[...]... Properties of stochastic integrals with respect to continuous local martingales 2.d Integration with respect to continuous semimartingales 2.e The stochastic integral as a limit of certain Riemann type sums 2.f Integration with respect to vector valued continuous semimartingales 135 140 142 147 150 153 Table of Contents 3 Ito’s Formula xi 157 3.a Ito’s... 112 116 118 118 120 127 128 Chapter III Stochastic Integration 1 Measurability Properties of Stochastic Processes 131 1.a The progressive and predictable σ-fields on Π 131 1.b Stochastic intervals and the optional σ-field 134 2 Stochastic Integration with Respect to Continuous Semimartingales 135 2.a Integration with respect to continuous local martingales 2.b M... expectation of X with respect to G) Processes Let X = (Xt )t≥0 be a stochastic process and T : Ω → [0, ∞] an optional time Then XT denotes the random variable (XT )(ω) = XT (ω) (ω) (sample of X T along T , I.3.b, I.7.a) X T denotes the process Xt = Xt∧T (process X stopped at n time T ) S, S+ and S denote the space of continuous semimartingales, continuous positive semimartingales and continuous Rn -valued... the set D in the proof of (g) is necessary since the σ-field G is not assumed to contain the null sets Since E(P) is not a vector space, EG : X ∈ E(P ) → EG (X) is not a linear operator However when its domain is restricted to L1 (P ), then EG becomes a nonnegative linear operator 14 2.b Conditional expectation 2.b.6 Monotone Convergence Let Xn , X, h ∈ E(P ) and assume that Xn ≥ h, n ≥ 1, and Xn ↑... Martingale Theory 19 3 SUBMARTINGALES 3.a Adapted stochastic processes Let T be a partially ordered index set It is useful to think of the index t ∈ T as time A stochastic process X on (Ω, F, P ) indexed by T is a family X = (Xt )t∈T of random variables Xt on Ω Alternatively, defining X(t, ω) = Xt (ω), t ∈ T , ω ∈ Ω, we can view X as a function X : T ×Ω → R with F-measurable sections Xt , t ∈ T A T -f... function of c ≥ 0 Consequently, to show that the family F = { Xi | i ∈ I } is uniformly integrable it suffices to show that for each > 0 there exists a c ≥ 0 such that supi∈I E |Xi |; [|Xi | ≥ c] ≤ (b) To show that the family F = { Xi | i ∈ I } is uniformly P -continuous we must show that for each > 0 there exists a δ > 0 such that supi∈I E 1A |Xi | < , for all sets A ∈ F with P (A) < δ This means that... ∈ I, is uniformly absolutely continuous with respect to the measure P (c) From 1.b.0 it follows that each finite family F = { f1 , f2 , , fn } ⊆ L1 (P ) of integrable functions is both uniformly integrable (increase c) and uniformly P continuous (decrease δ) 1.b.2 A family F = { Xi | i ∈ I } of random variables is uniformly integrable if and only if F is uniformly P -continuous and L1 -bounded Proof... distribution with mean m ∈ Rk and covariance matrix C (II.1.a) N (d) = P (X ≤ d) X a standard normal variable in R1 nk (x) = (2π)−k/2 exp − x 2 2 Standard normal density in Rk (II.1.a) xiv Notation Stochastic integrals, spaces of integrands H • X denotes the integral process t (H • X)t = 0 Hs · dXs and is defined for X ∈ S n and H ∈ L(X) L(X) is the space of X-integrable processes H If X is a continuous. .. L(X) The loc integral processes H • X and associated spaces of integrands H are introduced step by step for increasingly more general integrators X: Scalar valued integrators Let M be a continuous local martingale Then µM Doleans measure on (Π, B × F) associated with M (III.2.a) ∞ µM (∆) = EP 0 1∆ (s, ω)d M s (ω) , ∆ ∈ B × F L2 (M ) space L2 (Π, Pg , µM ) of all progressively measurable processes... P -ae ω ∈ Ω t 0 Hs dAs is defined pathwise Assume now that X is a continuous semimartingale with semimartingale decomposition X = A + M (A = uX , M a continuous local martingale, I.11.a) Then L(X) = L1 (A) ∩ L2 (M ) Thus L(X) = L2 (X), if X is a local martingale loc loc loc For H ∈ L(X) set H • X = H • A+H • M Then H • X is the unique continuous semimartingale satisfying (H • X)0 = 0, uH • X = H • uX . HALL/CRC MICHAEL MEYER, Ph.D. Continuous Stochastic Calculus with Applications to Finance Boca Raton London New York Washington, D.C. This book contains. Properties of stochastic integrals with respect to continuous local martingales 142 2.d Integration with respect to continuous semimartingales 147 2.e The stochastic