Appendix B

Answers to Selected Problems

Answers to most of the problems can be found in Appendix C, which is available

on the web via http://www.ma.utexas.edu/users/cup/Answers.

2.1.6 Define T_{n+1} = inf{t > T_{n} : X_{t+} − X_{t} ≠ 0} ∧ t, where t is an instant past which X vanishes. The T_{n} are stopping times (why?), and X is a linear combination of X_{0} · [[0]] and the intervals ((T_{n}, T_{n+1}]]. 2.3.6 For N = 1 this amounts to ζ_{1} ≤ L_{q}ζ_{1} , which is evidently true. Assuming that the inequality holds for N − 1, estimate

ζ_{N}^{2} = ζ_{N−1}^{2} + (ζ_{N} − ζ_{N−1})(ζ_{N} + ζ_{N−1})

≤ L_{q}^{2} ∑_{n=1}^{N}(ζ_{n} − ζ_{n−1})^{2} + (ζ_{N} − ζ_{N−1})(ζ_{N} + ζ_{N−1})

− L_{q}^{2}(ζ_{N} − ζ_{N−1})(ζ_{N} − ζ_{N−1})

= L_{q}^{2} ∑_{n=1}^{N}(ζ_{n} − ζ_{n−1})^{2} + (ζ_{N} − ζ_{N−1})((1 − L_{q}^{2})ζ_{N} + (L_{q}^{2} + 1)ζ_{N−1}) .

Now with L_{q}^{2} = (q+1)/(q−1) we have 1−L_{q}^{2} = −2/(q−1) and L_{q}^{2} +1 = 2q/(q−1), and therefore

(1 − L_{q}^{2})ζ_{N} + (L_{q}^{2} + 1)ζ_{N−1} = \frac{2}{q − 1}( − ζ_{N} + qζ_{N−1}) ≤ 0 .

2.5.17 Replacing F_{n} by F_{n} − p, we may assume that p = 0. Then S_{n} =^{def} ∑_{ν=1}^{n}F_{ν} has expectation zero. Let F_{n} denote the σ-algebra generated by F_{1}, F_{2}, …, F_{n−1} . Clearly S_{n} is a right-continuous square integrable martingale on the filtration {F_{n}}. More precisely, the fact that 𝔼[F_{ν}|F_{μ}] = 0 for μ < ν implies that

𝔼[S_{n}^{2} ] = 𝔼[(∑_{ν=1}^{n}F_{ν} )^{2} ] = 𝔼[∑_{ν=1}^{n}F_{ν}^{2} ] + 2∑_{1≤μ<ν≤n}𝔼[F_{μ}𝔼[F_{ν}|F_{μ}]] ≤ nσ^{2} .

Therefore Z_{n} =^{def} S_{n}/n has ∥Z_{n} ∥_{L^{2}} ≤ σ\sqrt{1/n} ⟶_{n→∞} 0. (Using Chebyscheff’s inequality here we may now deduce the weak law of large numbers.) The strong law says that Z_{n} ⟶_{n→∞} 0 almost surely and evidently follows if we can show that Z_{n} converges almost surely to something as n → ∞. To see that it does we write

Z_{ν} − Z_{ν−1} = \frac{F_{ν}}{ν} − \frac{1}{ν(ν−1)} ∑_{1≤i<ν} F_{i} ,  ν = 2, 3, …,

and set  Z̃_{n} =^{def} ∑_{1≤ν≤n} \frac{F_{ν}}{ν} ,  n = 1, 2, 3, …,

and Ẑ_{1} = 0,  Ẑ_{n} =^{def} ∑_{1<ν≤n} \frac{S_{ν−1}}{ν(ν−1)} ,  n = 2, 3, ….

Then  Z_{n} = Z̃_{n} − Ẑ_{n} ,

and it suffices to show that both Z̃. and Ẑ. converge almost surely. Now Z̃. is an L^{2}-bounded martingale and so has a limit almost surely. Namely, because of the cancellation as above,

𝔼\left[{ \atop Z_{n}^{2}}\right] ≤ ∑_{1≤ν≤n} \frac{𝔼[F_{ν}^{2}]}{ν^{2}} < ∑_{ν<∞} \frac{σ^{2}}{ν^{2}} < ∞ .

As to Ẑ, since Ẑ _{∞} =^{def} ∑_{2≤ν<∞} \frac{|S_{ν−1}|}{ν(ν−1)}

has  𝔼[ Ẑ _{∞}] ≤ ∑_{2≤ν<∞} \frac{∥S_{ν−1}∥_{L^{2}}}{ν(ν−1)} ≤ ∑_{2≤ν<∞} \frac{σ\sqrt{ν−1}}{ν(ν−1)} < ∞ ,

the sum defining Ẑ_{∞} =^{def} lim Ẑ_{n} almost surely converges, even absolutely. 3.8.10 (i) Both (Y, Z) → [Y, Z](ϖ) and (Y, Z) → ^{c}[Y, Z](ϖ) are inner products with associated lengths S[Z](ϖ), σ[Z](ϖ). The claims are Minkowski’s inequalities and follow from a standard argument.

(ii) Take U = V = [[0, T]] in theorem 3.8.9 and apply Hölder’s inequality. 3.8.18 (i) Choose numbers α_{i} > 0 satisfying inequality (3.7.12):

∑_{i=1}^{∞} iα_{i} · X^{i} _{I^{0}} < ∞

(X^{i} is X stopped at i). With each α_{i} goes a δ_{i} > 0 so that |x − x′| ≤ δ_{i} implies

|F (x) − F (x′)| ≤ α_{i}. Set T_{0}^{i} = 0 and T_{k+1}^{i} =^{def} inf{t > T_{k}^{i} : |Z_{t} − Z_{T_{k}^{i}}| ≥ α_{i}}. Then

set

iY_{t}^{η} =^{def} ∑_{k,ν} F_{ν}^{η}(X_{T_{k}^{i}})(X_{T_{k+1}^{i}∧t}^{ν} − X_{T_{k}^{i}∧t}^{ν}) ,  η = 1, …, d .

According to theorem 3.7.26, nearly everywhere ^{i}Y. ⟶_{i→∞} (F (X)∗X). uniformly on bounded time-intervals. Let us toss out the nearly empty set where the limit does not exist and elsewhere select this limit for F (X)∗X. Now let ω, ω′ be such that X.(ω) and X.(ω′) describe the same arc via t ↦ t′. The times T_{k}^{i} may well differ at ω and ω′, in fact T_{k}^{i}(ω′) = (T_{k}^{i}(ω))′, but the values X_{T_{k}^{i}}(ω) and X_{T_{k}^{i}}(ω′) clearly agree. Therefore ^{i}Y_{t}(ω) = ^{i}Y_{t′}(ω′) for all n ∈ ℕ and all t ≥ 0. In the limit (F (X)∗X)_{t}(ω) = (F (X)∗X)_{t′}(ω′).

(ii) We start with the case d = 1. Apply (i) with f(x) = 2x and toss out in addition the nearly empty set of ω ∈ Ω where [W, W]_{t}(ω) ≠ t for some t. Now if W.(ω) and W.(ω′) describe the same arc via t ↦ t′, then [W, W].(ω) = W.^{2}(ω) − 2(W.∗W).(ω) and [W, W].(ω′) = W.^{2}(ω′) − 2(W.∗W).(ω′) describe the same arc also via t ↦ t′. This reads t = t′.

In the case d > 1 apply the foregoing to the components W^{η} of W separately. 3.9.10 (i) Set M_{t} =^{def} ⟨ξ|X_{T +t} − X_{T} ⟩ and assume without loss of generality that M_{0} ≤ a. This continuous local martingale has a continuous strictly increasing square function [M, M]_{t} = ξ_{μ}ξ_{ν}([X^{μ}, X^{ν}]_{T +t} − [X^{μ}, X^{ν}]_{T}) ⟶_{t→∞} ∞ . The time transformation T^{λ} = T^{λ+} =^{def} inf{t : [M, M]_{t} ≥ λ} of exercise 3.9.8 turns M into a Wiener process. The zero-one law of exercise 1.3.47 on page 41 shows that T = T^{±} almost surely.

(ii) In the previous argument replace T by S. The continuous time transformation T^{λ} = T^{λ+} =^{def} inf{t : [M, M]_{t} ≥ λ} turns M_{t} =^{def} ⟨ξ|X_{S+t} −X_{S}⟩ into a standard Wiener process, which exceeds any number, including a±⟨ξ|X_{S}⟩, in finite time (exercise 1.3.48 on page 41). Therefore the stopping times T and T^{±} are almost surely finite. Consider the set X^{−} of paths [S(ω), ∞) ∋ t ↦ X_{t}(ω) that have T (ω) > τ. Each of them stays on the same side of H as X_{S}(ω) for t ∈ [S(ω), τ], in fact by continuity it stays a strictly positive distance away from H during this interval. Any other path X.(ω′) sufficiently close to a path in X^{−} will not enter H during [S(ω′), τ] either and thus will have T (ω′) > τ: the set X^{−} is open.

Next consider the set X^{+} of paths [S(ω), ∞) ∋ t ↦ X_{t}(ω) that have T^{+}(ω) < τ. If X.(ω) ∈ X^{+} , then there exists a σ < τ with ⟨ξ|X_{σ}⟩ > a: X_{S}(ω) and X_{σ}(ω) lie on different sides of H. Clearly if ω′ is such that the path X.(ω′) is sufficiently close to that of ω, then X_{σ}(ω′) will also lie on the other side of H: X^{+} is open as well. After removal of the nearly empty set [T ≠ T^{+}] we are in the situation that the sets {X.(ω) : T (ω)^{>}_{<}τ} are open for all τ: T depends continuously on the path. 3.10.6 (i) Let K^{1}, K^{2} ⊂ H be disjoint compact sets. There are ϕ_{n}^{i} ∈ C_{00}[H] with ϕ_{n}^{i} ↓_{n} K^{i} pointwise and such that ϕ_{1}^{1} and ϕ_{1}^{2} have disjoint support. Then ϕ_{n}^{i}∗β → K^{i}∗β as L^{2}-integrators and therefore uniformly on bounded time-intervals (theorem 2.3.6) and in law. Also, clearly K^{1}∗β and K^{2}∗β are independent, inasmuch as ϕ_{m}^{1}∗β and ϕ_{n}^{2}∗β are. In the next step exhaust disjoint relatively compact Borel subsets B^{1}, B^{2} of H by compact subsets with respect to the Daniell means B ↦ ∥B×[0, n]∥_{β−2}^{∗}, n ∈ ℕ, which are K-capacities (proposition 3.6.5), to see that B^{1}∗β and B^{2}∗β are independent Wiener processes. Clearly ν(B) =^{def} 𝔼[(B∗β)_{1}^{2} ] defines an additive measure on the Borels of H. It is σ-continuous, beacause it is majorized by B ↦ [∥B×[0, 1]∥_{β−2}^{∗}]^{2} , even inner regular. 3.10.9 Let I denote the image in L^{2} =^{def} L^{2}(F_{∞}^{0}[β], ℙ) of 𝔏^{1}[β−2] under the map X̌ ↦ ∫_{0}^{∞} X̌ dβ, and \overline{I} the algebraic sum I ⊕ ℝ. By theorem 3.10.6 (ii), I and then \overline{I} are closed in L^{2} , and their complexifications I_{ℂ}, \overline{I}_{ℂ} are closed in L_{ℂ}^{2}. By the Dominated Convergence Theorem for the L^{2}-mean both vector spaces are closed under pointwise limits of bounded sequences.

For h ∈ E[Ȟ] =^{def} E[H]⊗C_{00}(ℝ_{+}) set M^{h} =^{def} h∗β, which is a martingale with square bracket [M^{h}, M^{h}]_{t} = ∫_{0}^{t} h^{2}(η, s) ν(dη)ds. Then let G^{h} = exp (iM^{h} + [M^{h}, M^{h}]/2) be the Doléans–Dade exponential of iM^{h}. Clearly G_{∞}^{h} = 1 + ∫_{0}^{∞} iG^{h} dM^{h} belongs to \overline{I}_{ℂ}, and so does its scalar multiple exp (iM_{∞}^{h}). The latter form a multiplicative class M contained in \overline{I}_{ℂ} and generating F_{∞}^{0}[β] (page 409). By exercise A.3.5 the vector space \overline{I}_{ℂ} contains all bounded F_{∞}^{0}[β]-measurable functions. As it is mean closed, it contains all of L_{ℂ}^{2}. Thus \overline{I} = L^{2} . 4.3.10 Use exercise 3.7.19, proposition 3.7.33, and exercise 3.7.9. 4.5.8 Inequality (4.5.9) and the homogeneity argument following it show that for any bounded previsible X ≥ 0

μ^{⟨σ⟩}(X) ≤ (μ^{⟨ρ⟩}(X))^{\frac{τ−σ}{τ−ρ}} · (μ^{⟨τ⟩}(X))^{\frac{σ−ρ}{τ−ρ}} ≤ (μ^{⟨ρ⟩} ∨ μ^{⟨τ⟩})(X) .

4.5.9 Since z ↦ (z + ζ)^{p} − z^{p} increases, inequality (4.5.16) can with the help of theorem 2.3.6 be continued as

∥Z_{∞} ∥_{L^{p}}^{p} ≤ (C_{p}^{⋆} Z _{I^{p}} + ζ[Z])^{p} − C_{p}^{⋆p} Z _{I^{p}}^{p} .

Replace Z by X∗Z and take the supremum over X with |X| ≤ 1 to obtain

Z _{I^{p}}^{p} ≤ (C_{p}^{⋆} Z _{I^{p}} + ζ[Z])^{p} − C_{p}^{⋆p} Z _{I^{p}}^{p} ,
or  (1 + C_{p}^{⋆p})^{1/p} Z _{I^{p}} ≤ C_{p}^{⋆} Z _{I^{p}} + ζ[Z]
and  Z _{I^{p}} ≤ c_{p}^{⋄}ζ[Z]
with  c_{p}^{⋄} ≤ ((1 + C_{p}^{⋆p})^{1/p} − C_{p}^{∗})^{−1} ≤ 0.6 · 4^{p} .

4.5.21 Let T = inf{t : A_{t} ≥ a} and S = inf{t : Y_{t} ≥ y}. Both are stopping times, and T is predictable (theorem 3.5.13); there is an increasing sequence T_{n} of finite stopping times announcing T. On the set [Y_{∞} > y, A_{∞} < a], S is finite, Y_{S} ≥ y, and the T_{n} increase without bound. Therefore

[Y_{∞} > y, A_{∞} < a] ≤ \frac{1}{y} · sup_{n} Y_{S∧T_{n}}

and so  ℙ[Y_{∞} > y, A_{∞} < a] ≤ \frac{1}{y} · sup_{n} 𝔼[Y_{S∧T_{n}}]

≤ \frac{1}{y} · sup_{n} 𝔼[A_{S∧T_{n}}] ≤ \frac{1}{y} · 𝔼[A_{∞} ∧ a] .

Applying this to sequences y_{n} ↓ y and a_{n} ↑ a yields inequality (4.5.30). This then implies ℙ[Y = ∞, A ≤ a] = 0 for all a < ∞; then ℙ[Y = ∞, A < ∞] = 0, which is (4.5.31). 4.5.24 Use the characterizations 4.5.12, 4.5.13, and 4.5.14. Consider, for instance, the case of Z^{⟨q⟩}. Let ^{q}′X,^{q}′Ȟ be the quantities of exercise 4.5.14 and its answer

for ′Z. Then Ȟ(y, s) =^{def} ^{q}′Ȟ ∘ C(y, s) = ^{q}′Ȟ(Cy, s) = ⟨^{q}′X_{s}|Cy⟩ = ⟨C^{T}^{q}′X_{s}|y⟩,

where C^{T} : ℓ^{∞}(′d) → ℓ^{∞}(d) denotes the (again contractive) transpose of C. By exercise 4.5.14, the Doléans–Dade measure ′μ of |Ȟ|^{q}∗ȷ_{Z} is majorized by that of Λ^{⟨q⟩}[Z]. But ′μ is the Doléans–Dade measure of Λ^{⟨q⟩}[′Z]! Indeed, the compensator of |Ȟ|^{q}∗ȷ_{Z} = |^{q}′Ȟ ∘ C|^{q}∗ȷ_{Z} = |^{q}′Ȟ|^{q}∗C[ȷ_{Z}] = |^{q}′Ȟ|^{q}∗ȷ_{′Z} is Λ^{⟨q⟩}[′Z]. The other cases are similar but easier. 5.2.2 Let S < T^{μ} on [T^{μ} > 0]. From inequality (4.5.1)

∥|Δ∗Z|_{S}^{⋆} ∥_{L^{p}} ≤ C_{p}^{⋄} · max_{ρ=1,p^{⋄}} ∥(∫_{0}^{S} |Δ|^{ρ} dΛ)^{1/ρ}∥_{L^{p}}

≤ C_{p}^{⋄} · max_{ρ=1,p^{⋄}} ∥(∫_{0}^{μ} δ^{ρ} dλ)^{1/ρ}∥_{L^{p}} = δ · C_{p}^{⋄} max_{ρ=1,p^{⋄}} μ^{1/ρ} .

Letting S run through a sequence announcing T^{μ}, multiplying the resulting inequality ∥|Δ∗Z|_{T^{μ}−}^{⋆}∥_{L^{p}} ≤ δ · C_{p}^{⋄} max_{ρ=1,p^{⋄}} μ^{1/ρ} by e^{−Mμ}, and taking the supremum over μ > 0 produces the claim after a little calculus. 5.2.18 (i) Since e^{pmW_{t}−p^{2}m^{2}t/2} = ℰ_{t}[pmW] is a martingale of expectation one

we have |ℰ_{t}[mW]|^{p} = e^{pmW_{t}−pm^{2}t/2} = ℰ_{t}[pmW] · e^{(p^{2}−p)m^{2}t/2} ,
𝔼[|ℰ_{t}[mW]|^{p}] = e^{(p^{2}−p)m^{2}t/2} , and ∥ℰ_{t}[mW]∥_{L^{p}} = e^{m^{2}(p−1)t/2} .

Next, from e^{|x|} ≤ e^{x} + e^{−x} we get

e^{|mW_{t}|} ≤ e^{mW_{t}} + e^{−mW_{t}} = e^{m^{2}t/2} × (ℰ_{t}[mW] + ℰ_{t}[−mW]) ,

e^{|mW |_{t}^{⋆}} ≤ e^{m^{2}t/2} × (ℰ_{t}^{⋆}[mW] + ℰ_{t}^{⋆}[−mW]) ,
and  ∥e^{|mW |_{t}^{⋆}} ∥_{L^{p}} ≤ e^{m^{2}t/2} × (∥ℰ_{t}^{⋆}[mW]∥_{L^{p}} + ∥ℰ_{t}^{⋆}[−mW]∥_{L^{p}})
by theorem 2.5.19:  ≤ e^{m^{2}t/2} × 2p′ · e^{m^{2}(p−1)t/2} = 2p′ · e^{m^{2}pt/2} .

(ii) We do this with | | denoting the ℓ^{1}-norm on ℝ^{d}. First,

∥e^{|mZ^{⋆}|_{t}} ∥_{L^{p}} = e^{|m|t} × ∏_{η} ∥e^{m|W^{η⋆}|_{t}} ∥_{L^{p}}
by independence of the W^{η}: ≤ e^{|m|t} × (2p′ · e^{m^{2}pt/2})^{d−1}
= (2p′)^{d−1} × e^{(|m|+(d−1)m^{2}p/2)·t} .
Thus  ∥e^{|mZ^{⋆}|_{t}} ∥_{L^{p}} ≤ A_{p,d} × e^{M_{d,m,p}·t} .  (1)
Next,  ∥|Z^{⋆}|_{t}^{r} ∥_{L^{p}} = ∥|Z^{⋆}|_{t}∥_{L^{rp}}^{r} ≤ (t + ∑_{η}∥|W^{η⋆}|_{t}∥_{L^{rp}})^{r}
= (t + (d−1) · ∥|W |_{t}^{⋆}∥_{L^{rp}})^{r}
by theorem 2.5.19:  ≤ (t + (d−1)(rp)′ · ∥|W |_{t}∥_{L^{rp}})^{r}
≤ 2^{r′}(t^{r} + (d−1)^{r}(rp)^{′r} · ∥|W |_{t}×∥_{L^{rp}}^{r})
by exercise A.3.47 with σ = \sqrt{t}:= 2^{r′}(t^{r} + (d−1)^{r}(rp)^{′r}Γ_{p,r} · t^{r/2}) .
Thus  ∥|Z^{⋆}|_{t}^{r} ∥_{L^{p}} ≤ B_{r}t^{r} + B_{d,r,p}t^{r/2} .  (2)

Applying Hölder’s inequality to (1) and (2), we get

∥|Z^{⋆}|_{t}^{r} · e^{|mZ^{⋆}|_{t}} ∥_{L^{p}} ≤ (B_{r}t^{r} + B_{d,r,2p}t^{r/2}) × (A_{2p,d}e^{M_{d,m,p}t})
= t^{r/2}A_{2p,d}(B_{d,r,2p} + B_{r}t^{r/2}) × e^{M_{d,m,2p}t} :

we get, for suitable B′ = B′_{d,p,r}, M′ = M′_{d,m,p,r}, the desired inequality

∥|Z^{⋆}|_{t}^{r} · e^{|mZ^{⋆}|_{t}} ∥_{L^{p}} ≤ B′ · t^{r/2} e^{M′t} .

5.3.1 𝔖_{p,M}^{⋆n} is naturally equipped with the collection 𝔑^{∘} of seminorms _{p^{∘},M^{∘}}, where 2 ≤ p^{∘} < p and M^{∘} > M. 𝔑^{∘} forms an increasing family with pointwise limit _{p,M}. For 0 ≤ σ ≤ 1 set u^{σ} =^{def} u + σ(v−u) and X^{σ} =^{def} X + σ(Y −X). Write F for F_{η}, etc. Then the remainder F [v, Y] − F [u, X] − D_{1}F [u, X]·(v−u) − D_{2}F [u, X]·(Y −X) becomes, as in example A.2.48,

RF [u, X; v, Y] = ∫_{0}^{1} (Df(u^{σ}, X^{σ}) − Df(u, X)) · ( \begin{matrix}v−u \\ Y −X \end{matrix} ) dσ .

With R^{σ}f =^{def} Df(u^{σ}, X^{σ}) − Df(u, X), 1/p^{∘} = 1/p + 1/r, and R _{pp^{∘}} denoting the operator norm of a linear operator R : ℓ^{p}(k+n) → ℓ^{p^{∘}}(n) we get

|RF [u, X; v, Y]_{T^{λ}−}^{⋆}|_{p^{∘}} ≤ C · r_{λ} · (|v−u| + ||Y −X|_{T^{λ}−}^{⋆}|_{p}) ,
where  r_{λ} =^{def} sup_{t<T^{λ}} sup_{0≤σ≤1} R^{σ}f _{pp^{∘}} ,

and where C is a suitable constant depending only on k, n, p, p^{∘}. Now r_{λ} is a bounded random variable and converges to zero in probability as |v − u| + Y −X _{p,M}^{⋆} → 0. (Use that the T^{λ} of definition (5.2.4) on page 283 are bounded; then X_{t}^{σ} ranges over a relatively compact set as [0 ≤ σ ≤ 1] and 0 ≤ t ≤ T^{λ}.) In other words, the uniformly bounded increasing functions λ ↦ ∥r_{λ} ∥_{r} converge pointwise – and thus uniformly on compacta – to zero as |v −u|+ Y −X _{p,M}^{⋆} → 0. Therefore the first factor on the right in

RF [u, X; v, Y] _{p^{∘},M^{∘}} ≤ C sup_{λ} e^{(M−M^{∘})λ}∥r_{λ} ∥_{r} · (|v−u| + Y −X _{p,M}^{⋆})

converges to zero as |v−u|+ Y −X _{p,M}^{⋆} → 0, which is to say RF [u, X; v, Y] _{p^{∘},M^{∘}} = o(|v − u| + Y −X _{p,M}^{⋆}). 5.4.19 (ii) For fixed μ and δ set k =^{def} ⌈μ/δ⌉, and λ_{i} =^{def} iδ and T_{i} =^{def} T^{λ_{i}} for i = 0, 1, …, k. Then λ_{k−1} < μ ≤ λ_{k}. Let Δ_{i}^{⋆} denote the maximal function of the difference of the global solution at T_{i}, which is X_{T_{i}} = Ξ[C, Z]_{T_{i}}, from its Ξ′-approximate X′_{T_{i}}. Consider an s ∈ [T^{λ_{i}}, T^{λ_{i+1}}].

Since  X′_{s} − X_{s} = Ξ′[X′_{T_{i}}, Z_{s}−Z_{T_{i}}] − Ξ′[X_{T_{i}}, Z_{s}−Z_{T_{i}}]
+ Ξ′[X_{T_{i}}, Z_{s}−Z_{T_{i}}] − Ξ[X_{T_{i}}, Z_{s}−Z_{T_{i}}] ,
5.4.17 gives  ∥Δ_{i+1}^{⋆}∥_{L^{p}} ≤ ∥Δ_{i}^{⋆}∥_{L^{p}} × e^{L′δ} + (∥|X|_{T_{i}}^{⋆}+1∥_{L^{p}}) × (\underline{M}δ)^{r}e^{\underline{M}δ} ,
which implies  |Δ_{k}^{⋆}| ≤ (|X|_{T_{k}}^{⋆}+1) × (\underline{M}δ)^{r}e^{\underline{M}δ} · ∑_{0≤i<k}e^{iL′δ}
for X. ∈ 𝔖_{p,M}, as λ_{k} = kδ: ≤ ( X. _{M}^{⋆}e^{Mλ_{k}} + 1) × (\underline{M}δ)^{r}e^{\underline{M}δ} · \frac{e^{L′kδ} − 1}{e^{L′δ} − 1}

by (5.2.23), as δ → 0:  ≤ \frac{2}{1−γ} ( ^{0}C _{M}^{⋆} + 1) e^{Mλ_{k}} × (\underline{M}δ)^{r}e^{\underline{M}δ} · k e^{L′λ_{k}}
since k = λ_{k}/δ:  ≤ const ( C _{M}^{⋆} + 1)\underline{M}^{r}e^{\underline{M}δ}δ^{r−1} × λ_{k}·e^{(M+L′)λ_{k}}
≤ \overline{B}·(∥C∥_{L^{p}}+1) × δ^{r−1}·e^{\overline{M}λ_{k}}

for suitable \overline{B} = \overline{B}[f; Ξ′] and \overline{M} = \overline{M}[f; Ξ′] > M+L′. 5.5.7 Let ℙ, \overline{ℙ} ∈ 𝔓. Thanks to the uniform ellipticity (5.5.17) there exist bounded functions h^{η} so that f_{0} = − ∑_{η} f_{η} · h^{η}. Then M =^{def} h∗W is a martingale under both ℙ and \overline{ℙ}, and by exercise 3.9.12 so is the Doléans–Dade exponential G′ of M. The Girsanov theorem 3.9.19 asserts that W′ =^{def} W_{t} + ∫._{0} h_{s} ds is a Wiener process under the probabilities ℙ′ and \overline{ℙ}′ that on every F_{t} agree with G′_{t}ℙ and G′_{t}\overline{ℙ}, respectively. Now X satisfies equation (5.5.20) with W′ replacing W , and therefore by assumption ℙ′ = \overline{ℙ}′. This clearly implies ℙ = \overline{ℙ}.

5.5.13 Let s ≤ t ≤ t′. Then by Itô’s formula

u(t′−t, X_{t}) = u(t′−s, X_{s}) − ∫_{s}^{t} u̇(t′−σ, X_{σ}) dσ

+ ∫_{s}^{t} u_{;ν}(t′−σ, X_{σ}) dX_{σ}^{xν} + ∫_{s}^{t} Au(t′−σ, X_{σ}) dσ

= u(t′−s, X_{s}) + ∫_{s}^{t} u_{;ν}(t′−σ, X_{σ}) dX_{σ}^{xν} .

Taking the conditional expectation under F_{s} exhibits t ↦ u(t′−t, X_{t}) as a bounded local martingale. 5.5.14 It suffices to consider equation (5.5.20). Recall that 𝔓 is the collection of all probabilities on 𝒞^{n} under which the process W_{t} of (5.5.19) is a standard Wiener process. Let ℙ, \overline{ℙ} ∈ 𝔓. From ϕ_{0}, ϕ_{1}, …, ϕ_{k} ∈ C_{b}^{∞}(ℝ^{n}) and 0 = t_{0} < t_{1} < ⋯ < t_{k} make the function Φ =^{def} ϕ_{0}(X_{0}^{x}) · ϕ_{1}(X_{t_{1}}^{x}) ⋯ ϕ_{k}(X_{t_{k}}^{x}) on the path space 𝒞^{n}. Their collection forms a multiplicative class that separates the points of 𝒞^{n}. Since path space is polish and consequently every probability on it is tight (proposition A.6.2), or simply because the functions Φ generate the Borel σ-algebra on path space, ℙ = \overline{ℙ} will follow if we can show that 𝔼 = \overline{𝔼} on the functions Φ (proposition A.3.12). This we do by induction in k. The case k = 0 is trivial. Note that the equality 𝔼[Φ] = \overline{𝔼}[Φ] on Φ made from smooth functions ϕ_{i} persists on functions Φ made from continuous, even bounded Baire, functions ϕ_{i}, by the usual sequential closure argument. To propel ourselves from k to k + 1 let u denote a solution to the initial value problem u̇ = Au with u(0, x) = ϕ_{k+1}(x) and write

Φ =^{def} ϕ_{0}∘X_{0}^{x} · ϕ_{1}∘X_{t_{1}}^{x} ⋯ ϕ_{k}∘X_{t_{k}}^{x}. Then, with t = t′ = t_{k+1} and s = t_{k}, exer-

cise 5.5.13 produces

𝔼[ϕ_{k+1}(X_{t_{k+1}}^{x})|F_{t_{k}}] = 𝔼[u(0, X_{t_{k+1}}^{x})|F_{t_{k}}]
= 𝔼[u(t_{k+1}−t_{k}, X_{t_{k}}^{x})]
and so  𝔼[Φ · ϕ_{k+1}(X_{t_{k+1}}^{x})] = 𝔼[Φ · u(t_{k+1}−t_{k}, X_{t_{k}}^{x})]
= \overline{𝔼}[Φ · u(t_{k+1}−t_{k}, X_{t_{k}}^{x})]  (∗)
by the same token:  = \overline{𝔼}[Φ · ϕ_{k+1}(X_{t_{k+1}}^{x})] .

At (∗) we used the fact that the argument of the expectation is a k-fold product of the same form as Φ, so that the induction hypothesis kicks in. A.2.23 Let U_{n} be the set of points x in F that have a neighborhood V_{n}(x) whose intersection with E has ρ-diameter strictly less than 1/n. The U_{n} clearly are open and contain E. Their intersection is E. Indeed, if x ∈ ⋂ U_{n}, then the sets E ∩ V_{n}(x) form a Cauchy filter basis in E whose limit must be x. A.3.5 The family of all complex finite linear combinations of functions in M ∪ {1} is a complex algebra A of bounded functions in V that is closed under complex conjugation; the σ-algebra it generates is again M^{Σ} . The real-valued functions in A form a real algebra A_{0} of bounded functions that again generates M^{Σ} . It is a multiplicative class contained in the bounded monotone class V_{0} of real-valued functions in V. Now apply theorem A.3.4 suitably.

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Index of Notations

1_{A} = A  the indicator function of A  365
Φ[μ] = μ ∘ Φ^{−1}  the image of the measure μ under Φ  405
A[F]  the F-analytic sets  432
A_{∞}  the algebra ⋃_{0≤t<∞} F_{t}  22
A_{∞σ}  the countable unions of sets in A_{∞}  35
μ⋆ν  the convolution of μ and ν  413
V^{∗}  dual of the topological vector space V  381
X∗Z  the indefinite Itô integral  133
Z^{⋆}  the maximal process of Z  26
* ϕ, * μ  ϕ , μ reflected through the origin  410
E_{e}^{∗}  the universal completion of E  407
B  the base space [0, ∞) × Ω  22
B̌  product of auxiliary space with base space B  109
η̌ = (η, ϖ)  = (η, s, ω), the typical point of B̌ =^{def} H × B  172
B^{∗}(E) , B^{•}(E)  the Baire & Borel sets or functions of E  391
b(p)  = \frac{2}{π} ∫_{0}^{∞} ξ^{p−1} sin ξ dξ  458
[[[0, t]]]  =^{def} ℝ_{∗}^{d} × [[0, t]]  181
[[[0, T]]]  =^{def} H × [[0, T]]  172
A^{c}  the complement of A  373
C_{b}(E)  the continuous bounded functions on E  376
C_{0}(B)  the continuous functions vanishing at infinity  366
C_{00}(B)  the continuous functions with compact support  370
μ^{Γ}  the characteristic function of μ for Γ  410
C_{b}^{k}  the bounded functions with k bounded cont. partials 281
C^{k}(D)  the k-times continuously diff’ble functions on D  372
𝒞^{d}  the path space C_{ℝ^{d}}[0, ∞)  20
𝒞 = 𝒞^{1}  the path space C[0, ∞)  14
δ^{ηθ}  the Kronecker delta  19
δ_{s}  the Dirac measure, or point mass, at s  398

ΔX  the jump process of X ∈ 𝔇 (ΔX_{0} =^{def} 0)  25
D^{λ}F [u]  the λth (weak) derivative of F at u  305
dF  the measure with distribution function F  406
d F = |dF |  its variation  406
𝔇 = 𝔇[F.]  the càdlàg adapted processes  24
𝒟, 𝒟^{d}, 𝒟_{E}  canonical space of càdlàg paths  66
𝔼^{x}  𝔼^{x}[F] = ∫ F dℙ^{x}  351
𝔼, 𝔼^{ℙ}  expectation under the prevailing probability, ℙ  32
𝔼[f|Φ], 𝔼[f|Y]  the conditional expectation of f given Φ, Y  407
E = E[F.]  the elementary stochastic integrands  46
E_{1}  the unit ball {X ∈ E : −1 ≤ X ≤ 1} of E  51
E_{+}^{↑}  the pointwise suprema of sequences in E  88
E^{σ}, (E^{d})^{σ}  the sequential closure of E, E^{d}  392
E_{00}  the E-confined functions in E  369
\overline{E}_{00}  the confined uniform closure of E_{00}  370
E_{00}^{σ}  the E-confined functions in E^{σ}  393
E^{ℙ} = E[F.^{ℙ}]  the elementary integrands of the natural enlargement 57
ϵ  denoting measurability, as in f ∈ F/G  391
ϵ  “is member of” or “is measurable on”  23
=  denotes near equality and indistinguishability  35
0F  coupling coefficients adjusted so that ^{0}F [0] = 0  272
0C  initial condition adjusted correspondingly  272
F_{+}  the positive elements of F  363
F_{t}  the past or history at time t  21
F_{T}  the past at the stopping time T  28
F.  the filtration {F_{t}}_{0≤t≤∞}  21
F_{.−}  the left-continuous version of F.  123
F_{.+}  the right-continuous version of F.  37
F.^{0}[Z]  the basic filtration of the process Z  23
F.[Z]  the natural filtration of the process Z  39
F.^{0}[𝒟_{E}], F_{.+}^{0}[𝒟_{E}]  basic, canonical filtration of path space  66
F.[𝒟_{E}]  the natural filtration of path space  67
F_{δ}  the intersections of countable subfamilies of F  432
F_{∞}, F_{∞}^{∗}  the σ-algebra ⋁_{0≤t<∞} F_{t}, its universal completion  22
F_{T−}  the strict past of T  120
𝔉[⌈⌈.⌉⌉^{∗}]  the processes finite for ⌈⌈.⌉⌉^{∗}  97
F_{σ}  the unions of countable subfamilies of F  432
F_{σδ}  the intersections of countable subfamilies of F_{σ}  432
F.^{ℙ}, F.^{𝔓}  the ℙ, 𝔓-regularization of F.  37

F_{.+}^{𝔓}  the natural enlargement of a filtration  39
G_{δ}  the intersections of countable subfamilies of G  441
G  the open sets of the topological space at hand  441
K_{σ}  the unions of countable subfamilies of K  441
K  the compact sets of the topological space at hand  441
F̃  a predictable envelope of F.  125
S^{Δ} =^{def} S ˙∪ {Δ}  the one-point compactification of S  374
Ȟ =^{def} H × [0, ∞)  the product of auxiliary space with time  177
h_{0}  prototypical sure Hunt function y ↦ |y|^{2} ∧ 1  180
h′_{0}  y ↦ ∫_{[|ζ|≤1]} |e^{i⟨ζ|y⟩} − 1|^{2} dζ, another one  182
η_{p,q}(I)  a factorization constant  192
∫ X dZ  the elementary integral for vectors  56
∫ X dZ  the elementary integral  47
∫_{A} F = ∫ A·F  the integral over the set A of the function F  105
∫ X dZ  the (extended or Itô) integral for vectors  110
∫ X dZ  the (extended or Itô) stochastic integral  99
∫ X dZ  the (extended) stochastic integral revisited  134
∫_{0}^{T} X dZ  its value at T ∈ 𝔗  134
X∗Z  the indefinite Itô integral for vectors  134
∫ XδZ  the Stratonovich integral  169
X∘Z  the indefinite Stratonovich integral  169
∫_{0}^{T} G dZ  ∫_{0}^{T} G dZ =^{def} ∫ G dZ^{T}  131
∫_{S+}^{T} G dZ  ∫_{0}^{T} G · ((S, ∞)) dZ  131
ȷ_{Z}  the jump measure of Z  181
H∗ȷ_{Z}  the indefinite integral of H against jump measure  181
L^{∞}  the essentially bounded measurable functions  448
L^{p} = L^{p}(ℙ)  the space of p;ℙ-integrable functions  33
L^{0}, L^{0}(F_{t}, ℙ)  (classes of) measurable a.s. finite functions  33
M_{∞}  the limit of the martingale M at infinity  75
| |_{p} = ∥ ∥_{ℓ^{p}}  |x|_{p} =^{def} ( ∑_{ν} |x^{ν}|^{p})^{1/p}, 0 < p < ∞  364
| |_{∞} = ∥ ∥_{ℓ^{∞}}  |x|_{∞} =^{def} sup_{ν} |x^{ν}|  364
| |  any of the norms | |_{p} on ℝ^{n}  364
⟨x|y⟩  the inner product of vectors x and y  238
ℓ^{p}  the vectors or sequences x = (x^{ν}) with |x|_{p} < ∞  364
ℓ^{0} =^{def} ℝ^{ℕ}  the Fréchet space of scalar sequences  364
ℒ  the left-continuous paths with finite right limits  24
𝔏 = 𝔏[F.]  the collection of adapted maps Z : Ω → ℒ  24
𝔏^{1}[⌈⌈.⌉⌉^{∗}] & 𝔏^{1}[Z−p] the ⌈⌈.⌉⌉^{∗}- & Z−p-integrable processes  99

𝔏^{1}[ζ−p] = 𝔏^{1}[\begin{matrix}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{matrix}_{ζ−p}^{∗}] the ζ−p-integrable processes  175
Λ^{⟨q⟩}[Z]  the previsible controller for Z  238
𝔐^{∗}(E) & 𝔐.(E)  the σ-additive & order-continuous measures on E  421
𝔐^{∗}[E]  the σ-additive measures on E  406
⋁  ⋁ F = supremum or span of the family F  22
∧ & ∨  a ∨ b & a ∧ b: smaller & larger of a, b  364
∥ ∥_{E}  the quasinorm on the quasinormed space E  381
∥ ∥_{E}  the sup–norm on the space E of functions  188
∥I ∥ = ∥I ∥_{L(E,F)}  = sup{∥I(x)∥_{F} : x ∈ E, ∥x∥_{E} ≤ 1}  381
∥u∥ = ∥u∥_{L(E,F)}  = sup{∥u(x)∥_{F} : x ∈ E, ∥x∥_{E} ≤ 1}  381
M^{g}  the martingale M.^{g} = 𝔼[g|F.]  72
∥f ∥_{p}^{∗} = ∥f ∥_{p;ℙ}^{∗}  its mean ∥f ∥_{L^{p}(ℙ)}^{∗} =^{def} (∫^{∗} |f|^{p} dℙ)^{1/p}  452
∥f ∥_{p} = ∥f ∥_{L^{p}(ℙ)}  =^{def} (∫ |f|^{p} dℙ)^{1/p}  33
\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z−p}  the semivariation for \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{p}  53
∥ ∥_{Z−p}^{∗}  the Daniell extension of ∥.∥_{Z−p}  88
\begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{p}^{∗} = \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{p;ℙ}^{∗}  its mean \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{L^{p}(ℙ)}^{∗} =^{def} (∫^{∗}|f|^{p} dℙ)^{1/p ∧1}  452
\begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{p} = \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{L^{p}(ℙ)}  = \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{L^{p}(ℙ)} =^{def} (∫ |f|^{p} dℙ)^{1/p ∧1}  33
\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}  a subadditive mean  95
ζ^{h,t} _{I^{p}}= ζ^{h,t} _{I^{p}[ℙ]}  integrator (quasi)norms of the random measure ζ 173
Z _{I^{p}}= Z _{I^{p}[ℙ]}  integrator (quasi)norms of the integrator Z  55
\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z−p}^{∗} = \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z−p;ℙ}^{∗}  the Daniell mean  88
Z _{I^{p}} = Z _{I^{p}[ℙ]}  =^{def} sup {∥∫ X dZ ∥_{L^{p}(ℙ)} : X ∈ E , |X| ≤ 1}  55
Z _{I^{p}} = Z _{I^{p}[ℙ]}  =^{def} sup {∥∫ X dZ ∥_{L^{p}(ℙ)} : X ∈ E , |X| ≤ 1}  56
∥ ∥_{[α]}^{∗}  the corresponding mean  452
∥ ∥_{[α]}  ∥f ∥_{[α]} =^{def} inf{λ > 0 : ℙ[|f| > λ] ≤ α}  34
∥ ∥_{Z−[α]}  the corresponding semivariation  53
∥ ∥_{Z−[α]}^{∗}  the Daniell extension of ∥.∥_{Z−[α]}  88
[α]  the integrator size according to ∥ ∥_{[α]}  55
p,M, _{p,M}^{⋆}  Picard Norms  283
μ_{Z}  the Doléans–Dade measure of Z  222
\begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{0} = \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{0;ℙ}  the metric inf\{λ : ℙ[|f| ≥ λ] ≤ λ\} on L^{0}  34
F^{μ}  μ-completion of F  414
O( . ), o( . )  big O and little o  388
(Ω, F.)  the underlying filtered measurable space  21
ϖ  the typical point (s, ω) of ℝ_{+} × Ω  22
𝔓  the pertinent probabilities  32
P_{00}  the bdd. predictable processes with bdd. carrier  128
P_{b}  the bounded predictable processes  135
𝔓[Z]  the probabilities for which Z is an L^{0}-integrator  61

P̌ =^{def} B^{•}(H) ⊗ P  = (E[H]⊗E[F.])^{σ}, predictable random functions  172
𝔓^{∗}(E) = 𝔐_{1,+}^{∗}(E)  the probabilities on C_{b}(E)  421
𝔓.(E) = 𝔐._{1,+}(E)  the order-continuous probabilities on E  421
p^{⋄} = p^{⋄}[Z]  p if Z jumps, 2 otherwise  238
1^{⋄} = 1^{⋄}[Z]  2 if Z is a martingale, 1 otherwise  238
P = P[F.]  the predictable processes or σ-algebra  115
P^{ℙ}  the processes previsible with ℙ  118
ℝ_{+}  the positive reals, i.e., the reals ≥ 0  363
ℝ_{∗}^{d}  punctured d-space ℝ^{d} \ {0}  363
T_{A}  the reduction of T ∈ 𝔗 to A ∈ F_{T}  31
0_{A}  the stopping time 0 reduced to A ∈ F_{0}  48
\overline{ρ}(r, s)  the arctan metric on \overline{ℝ}  364
\overline{ℝ}  the extended reals \{−∞\} ∪ ℝ ∪ \{∞\}  363
σ, E^{σ}, E σ ; \overline{ℝ}  ^{σ} denotes sequential closure, of E  392
cV,^{j}V  continuous & jump part of finite variation process V  69
c̃Z,^{r}Z  continuous martingale part of Z, rest Z −^{c̃}Z  235
s̃Z &^{l}Z  small-jump martingale part & large-jump part of Z 237
v̂Z  a continuous finite variation part of Z  237
pZ  the part of Z supported on a sparse previsible set  235
qZ  the quasi-left-continuous rest Z −^{p}Z  235
[Z, Z],^{c}[Z, Z] &^{j}[Z, Z] square bracket, its continuous & jump parts  148
[Y, Z],^{c}[Y, Z] &^{j}[Y, Z] square bracket, its continuous & jump parts  150
𝔖_{p,M}^{⋆n}  processes with finite Picard Norm _{p,M}^{⋆}  283
S[Z] & σ[Z]  square function & continuous square function of Z  148
S  Schwartz space of C^{∞}-functions of fast decay  269
σ(C_{b}^{∗}(E), C_{b}(E))  the topology of weak convergence of measures  421
Z^{T}  Z_{s}^{T} = Z_{T ∧s} is the process Z stopped at T ∈ 𝔗  28
Z^{S}  the S-scalæfication of Z  139
𝔗 = 𝔗[F.]  the F.-stopping times  27
[[T]] = [[T, T]]  the graph of the stopping time T  28
T. : λ ↦ T^{λ}  the time transformation for Z  239
X_{.−} & X_{.+}  left- & right-continuous version of X  24
V^{μ}  the Doléans–Dade process of μ  222
z , μ  variation of distribution function z, or measure μ  45
dz = |dz|  the variation measure of the measure dz  45
ĵ_{Z} & j̃_{Z}  compensator & compensatrix of jump measure  232
𝕎  Wiener measure  16
\underline{W}  Wiener process as a C[0, ∞)-valued random variable  14
ḟ  the equivalence class of f mod negligible functions  13

Z_{∞} = Z_{∞−}  the limit (possibly ±∞) of Z at ∞  27
≪. & ≈.  local absolute continuity & local equivalence  40
≪  denoting absolute continuity  407
Z.(ω)  the path s ↦ Z_{s}(ω)  23
⇒  denoting weak convergence of measures  421
Z_{T}  the random variable ω ↦ Z_{T (ω)}(ω).  28
Z_{s}  the function ω ↦ Z(s, ω)  23
ζ^{T} = [[[0, T]]]ζ  the random measure ζ stopped at T  173
ζ̂, ζ̃  previsible & martingale parts of random measure ζ  231
Ẑ, Z̃  previsible & martingale parts of the integrator Z  221

Symbols

Index

The page where an item is defined appears in boldface.

A full index is at http://www.ma.utexas.edu/users/cup/Indexes

A

absolute-homogeneous, 33, 54, 380 absolutely continuous, 407

locally, 40 accessible stopping time, 122 action, differentiable, 279, 326 adapted, 23, 25

map of filtered spaces, 64, 317, 348 process, 23, 28

adaptive, 7, 141, 281, 317, 319 a.e., almost everywhere, 32, 96

meaning Z-a.e., 129 algorithm

computing integral pathwise, 140 solving a markovian SDE, 311 solving a SDE pathwise, 7, 314

a.s., almost surely, ℙ-a.s., 32 ambient set, 90 analytic set, 432 analytic space, 441 approximate identity, 447 approximation, numerical – see

method, 280 arctan metric, 110, 364, 369, 375 Ascoli–Arzela theorem, 7, 385, 428 ASSUMPTIONS

on the data of a SDE, 272

the filtration is natural (regular and right-continuous), 58, 118, 119, 123, 130, 162, 221, 254, 437

the filtration is right-continuous,

140 atom of an algebra of sets, 391 auxiliary base space, 172 auxiliary space, 171, 172, 239, 251 B

backward equation, 359

Baire & Borel, function, set & σ-algebra, 391, 393

Banach space

-valued integrable functions, 409 base space

auxiliary, 172 , 22, 40, 90, 172 basic filtration, 18

of a process, 19, 23 of Wiener process, 18 on path space, 66, 445

basis of neighborhoods, 374 of a uniformity, 374 bigger than (≥), 363 Big O, 388 Blumenthal, 352 Borel σ-algebra on path space, 15 bounded

I^{p}[ℙ]-bounded process, 49

linear map, 452 polynomially, 322 process of — variation, 67 stochastic interval, 28 subset of a topological vector space,

379, 451 subset of L^{p}, 451 variation, 45

(B-p), 50, 53 bracket

previsible, oblique or angle —, 228 square —, 148, 150

Brownian motion, 9, 12, 18, 80 Burkholder, 84, 213 C càdlàg, càglàd, 24 canonical

decompositions of an integrator,

221, 234, 235 filtration on path space, 66, 335 Wiener process, 16, 58

canonical representation, 66 of an integrator, 14, 67 of a random measure, 177 of a semigroup, 358

capacity, 124, 252, 398, 434 Caratheodory, 395 carrier of a function, 369 Cauchy

filter, 374

random variable & distribution, 458 Cauchy–Schwarz inequality, 150 Central Limit Theorem, 10, 423 Chapman–Kolmogorov equations, 465 characteristic function, 13, 161, 254,

410, 430, 458 of a Gausssian, 419 of a Lévy process, 263 of Wiener process, 161

characteristic triple, 232, 259, 270 chopping, 47, 366 Choquet, 435, 442 class C_{b}^{k} etc, 281 closure

sequential, 392

topological of a set, 373 commuting flows, 279 commuting vector fields, 279, 326 compact, 373, 375

paving, 432 compensated part of an integrator, 221 compensator, 185, 221, 230, 231 compensatrix, 221 complete

uniform space, 374

universally — filtration, 22 completely regular, 373, 376, 421 completion

of a filtration, 39

of a uniform space, 374

universal, 22, 26, 407, 436

with respect to a measure, 414 E-completion, 369, 375 completely (pseudo)metrizable, 375 compound Poisson process, 270 conditional expectation, 71, 72, 408 condition(s) (IC-p), 89 (B-p), 50, 53 (RC-0), 50 the natural, 39

confined, 394 E-confined, 369 uniform limit, 370 conjugate exponents, 76, 449 conservative

generator, 466, 467

semigroup, 268, 465, 466, 467 consistent family, 164, 166, 401

full, 402 continuity along increasing sequences, 95, 124, 137, 398, 452

of a capacity, 434

of E, 89 of E_{+} , 94 of E, 106 of E_{+}^{↑}, 90, 95, 125

Continuity Theorem, 254, 423 continuous martingale part of a Lévy process, 258 of an integrator, 235 contractivity, 235, 251, 407

modulus, 274, 290, 299, 407

strict, 274, 275, 282, 291, 297, 407 control

previsible, of integrators, 238

previsible, of random measures, 251 sure, of integrators, 292

controller the, for random measures, 251 the previsible, 238, 283, 294

CONVENTIONS, 21 about ±∞, 363 about positivity, negativity etc., 363 = denotes near equality and indis-

tinguishability, 35 Einstein convention, 8, 146 ∈ meaning “measurable on”, 23,

391 ∫_{0}^{T} G dZ = ∫ G dZ^{T} , 131 ∫_{0}^{T} G dZ is the random variable

(G∗Z)_{T} , 134 integrators have right-continuous paths with left limits, 62

ℙ is understood, 55 ΔX_{0} = X_{0} , 25 X_{0−} = 0, 24 processes are only a.e. defined, 97 CONVENTIONS (cont’d)

re Z-negligibility, Z-measurability,

Z-a.e., 129 sets are functions, 364 [Y, Z]_{0} = ^{j}[Y, Z]_{0} = Y_{0}·Z_{0} , 150 [Z, Z]_{0} = ^{j}[Z, Z]_{0} = Z_{0}^{2} , 148

convergence

\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-a.e. & Z−p-a.e., 96

in law or distribution, 421 in mean, 6, 96, 99, 102, 103, 449 in measure or probability, 34 largely uniform, 111 of a filter, 373 uniform — on large sets, 111 weak, of measures, 421

convex function, 147, 408 function of an L^{0}-integrator, 146 set, 146, 188, 197, 379, 382

convolution, 117, 196, 199, 275, 288,

373, 413 semigroup, 254

core, 270, 464 countably generated

algebra or vector lattice, 367 , 100, 377

countably subadditive, 87, 90, 94, 396 coupling coefficient

autologous, 288, 299, 305 autonomous, 287 , 1, 2, 4, 8, 272 depending on a parameter, 302, 305 endogenous, 289, 314, 316, 331 instantaneous, 287 Lipschitz, 285 Lipschitz in Picard norm, 286 markovian, 272, 321 non-anticipating, 272, 333 of linear growth, 332 randomly autologous, 288, 300 strongly Lipschitz, 285, 288, 289

Courrège, 232 covariance matrix, 10, 19, 161, 420 cross section, 204, 331, 378, 436, 438,

440 cumulative distr. function, 69, 406

D Daniell mean

the, 89, 95, 109, 124, 174 , 87, 397 maximality of, 123, 135, 217, 398

Daniell’s method, 87, 395 Davis, 213 DCT: Dominated Convergence Theo-

rem, 5, 52, 103 debut, 40, 127, 437 decompositions, canonical,

of an integrator, 221, 234, 235

of a Lévy process, 265 decreasingly directed, 366, 398 Dellacherie, 432 derivative, 388

higher order weak, 305

tiered weak, 306 weak, 302, 390

deterministic, not depending on chance

ω, 7, 47, 123, 132 differentiable

action, 279, 326

, 388

l-times weakly, 305 weakly, 278, 390 diffusion coefficient, 11, 19 Dirac measure, 41, 398 disintegration, 231, 258, 418 dissipative, 464 distinguish processes, 35 distribution

cumulative — function, 406

, 14, 405

Gaussian or normal, 419 distribution function, 45, 87, 406 of Lebesgue measure, 51 random, 49 stopped, 131 Doléans–Dade

exponential, 159, 163, 167, 180, 186,

219, 326 measure, 222, 226, 241, 245, 247,

248, 249, 251 process, 222, 226, 231, 245, 247, 249

Donsker, 426

Doob, 60, 75, 76, 77, 223

Doob–Meyer decomposition, 187, 202, 221, 228, 231, 233, 235, 240, 244, 247, 254, 266 δ-ring, 104, 394, 409 drift, 4, 72, 272 driver, 11, 20, 169, 188, 202, 228, 246

of a stochastic differential equation,

4, 6, 17, 271, 311 d-tuple of integrators see vector of —, 9

dual of a topological vector space, 381 E effect, of noise, 2, 8, 272 Einstein convention, 146 elementary

integral, 7, 44, 47, 99, 395 integrand, 7, 43, 44, 46, 47, 49, 56,

79, 87, 90, 95, 99, 111, 115, 136, 155, 395

stochastic integral, 47 stopping time, 47, 61

endogenous coupling coefficient, 314,

316, 331 enlargement

natural, of a filtration, 39, 57, 101,

440 usual, of a filtration, 39

envelope, 125, 129 Z-envelope, 129 measurable, 398 predictable, 125, 129, 135, 337

equicontinuity, uniform, 384, 387 equivalence class ḟ of a function, 32 equivalent, locally, 40, 162 equivalent probabilities, 32, 34, 50, 187,

191, 206, 208, 450 Euler–Peano approximation, 7, 311,

312, 314, 317 evaluation process, 66 evanescent, 35 exceed, 363 expectation, 32 exponential, stochastic, 159, 163, 167,

180, 186, 219 extended reals, 23, 60, 74, 90, 125, 363,

364, 392, 394, 408 extension, integral, of Daniell, 87

F factorization, 50, 192, 233, 282, 291,

293, 295, 298, 304, 320, 458 of random measures, 187, 208, 297

factorization constant, 192 Fatou’s lemma, 449 Feller semigroup

canonical representation of, 358 conservative, 268, 465 convolution, 268, 466 , 269, 350, 351, 465, 466 natural extension of, 467 stochastic representation, 351 stochastic representation of, 352

filter, convergence of, 373 filtered measurable space, 21, 438 filtration

basic, of a process, 18, 19, 23, 254 basic, of path space, 66, 166, 445 basic, of Wiener process, 18 canonical, of path space, 66, 335 , 5, 22 full, 166, 176, 177, 185 full measured, 166 𝔓-regular, 38, 437 is assumed right-continuous and

regular, 58, 60, 62 measured, 32, 39 natural enlargement of, 39, 57, 254 natural, of a process, 39 natural, of a random measure, 177 natural, of path space, 67, 166 regular, 38, 437 regularization of a, 38, 135 right-continuous, 37 right-continuous version of, 37, 38,

117, 438 universally complete, 22, 437, 440

finite for the mean, 90, 94 function — in p-mean, 448 process — for the mean, 97, 100 process of — variation, 68 stochastic interval, 28 variation, 45 variation, process of —, 23

finite intersection property, 432 finite variation, 394, 395 flow, 278, 326, 359, 466 λ-form, 305 Fourier transform, 410 Fréchet space, 364, 380, 400, 467 French School, 21, 39 Friedman, 459 Fubini’s theorem, 403, 405 full

filtration, 166, 176, 185 measured filtration, 166 projective system, 402, 447

function finite in p-mean, 448 p-integrable, 33 idempotent, 365 lower semicontinuous, 194, 207, 376 numerical, 110, 364 simple measurable, 448 universally measurable, 22, 23, 351,

407, 437 upper semicont., 108, 194, 376, 382 vanishing at infinity, 366, 367 functional, 370, 372, 379, 381, 394 functional, solid, 34

G

Garsia, 215 gauge, 50, 89, 95, 130, 380, 381, 400,

450, 468 gauges defining a topology, 95, 379 Gauß kernel, 420 Gaussian

distribution, 419

normalized, 12, 419, 423

semigroup, 466

with covariance matrix, 270, 420 G_{δ}-set, 379 Gelfand transform, 108, 174, 370 generate

a σ-algebra from functions, 391 a topology, 376, 411

generator, conservative, 467 Girsanov’s theorem, 39, 162, 164, 167,

176, 186, 338, 475 global L^{p}(ℙ)-integrator, 50 global L^{p}-random measure, 173 graph of an operator, 464 graph of a random time, 28 Gronwall, 383 Gundy, 213

H

Haagerup, 457 Haar measure, 196, 394, 413 Hardy mean, 179, 217, 219, 232, 234,

250 Hardy space, 220 harmonic measure, 340 Hausdorff space or topology, 367, 370,

373 heat kernel, 420 Heun method, 322 heuristics, 10, 11, 12 history

course of, 3, 318

, 5, 21 hitting distribution, 340 Hölder’s inequality, 33, 188, 449 homeomorphism, 364, 379, 441, 460 Hunt function, 181, 185, 257, 258 I (IC-p), 89 idempotent function, 365 iff, meaning if and only if, 37 L^{p}(ℙ)-integrator, 50 global, 50 vector of, 56

image of a measure, 405

increasingly directed, 72, 366, 376, 398,

417 increasing process, 23 increments, 10, 19

independent, 10, 253 indefinite integral

against an integrator, 134 against jump measure, 181

independent increments, 10, 253 , 10, 16, 263, 413

indistinguishable, 35, 62, 226, 387 induced topology, 373 induced uniformity, 374 inequality

Cauchy–Schwarz —, 150

Hölder’s —, 449 of Hölder, 33, 188 initial condition, 8, 273 initial value problem, 342 inner measure, 396 inner regularity, 400 instant, 31 integrability criterion, 113, 397 integrable

function, Banach space-valued, 409 process, on a stochastic interval, 131 set, 104

integral against a random measure, 174 elementary, 7, 44, 99, 395 elementary stochastic, 47 extended, 100 extension of Daniell, 87 Itô stochastic, 99, 169 Stratonovich stochastic, 169, 320 upper, of Daniell, 87 Wiener, 5 integral equation, 1 integrand

elementary, 7, 43, 44, 46, 47, 49, 56, 79, 87, 90, 95, 99, 111, 115, 136, 155

elementary stochastic, 46 integrator

d-tuple of, see vector of, 9 , 7, 20, 43, 50 local, 51, 234 slew of, see vector of, 9

integrators

vector of, 56, 63, 77, 134, 149, 187,

202, 238, 246, 282, 285 intensity, 231

measure, 231 of jumps, 232, 235 rate, 179, 185, 231

interpolation, 33, 85, 453 interval

finite stochastic, 28 stochastic, 28

intrinsic time, 239, 318, 321, 323 iterates

Picard, 2, 3, 6, 273, 275 Itô, 5, 24 Itô’s theorem, 157 Itô stochastic integral, 99, 169

J Jensen’s inequality, 73, 243, 408 jump intensity

continuous, 232, 235 , 232 measure, 258 rate, 232, 258

jump measure, 181 jump process, 148

K Khintchine, 64, 455 Khintchine constants, 457 Kolmogorov, 3, 384 Kronecker delta, 19 Kunita–Watanabe, 212, 228, 232 Ky–Fan, 34, 189, 195, 207, 382

L largely

smooth, 111, 175 uniform convergence, 111 uniformly continuous, 405

lattice algebra, 366, 370 law, 10, 13, 14, 161, 405

Newton’s second, 1 of Wiener process, 16 strong law of large numbers, 76, 216

Lebesgue, 87, 395 Lebesgue measure, 394 Lebesgue–Stieltjes integral, 4, 43, 54,

87, 142, 144, 153, 158, 170, 232, 234

left-continuous process, 23 version of a process, 24 version of the filtration, 123

ℓ^{p}-length of a vector or sequence, 364 less than (≤), 363 Lévy

measure, 258, 269

process, 239, 253, 255, 292, 349, 466 Lie bracket, 278 lifetime of a solution, 273, 292 lifting, 415

strong, 419 Lindeberg condition, 10, 423 linear functional, positive, 395 linear map, bounded, 452 Lipschitz

constant, 144, 274

coupling coefficients, 285

function, 2, 4, 144

in p-mean, 286

in Picard norm, 286

locally, 292

strongly, 285, 288, 289

vector field, 274, 287 Little o, 388 local

L^{1}-integrator, 221

L^{p}-random measure, 173

martingale, 75, 84, 221

property of a process, 51, 80 local E-compactification, 108, 367, 370 local martingale, 75

square integrable, 84 locally absolutely continuous, 40, 52,

130, 162 locally bounded, 122, 225 locally compact, 374 locally convex, 50, 187, 379, 467 locally equivalent probabilities, 40, 162,

164 locally Z−p-integrable, 131 locally Lipschitz, 292 Lorentz space, 51, 83, 452 lower integral, 396 lower semicontinuous

function, 194

, 207, 376 L^{p}-bounded, 33 L^{p}(ℙ)-integrator, 50 L^{p}-integrator, 50, 62

complex, 152 vector of, 56

Lusin, 110, 432 space, 447 space or subset, 441

L^{0}-integrator convex function of a, 146

M (M), condition, 91, 95 markovian coupling coefficient, 272,

287, 311, 321 Markov property, 349, 352

strong, 349, 352 martingale

continuous, 161 locally square integrable, 213 local, 75, 84 , 5, 19, 71, 74, 75 of finite variation, 152, 157 previsible, 157 previsible, of finite variation, 157 square integrable, 78, 163, 186, 262 uniformly integrable, 72, 75, 77, 123,

164 martingale measure, 220 martingale part, continuous, of an

integrator, 235 martingale problem, 338 martingale random measure, 180 Maurey, 195, 205 maximal inequality, 61, 337 maximality

of a mean, 123 of Daniell’s mean, 123, 135, 217, 398

maximal lemma of Doob, 76 maximal path, 26, 274 maximal process, 21, 26, 29, 38, 61, 63,

122, 137, 159, 227, 360, 443 maximum principle, 340 MCT: Monotone Convergence Theo-

rem, 102 mean

the Daniell —, 89, 124, 174 controlling a linear map, 95 controlling the integral, 95, 99, 100,

101, 123, 124, 217, 247, 249, 250 Hardy—, 217, 219, 232, 234, 250 σ-finite –, 105 majorizing a linear map, 95 maximality of a —, 123 mean (cont’d)

maximality of Daniell’s —, 123, 135,

217 , 94, 399 pathwise, 216

measurable cross section, 436, 438, 440 filtered — space, 21, 438 Z-measurable, 129 Z−p-measurable, 111 , 110 on a σ-algebra, 391 process, 23, 243 set, 114 simple function, 448 space, 391

measure σ-finite, 449 , 394 of totally finite variation, 394 support of a, 400 tight, 21, 165, 399, 407, 425, 441

measured filtration, 39 measure space, 409 method

adaptive, 281 Euler, 281, 311, 314, 318, 322 Heun, 322 order of, 281, 324 Runge–Kutta, 282, 321, 322 scale-invariant, 282, 329 single-step — ξ′, 280 single-step — Ξ′, 322, 325 Taylor, 281, 282, 321, 322, 327

metric, 374 metrizable, 367, 375, 376, 377 Meyer, 228, 232, 432 minimax theorem, 382 μ-integrable, 397 μ-measurable, 397 μ-negligible, 397 model, 1, 4, 6, 10, 11, 18, 20, 21, 27, 32 modification, 58, 61, 62, 75, 101, 132,

134, 136, 147, 167, 223, 268 ℙ-modification, 34, 384 modulus

of continuity, 58, 191, 206, 209, 444 of contractivity, 290, 299, 407

monotone class, 16, 145, 155, 218, 224,

393, 439 Monotone Class Theorem, 16 multiplicative class

complex, 393, 399, 410, 421, 423, 425, 427 multiplicative class (cont’d)

, 393, 399, 421 μ-measurable

map between measure spaces, 405

N

natural domain of a semigroup, 467 extension of a semigroup, 467 filtration of path space, 67, 166 increasing process, 228, 234

natural conditions, 29, 39, 60, 118, 119,

127, 130, 133, 162, 227, 437 natural enlargement of a filtration, 57,

101, 254, 440 nearly, ℙ-nearly, 35 nearly, 118, 119, 141 negative, (≤ 0), 363

strictly — (< 0), 363 negligible

meaning Z-negligible, 129 , 32, 95

ℙ-negligible, 32 neighborhood filter, 373 Nelson, 10 Newton, 1 Nikisin, 207 noise

background, 2, 8, 272 cumulative, 2 effect of, 2 , 2

nolens volens, meaning nilly–willy, 421 non-adaptive, 281, 317, 321 non-anticipating

process, 6, 144 random measure, 173

normal distribution, 419 normalized Gaussian, 12, 419, 423 Notations, see Conventions, 21 Novikov, 163, 165 numerical

function, 110, 364 , 23, 392, 394, 408

numerical approximation, see method, 280 O ODE, 4, 273, 326 one-point compactification, 374, 465 optional

σ-algebra O, 440 process, 440 projection, 440 stopping theorem, 77

order-bounded, 406 order-complete, 252, 406 order-continuity, 398, 421 order-continuous mean, 399 order interval, 45, 50, 173, 372, 449 order of an approximation scheme, 281,

319, 324 oscillation of a path, 444 outer measure, 32

P partition, random or stochastic, 138 past, 5, 21

strict, of a stopping time, 120, 225 path, 4, 5, 11, 14, 19, 23

—s describing the same arc, 153 path space

basic filtration of, 66 canonical filtration of, 66, 335 canonical, 66 natural filtration of, 67, 166 , 14, 20, 166, 380, 391, 411, 440

pathwise computation of the integral, 140 failure of the integral, 5, 17 mean, 216, 238 previsible control, 238 solution of an SDE, 311 stochastic integral, 7, 140, 142, 145

paving, 432 permanence properties

of fullness, 166 of integrable functions, 101 of integrable sets, 104 of measurable functions, 111, 114,

115, 137 of measurable sets, 114 , 165, 391, 397

Picard iterates, 2, 3, 6, 273, 275 , 4, 6

Picard norm Lipschitz in, 286 , 274, 283 p-integrable

function or process, 33 uniformly — family, 449

Pisier, 195 p-mean, 33 p-mean σ-additivity, 106 p;ℙ-mean σ-additivity, 90 point at infinity, 374 point process, 183

Poisson, 185 simple, 183

Poisson point process compensated, 185 , 185, 264

Poisson process, 270, 359 Poisson random variable, 420 Poisson semigroup, 359 polarization, 366 polish space, 15, 20, 440 polynomially bounded, 322 positive (≥ 0)

linear functional, 395 , 363 strictly — (> 0), 363

positive maximum principle, 269, 466 𝔓-regular filtration, 38, 437 precompact, 376 predictable

increasing process, 225 , 115 process of finite variation, 117 projection, 439 random function, 172, 175, 180 stopping time, 118, 438 transformation, 185

predict a stopping time, 118 previsible

bracket, 228 control, 238 dual — projection, 221 , 68 process of finite variation, 221 process, 68, 122, 138, 149, 156, 228 process — with ℙ, 118 set, 118 set, sparse, 235 square function, 228

previsible controller, 238, 283, 294 probabilities

locally equivalent, 40, 162 probability

on a topological space, 421 process

adapted, 23 basic filtration of a, 19, 23, 254 continuous, 23 defined \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-a.e., 97 evanescent, 35 finite for the mean, 97, 100 I^{p}[ℙ]-bounded, 49 L^{p}-bounded, 33 p-integrable, 33 \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-integrable, 99 \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-negligible, 96 Z−p-integrable, 99 Z−p-measurable, 111 increasing, 23 indistinguishable —es, 35 integrable on a stochastic interval,

131 jump part of a, 148 jumps of a, 25 left-continuous, 23 Lévy, 239, 253, 255, 292, 349 locally Z−p-integrable, 131 local property of a, 51, 80 maximal, 21, 26, 29, 61, 63, 122, 137,

159, 227, 360, 443 measurable, 23, 243 modification of a, 34 natural increasing, 228 non-anticipating, 6, 144 of bounded variation, 67 of finite variation, 23, 67 optional, 440 predictable increasing, 225 predictable of finite variation, 117 predictable, 115 previsible, 68, 118, 122, 138, 149,

156, 228 previsible with ℙ, 118 , 6, 23, 90, 97 right-continuous, 23 square integrable, 72 stationary, 10, 19, 253 stopped just before T, 159, 292 stopped, 23, 28, 51 variation — of another, 68, 226

product σ-algebra, 402 product of elementary integrals

infinite, 12, 404 , 402, 413

product paving, 402, 432, 434 progressively measurable, 25, 28, 35, 37,

38, 40, 41, 65, 437, 440 projection

dual previsible, 221 predictable, 439 well-measurable, 440

projective limit

of elementary integrals, 402, 404,

447 of probabilities, 164

projective system full, 402, 447 , 401

Prokhoroff, 425 ∥ ∥_{p}-semivariation, 53 pseudometric, 374 pseudometrizable, 375 punctured d-space, 180, 257

Q quasi-left-continuity

of a Lévy process, 258 of a Markov process, 352 , 232, 235, 239, 250, 265, 285, 292,

319, 350 quasinorm, 381

R Rademacher functions, 457 Radon measure, 177, 184, 231, 257, 263,

355, 394, 398, 413, 418, 442, 465, 469

Radon–Nikodym derivative, 41, 151,

223, 407, 450 random

interval, 28 partition, refinement of a, 138 sheet, 20 time, 27, 118, 436 vector field, 272

random function predictable, 175 , 172, 180

randomly autologous

coupling coefficient, 300 random measure

canonical representation, 177 compensated, 231 driving a SDE, 296 factorization of, 187, 208 local martingale–, 231 martingale–, 180 quasi-left-continuous, 232 , 109, 173, 188, 205, 235, 246, 251,

263, 296, 370 random measure (cont’d)

spatially bounded, 173, 296 stopped, 173 strict previsible, 231 strict, 183, 231, 232 vanishing at 0, 173 Wiener, 178, 219

random time, graph of, 28 random variable

nearly zero, 35 , 22, 391 simple, 46, 58, 391 symmetric stable, 458

(RC-0), 50 rectification

time — of a SDE, 280, 287 time — of a semigroup, 469

recurrent, 41 reduce

a process to a property, 51 a stopping time to a subset, 31, 118

reduced stopping time, 31 refine

a filter, 373, 428 a random partition, 62, 138

regular filtration, 38, 437 , 35 stochastic representation, 352

regularization

of a filtration, 38, 135 relatively compact, 260, 264, 366, 385,

387, 425, 426, 428, 447 remainder, 277, 388, 390 remove a negligible set from Ω, 165,

166, 304 representation

canonical of an integrator, 67 of a filtered probability space, 14,

64, 316 representation of martingales

for Lévy processes, 261 on Wiener space, 218

resolvent identity, 463 , 352, 463

right-continuous filtration, 37 process, 23 , 44 version of a filtration, 37 version of a process, 24, 168

ring of sets, 394 Runge–Kutta, 281, 282, 321, 322, 327 S σ-additive, 394

in p-mean, 90, 106 marginally, 174, 371

σ-additivity, 90 σ-algebra, 394

Baire vs. Borel, 391 function measurable on a, 391 generated by a family of functions,

391 generated by a property, 391

σ-algebras, product of, 402 σ-algebra

universally complete, 407 scalæfication, of processes, 139, 300,

312, 315, 335 scalae: ladder, flight of steps, 139, 312 Schwartz, 195, 205 Schwartz space, 269 σ-continuity, 90, 370, 394, 395 self-adjoint, 454 self-confined, 369 semicontinuous, 207, 376, 382 semigroup

conservative, 467 convolution, 254 Feller convolution, 268, 466 Feller, 268, 465 Gaussian, 19, 466 natural domain of a, 467 of operators, 463 Poisson, 359 resolvent of a, 463 time-rectification of a, 469

semimartingale, 232 seminorm, 380 semivariation, 53, 92 separable, 367

topological space, 15, 373, 377 sequential closure or span, 392 set

analytic, 432 ℙ-nearly empty, 60 \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-measurable, 114 identified with indicator function,

364 integrable, 104

σ-field, 394 shift

a process, 4, 162, 164 a random measure, 186

σ-algebra of \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-measurable sets, 114 σ-algebra (cont’d)

optional — O, 440 P of predictable sets, 115 O of well-measurable sets, 440

σ-finite

class of functions or sets, 392, 395,

397, 398, 406, 409 mean, 105, 112 measure, 406, 409, 416, 449

simple measurable function, 448 point processs, 183 random variable, 46, 58, 391

size of a linear map, 381 Skorohod, 21, 443

space, 391 Skorohod topology, 21, 167, 411, 443,

445 slew of integrators

see vector of integrators, 9 solid

functional, 34 , 36, 90, 94

solution strong, of a SDE, 273, 291 weak, of a SDE, 331

space analytic, 441 completely regular, 373, 376 Hausdorff, 373 locally compact, 374 measurable, 391 polish, 15 Skorohod, 391

span, sequential, 392 sparse, 69, 225 sparse previsible set, 235, 265 spatially bounded random measure,

173, 296 spectrum of a function algebra, 367 square bracket, 148, 150 square function

continuous, 148 of a complex integrator, 152 previsible, 228 , 94, 148

square integrable locally — martingale, 84, 213 martingale, 78, 163, 186, 262 process, 72

square variation, 148, 149 stability

of solutions to SDE’s, 50, 273, 293, 297 stability (cont’d)

under change of measure, 129 stable, 220 standard deviation, 419 stationary process, 10, 19, 253 step function, 43 step size, 280, 311, 319, 321, 327 stochastic

analysis, 22, 34, 47, 436, 443

exponential, 159, 163, 167, 180, 185,

219 flow, 343 integral, 99, 134 integral, elementary, 47 integrand, elementary, 46 integrator, 43, 50, 62 interval, bounded, 28 interval, finite, 28 partition, 140, 169, 300, 312, 318 representation of a semigroup, 351 representation, regular, 352

Stone–Weierstraß, 108, 366, 377, 393,

399, 441, 442 stopped

just before T, 159, 292

process, 23, 28, 51 stopping time

accessible, 122

announce a, 118, 284, 333

arbitrarily large, 51

elementary, 47, 61

examples of, 29, 119, 437, 438

past of a, 28

predictable, 118, 438

, 27, 51

totally inaccessible, 122, 232, 235,

258 Stratonovich equation, 320, 321, 326 Stratonovich integral, 169, 320, 326 strictly positive or negative, 363 strict past of a stopping time, 120 strict random measure, 183, 231, 232 strong law of large numbers, 76, 216 strong lifting, 419 strongly perpendicular, 220 strong Markov property, 352 strong solution, 273, 291 strong type, 453 subadditive, 33, 34, 53, 54, 130, 380 submartingale, 73, 74 supermartingale, 73, 74, 78, 81, 85, 356 support of a measure, 400 sure control, 292 Suslin

space or subset, 441

, 432 symmetric form, 305 symmetrization, 305 Szarek, 457

T

tail filter, 373

Taylor method, 281, 282, 321, 322, 327 Taylor’s formula, 387 THE

Daniell mean, 89, 109, 124 previsible controller, 238 time transformation, 239, 283, 296 threshold, 7, 140, 168, 280, 311, 314 tiered weak derivatives, 306 tight

measure, 21, 165, 399, 407, 425, 441 uniformly, 334, 425, 427 time, random, 27, 118, 436 time-rectification of a SDE, 287 time-rectification of a semigroup, 469 time shift operator, 359 time transformation the, 239, 283, 296 , 283, 444 topological space Lusin, 441 polish, 20, 440 separable, 15, 373, 377 Suslin, 441 , 373

topological vector space, 379

topology generated by functions, 411 generated from functions, 376 Hausdorff, 373 of a uniformity, 374 of confined uniform convergence, 50,

172, 252, 370 of uniform convergence on com-

pacta, 14, 263, 372, 380, 385, 411, 426, 467 Skorohod, 21 , 373 uniform — on E, 51

totally bounded, 376 totally finite variation measure of, 394 , 45

totally inaccessible stopping time, 122, 232, 235, 258 trajectory, 23 transformation, predictable, 185 transition probabilities, 465 transparent, 162 triangle inequality, 374 Tychonoff’s theorem, 374, 425, 428 type

map of weak —, 453

(p, q) of a map, 461

U

\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-a.e.

defined process, 97

convergence, 96 \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-integrable, 99 \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-measurable, 111

process, on a set, 110

set, 114 \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-negligible, 95

\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z−p}^{∗}-a.e., 96

\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z−p}^{∗}-negligible, 96 uniform

convergence, 111

largely — convergence, 111 uniform convergence on compacta

see topology of, 380 uniformity

generated by functions, 375, 405 induced on a subset, 374 , 374

E-uniformity, 110, 375 uniformizable, 375 uniformly continuous

largely, 405

, 374 uniformly differentiable

weakly l-times, 305

weakly, 299, 300, 390 uniformly integrable

martingale, 72, 77

, 75, 225, 449 uniqueness of weak solutions, 331 universal completeness of the regular-

ization, 38 universal completion, 22, 26, 407, 436 universal integral, 141, 331 universally complete

filtration, 437, 440

, 38 universally measurable

function, 22

set or function, 23, 351, 407, 437 universal solution of an endogenous SDE, 317, 347, 348 up-and-down procedure, 88 upcrossing, 59, 74 upcrossing argument, 60, 75 upper integral, 32, 87, 396 upper regularity, 124 upper semicontinuous, 107, 194, 207,

376, 382 usual conditions, 39, 168 usual enlargement, 39, 168

V

vanish at infinity, 366, 367

variation bounded, 45 finite, 45 function of finite, 45 measure of finite, 394 of a measure, 45, 394 process of bounded, 67 process of finite, 67 process, 68, 226 square, 148, 149 totally finite, 45 , 395 vector field

random, 272 , 272, 311

vector lattice, 366, 395

vector measure, 49, 53, 90, 108, 172,

448 vector of integrators

see integrators, vector of, 9 version

left-continuous, of a process, 24 right-continuous, of a process, 24 right-continuous, of a filtration, 37

W weak

derivative, 302, 390

higher order derivatives, 305

tiered derivatives, 306 weak convergence of measures, 421 , 421

weak∗ topology, 263, 381 weakly differentiable l-times, 305 , 278, 390 weak solution, 331 weak topology, 381, 411 weak type, 453 well-measurable

σ-algebra O, 440 process, 217, 440

Wiener integral, 5 measure, 16, 20, 426 random measure, 178, 219 sheet, 20 space, 16, 58 , 5

Wiener process as integrator, 79, 220 canonical, 16, 58 characteristic function, 161 d-dimensional, 20, 218 Lévy’s characterization, 19, 160 on a filtration, 24, 72, 79, 298 square bracket, 153 standard d-dimensional, 20, 218 standard, 11, 16, 18, 19, 41, 77, 153,

162, 250, 326 , 9, 10, 11, 17, 89, 149, 161, 162, 251,

426 with covariance, 161, 258

Z Zero-One Law, 41, 256, 352, 358 Z-measurable, 129 Z−p-a.e., 96, 123

convergence, 96 Z−p-integrable, 99, 123 ζ−p-integrable, 175 Z−p-measurable, 111, 123 Z−p-negligible, 96