and Dellacherie states that this family is in some sense the broadest possible family of good integrators. Namely, when we define for a given integrator M and every simple process the integral t KdM with the formula analogous to the formula (3), then the transformation K t KdM is continuous if and only if M is a semimartingale. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 17 / 29

36 Semimartingales The family of semimartingales is rich enough to encompass majority of processes used in stochastic modelling (except maybe fractional Brownian motions). Moreover, the deep result of Bichteler and Dellacherie states that this family is in some sense the broadest possible family of good integrators. Namely, when we define for a given integrator M and every simple process the integral t KdM with the formula analogous to the formula (3), then the transformation K t KdM is continuous if and only if M is a semimartingale. To deal with continuity we need to define topologies: the space of simple processes is endowed with uniform convergence in (t, ω) topology and the space of integrals is topologized by convergence in probability. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 17 / 29

37 Few properties of a stochastic integral The stochastic integral is now defined for any semimartingale M and any (predictable) càglàd process X. For example t B tdb t = 1 2 B2 t 1 2 t. We have important Theorem (counterpart of the Lebesgue dominated convergence) For any càglàd (predictable) processes X 1, X 2,..., such that X i X for i = 1, 2,..., where X is some càglàd (predictable) process and lim i + X i pointwise then we have sup s T s X i dm P. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 18 / 29

38 Few properties of a stochastic integral The stochastic integral is now defined for any semimartingale M and any (predictable) càglàd process X. For example t B tdb t = 1 2 B2 t 1 2 t. We have important Theorem (counterpart of the Lebesgue dominated convergence) For any càglàd (predictable) processes X 1, X 2,..., such that X i X for i = 1, 2,..., where X is some càglàd (predictable) process and lim i + X i pointwise then we have sup s T s X i dm P. For any p > we also have important Burkholder-Davis-Gundy s inequality E sup s T s X dm p C p E T X 2 d[m] p/2. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 18 / 29

39 Wong-Zakai s pathwise approach to the stochastic integral Since many years mathematicians tried to define the stochastic integral in a pathwise way. One of the earliest of such attemps is due to Wong and Zakai (1965). For T > they considered the following approximation of Brownian paths: (A) for all s [; T ], B n s B s pointwise as n +, where B n, n = 1, 2,..., are continuous and have locally bounded variation; (B) (A) and there exists such a bounded process Z that for all s [; T ], B n s Z s ; and stated the following approximation theorem. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 19 / 29

40 Wong-Zakai s pathwise approach to the stochastic integral Since many years mathematicians tried to define the stochastic integral in a pathwise way. One of the earliest of such attemps is due to Wong and Zakai (1965). For T > they considered the following approximation of Brownian paths: (A) for all s [; T ], B n s B s pointwise as n +, where B n, n = 1, 2,..., are continuous and have locally bounded variation; (B) (A) and there exists such a bounded process Z that for all s [; T ], B n s Z s ; and stated the following approximation theorem. Theorem (Wong-Zakai (1965)) Let ψ(t, x) has continuous partial derivatives ψ t (B), then for the Lebesgue-Stieltjes integrals T and ψ x T T lim ψ (t, B n n t ) dbt n = ψ (t, B t ) db t + 1 T 2 and let Bn satisfy ψ (t, Bn t ) db n t, a.s., ψ x (t, B t) dt. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 19 / 29

41 Wong-Zakai s pathwise approach to the stochastic integral, cont. The correction term in Wong-Zakai s theorem, T ψ x (t, B t) dt, is simply the quadratic covariation of the processes B t and ψ (t, B t ). For two semimartingales X and Y the quadratic covariation, [X, Y ], is the (unique) limit in the probability of the sums n(k) i=1 ) ) (X t X ki t (Y ki 1 t Y ki t, ki 1 } where the sequence of partitions π k = {a = t k < tk 1 <... < tk n(k) = b is such that mesh(π k ) as k +. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 2 / 29

42 Wong-Zakai s pathwise approach to the stochastic integral, cont. The correction term in Wong-Zakai s theorem, T ψ x (t, B t) dt, is simply the quadratic covariation of the processes B t and ψ (t, B t ). For two semimartingales X and Y the quadratic covariation, [X, Y ], is the (unique) limit in the probability of the sums n(k) i=1 ) ) (X t X ki t (Y ki 1 t Y ki t, ki 1 } where the sequence of partitions π k = {a = t k < tk 1 <... < tk n(k) = b is such that mesh(π k ) as k +. For two càdlàg semimartingales X and Y the integral (S) T Y dx := is called the Stratonovich integral. T Y dx [X, Y ] Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 2 / 29

43 Disadvantages of the Wong-Zakai construction The Wong-Zakai construction works only for very limited family of integrators: Brownian motions and processes of the form X t = t µ sds + t σ sdb s and integrands must be functions of the integrators. The generalisation for any pair of semimartingale integrator and integrand is impossible. It is relatively easy to give an example of two sequences of continuous, with locally finite variation, bounded (and adapted to the natural Brownian filtration) processes B n and B n such that B n B, B n B uniformly but diverges (Lochowski (213)). 1 B n db n Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 21 / 29

44 A modification of the Wong-Zakai construction In Lochowski (213) for any càdlàg process X the following construction is considered. For any c > we there exists a process X c such that (i) X c has locally finite total variation; (ii) X c has càdlàg paths; (iii) for every T there exists such K T < + that for every t [; T ], X t X c t K T c; (iv) for every T there exists such L T < + that for every t [; T ], X c t L T X t, where X c t = X c t X c t, X t = X t X t ; (v) the process X c is adapted to the natural filtration of X. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 22 / 29

45 A modification of the Wong-Zakai construction, cont. Further, in Lochowski (213) it was shown that if processes X and Y are càdlàg semimartingales then the sequence of pathwise Lebesgue-Stieltjes integrals T Y dx c P c T Y dx + [X, Y ] cont T. T Y dx denotes here the (semimartingale) stochastic integral and [X, Y ] cont denotes here the continuous part of [X, Y ], i.e. [X, Y ] cont T = [X, Y ] T <s T X s Y s. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 23 / 29

46 A modification of the Wong-Zakai construction, cont. Further, in Lochowski (213) it was shown that if processes X and Y are càdlàg semimartingales then the sequence of pathwise Lebesgue-Stieltjes integrals T Y dx c P c T Y dx + [X, Y ] cont T. T Y dx denotes here the (semimartingale) stochastic integral and [X, Y ] cont denotes here the continuous part of [X, Y ], i.e. [X, Y ] cont T = [X, Y ] T <s T X s Y s. Moreover, when c(n) > and + n=1 c(n)2 < + then the convergence of T Y dx c(n) to T Y dx + [X, Y ] cont T holds almost surely. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 23 / 29

47 Drawbacks of the construction presented Unfortunately, the construction presented does not work for any càglàd integrand Y. It is possible to construct a continuous, bounded (and adapted to the natural Brownian filtration) process Y and a sequence B c(n), n = 1, 2,..., satisfying all conditions (i)-(v) for X = B such that the integral diverges (cf. Lochowski (213)). 1 Y db c(n) Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 24 / 29

48 Bichteler s construction The remarkable Bichteler s approach provides pathwise construction for integration of any adapted càdlàg process Y with càdlàg semimartingale integrator X and is based on the approximation lim sup n s T Y X + i=1 Y τ n i 1 s ( X τ n i s X τ n i 1 s ) s Y dx = a.s., Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 25 / 29

49 Bichteler s construction The remarkable Bichteler s approach provides pathwise construction for integration of any adapted càdlàg process Y with càdlàg semimartingale integrator X and is based on the approximation lim sup n s T Y X + i=1 Y τ n i 1 s ( X τ n i s X τ n i 1 s ) s Y dx = a.s., where τ n = (τi n ), i =, 1, 2,..., is the following sequence of stopping times: τ n = and for i = 1, 2,..., { τn = inf t > τi 1 n } : Y t Y τ n i 1 2 n. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 25 / 29

50 Bichteler s construction The remarkable Bichteler s approach provides pathwise construction for integration of any adapted càdlàg process Y with càdlàg semimartingale integrator X and is based on the approximation lim sup n s T Y X + i=1 Y τ n i 1 s ( X τ n i s X τ n i 1 s ) s Y dx = a.s., where τ n = (τi n ), i =, 1, 2,..., is the following sequence of stopping times: τ n = and for i = 1, 2,..., { τn = inf t > τi 1 n } : Y t Y τ n i 1 2 n. Remark In fact, given c(n) >, n=1 c2 (n) < +, Bichteler s construction works for any sequence τ n = (τi n ), i =, 1, { 2,..., of stopping times, such that } τ n = and for i = 1, 2,..., τ n i = inf t > τi 1 n : Yt Y τ n i 1 c (n). Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 25 / 29

51 Norvaiša s integral for functions with finite quadratic variation In a long (171 pages!) paper, Norvaiša develops (among other results) a theory of an integral for deterministic functions with finite λ - quadratic variation. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 26 / 29

52 Norvaiša s integral for functions with finite quadratic variation In a long (171 pages!) paper, Norvaiša develops (among other results) a theory of an integral for deterministic functions with finite λ - quadratic variation. A function f : [a; b] R has finite λ - quadratic variation if the sums n(k) i=1 ( ) 2 f (ti k ) f (ti 1) k converge for any nested partitions i.e. π k π k+1 with t k = a, tk n(k) = b and mesh ( π k) as k +. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 26 / 29

53 Norvaiša s integral for functions with finite quadratic variation In a long (171 pages!) paper, Norvaiša develops (among other results) a theory of an integral for deterministic functions with finite λ - quadratic variation. A function f : [a; b] R has finite λ - quadratic variation if the sums n(k) i=1 ( ) 2 f (ti k ) f (ti 1) k converge for any nested partitions i.e. π k π k+1 with t k = a, tk n(k) = b and mesh ( π k) as k +. Sample paths of a Brownian motion have this property with probability P equal 1 - a result due to Paul Lévy. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 26 / 29

54 Norvaiša s integral for functions with finite quadratic variation, cont. Further, he defines Left-Cauchy λ integral of g with respect to f, (LC) b a gdf, as a limit of the sums n(k) i=1 ( ) g(ti 1) k f (ti k ) f (ti 1) k and proves its existence for the integrands of the form g(t) = ψ(f (t)), where ψ is a C 1 function. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 27 / 29

55 Norvaiša s integral for functions with finite quadratic variation, cont. Further, he defines Left-Cauchy λ integral of g with respect to f, (LC) b a gdf, as a limit of the sums n(k) i=1 ( ) g(ti 1) k f (ti k ) f (ti 1) k and proves its existence for the integrands of the form g(t) = ψ(f (t)), where ψ is a C 1 function. He manages to prove a version of Itô s formula for this integral. But still this is far from the theory of an integral for any (locally bounded) deterministic integrands and finite quadratic variation integrators. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 27 / 29

56 Norvaiša s integral for functions with finite quadratic variation, cont. Further, he defines Left-Cauchy λ integral of g with respect to f, (LC) b a gdf, as a limit of the sums n(k) i=1 ( ) g(ti 1) k f (ti k ) f (ti 1) k and proves its existence for the integrands of the form g(t) = ψ(f (t)), where ψ is a C 1 function. He manages to prove a version of Itô s formula for this integral. But still this is far from the theory of an integral for any (locally bounded) deterministic integrands and finite quadratic variation integrators. The reason for this might be the fact that the analogous construction for semimartingales does not require only finiteness of the quadratic variation of the integrator but also a centering property of the increments of a local martingale part. Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 27 / 29

57 Some references Bichteler, K., (1981) Stochastic integration and L p theory of semimartingales. Ann. Probab., 9(1): Karandikar, R. L., (1995) On pathwise stochastic integration. Stoch. Process. Appl., 57(1): Lochowski, R., M., (213) Pathwise stochatic integration with finite variation processes uniformly approximating càdlàg processes, submitted. Norvaiša, R., (28) Quadratic Variation, p-variation and Integration with Applications to Stock Price Modelling, arxiv. Wong, E. and Zakai, M., (1965) On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist., 36: Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 28 / 29

58 Thank you! Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic integration UWC seminar 29 / 29