A stochastic process is called measurable, if the mapping

$(t,\omega )\mapsto X_t(\omega ):([0,\infty )\times \mathrm{\Omega },\mathfrak{B}([0,\infty ))\otimes \mathcal{F})\to (\mathbb{R},\mathfrak{B}(\mathbb{R}))$

is measurable.

A stochastic process is called adapted to the filtration ${({\mathcal{F}}_t)}_{t⩾0}$, if the mapping

$\omega \mapsto X_t(\omega ):(\mathrm{\Omega },{\mathcal{F}}_t)\to (\mathbb{R},\mathfrak{B}(\mathbb{R}))$

is measurable for all $t⩾0$.

A stochastic process is called progressively measurable, if the mapping

$(s,\omega )\mapsto X_s(\omega ):([0,t]\times \mathrm{\Omega },\mathfrak{B}([0,t])\otimes {\mathcal{F}}_t)\to (\mathbb{R},\mathfrak{B}(\mathbb{R}))$

is measurable for all $t⩾0$.

Progressive measurability implies adaptedness.

If a stochastic process X is adapted to the filtration ${({\mathcal{F}}_t)}_{t⩾0}$ and a.e. path is left-continuous or right-continuous, then X is progressively measurable.

A Wiener process is a real valued stochastic process $W={(W_t)}_{t⩾0}$ with the following properties.

i) The increments of B are independent, i.e. for arbitrary $0⩽t_0<t_1<\cdots <t_n$ the random variables $W_{t_1}-W_{t_0},W_{t_2}-W_{t_1},\dots ,W_{t_n}-W_{t_{n-1}}$ are independent.

ii) For all $t>s⩾0$ we have $W_t-W_s\sim \mathcal{N}(0,t-s)$.

iii) There holds $W_0=0$ almost surely.

iv) The mapping $t\mapsto W_t(\omega )$ is continuous for a.e. $\omega \in \mathrm{\Omega }$.

A filtration ${({\mathcal{F}}_t)}_{t⩾0}$ is called right-continuous, if

${\mathcal{F}}_t=\bigcap _{\epsilon >0}{\mathcal{F}}_{t+\epsilon }\forall t⩾0$

and left-continuous, if

${\mathcal{F}}_t=\bigcup _{s<t}{\mathcal{F}}_s\forall t>0.$

A filtration ${({\mathcal{F}}_t)}_{t⩾0}$ satisfies the usual conditions, if it is right-continuous and ${\mathcal{F}}_0$ contains all the $\mathbb{P}$-nullsets of $\mathcal{F}$.

Let X be a stochastic process and $\mathcal{T}$ a $\mathcal{F}$-measurable random variable with values in $[0,\infty ]$. We define $X_{\mathcal{T}}$ in $\{\mathcal{T}<\infty \}$ by

$X_{\mathcal{T}}(\omega )=X_{\mathcal{T}(\omega )}(\omega ).$

If $X_{\infty }(\omega )=\underset{t\to \infty }{\lim }{X}_t(\omega )$ exists for a.e. $\omega \in \mathrm{\Omega }$ we set $X_{\mathcal{T}}(\omega )=X_{\infty }(\omega )$ in $\{\mathcal{T}=\infty \}$.

Let X be a stochastic process on a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ with filtration ${({\mathcal{F}}_t)}_{t⩾0}$. A random variable $\mathcal{T}$ is called a stopping time if the set $\{\mathcal{T}⩽t\}$ belongs to ${\mathcal{F}}_t$ for all $t⩾0$.

Then by T induced σ-algebra ${\mathcal{F}}_{\mathcal{T}}$ is given by

${\mathcal{F}}_{\mathcal{T}}:=\{A\in \mathcal{F}:A\cap \{\mathcal{T}⩽t\}\in {\mathcal{F}}_t\forall t⩾0\}.$

Let X be a progressively ${({\mathcal{F}}_t)}_{t⩾0}$-measurable stochastic process. Let $f:[0,\infty )\times \mathbb{R}\to \mathbb{R}$ be $\mathfrak{B}([0,\infty ))\otimes \mathfrak{B}(\mathbb{R})$-measurable. The process

$Y_t:={\int }_0^{t}f(s,X_s)\mathrm{d}s,t⩾0,$

is progressively ${({\mathcal{F}}_t)}_{t⩾0}$-measurable. If $\mathcal{T}$ is a stopping time then $Y_{\mathcal{T}}$ is a ${\mathcal{F}}_{\mathcal{T}}$-measurable random variable.

Let $M={(M_t)}_{t⩾0}$ be a ${({\mathcal{F}}_t)}_{t⩾0}$-adapted stochastic process with $\mathbb{E}[|M_t|]<\infty$ for all $t⩾0$. X is called a sub-martingale (super-martingale, respectively) if we have for all $0⩽s⩽t<\infty$ that $\mathbb{P}$-a.s. $\mathbb{E}[M_t|{\mathcal{F}}_s]⩾M_s$ ($\mathbb{E}[M_t|{\mathcal{F}}_s]⩽M_s$, respectively). M is called a martingale if it is a sub-martingale and a super-martingale.

Let A be an adapted stochastic process. A is called increasing, if we have for $\mathbb{P}$-a.e. $\omega \in \mathrm{\Omega }$

i) $A_0=0$;

ii) $t\mapsto A_t(\omega )$ is increasing and right-continuous;

iii) $\mathbb{E}\left[A_t\right]<\infty$ for all $t\in [0,\infty )$.

An increasing process is called integrable if $\mathbb{E}\left[A_{\infty }\right]<\infty$, where $A_{\infty }(\omega )=\underset{t\to \infty }{\lim }{A}_t(\omega )$ for $\omega \in \mathrm{\Omega }$.

Let X be a positive submartingale with a.s. continuous trajectories. There is a continuous martingale M and an a.s. increasing continuous and adapted process A, such that

$X_t^2=M_t+A_t.$

The decomposition is unique.

Let X be a right-continuous martingale. We call X quadratically integrable, if $\mathbb{E}\left[X_t^2\right]<\infty$, for all $t⩾0$. If we have in addition that $X_0=0$ $\mathbb{P}$-a.s., we write $X\in {\mathcal{M}}_2$ or $X\in {\mathcal{M}}_2^c$, if X is continuous.

For $X\in {\mathcal{M}}_2$ and $0⩽t<\infty$ we define

${\Vert X\Vert }_t:=\sqrt{\mathbb{E}\left[X_t^2\right]}.$

We set

$\Vert X\Vert :=\sum _{n=1}^{\infty }\frac{{\Vert X\Vert }_n\wedge 1}{2^n}$

which induces a metric d on ${\mathcal{M}}_2$ given by

$d(X,Y)=\Vert X-Y\Vert .$

A function $f:[0,t]\to \mathbb{R}$ is called of bounded variation if there is $M>0$ such that ${\sum }_{i=1}^n|f(x_i)-f(x_{i-1)}|⩽M$ for all finite partitions $\mathrm{\Pi }=\{x_0,x_1,\dots ,x_n\}\subset [0,t]$ ($n\in \mathbb{N}$) with $0=x_0<x_1<\cdots <x_n=t$. The quantity

$V(f):=\sup \{\sum _{i=1}^n|f(x_i)-f(x_{i-1)}|:0=x_0<x_1<\cdots <x_n=t,n\in \mathbb{N}\}$

is called total variation of f over $[0,t]$.

For $X\in {\mathcal{M}}_2$ we define the quadratic variation of X, as the process ${\langle \langle X\rangle \rangle }_t:=A_t$, where A is the increasing process from the Doob–Meyer-decomposition of X.

For $X,Y\in {\mathcal{M}}_2$ we define the covariation $\langle \langle X,Y\rangle \rangle$ by

${\langle \langle X,Y\rangle \rangle }_t:=\frac{1}{4}[{\langle \langle X+Y\rangle \rangle }_t-{\langle \langle X-Y\rangle \rangle }_t],$

for $t⩾0$. The process $XY-\langle \langle X,Y\rangle \rangle$ is a martingale. In particular, we have $\langle \langle X,X\rangle \rangle =\langle \langle X\rangle \rangle$.

Let X be a stochastic process, $p⩾1$, $t>0$ fixed and $\mathrm{\Pi }=\{t_0,t_1\dots t_n\}$, with $0=t_0<t_1<\cdots <t_n=t$, $n\in \mathbb{N}$, a partition of $[0,t]$. The p-th variation of X in Π is defined by

$V_t^{(p)}(\mathrm{\Pi })=\sum _{k=0}^n|X_{t_k}-X_{t_{k-1}}{|}^p.$

Let $X\in {\mathcal{M}}_2^c$, Π be a partition of $[0,t]$ and $\Vert \mathrm{\Pi }\Vert :={\max }_{1⩽k⩽n}|t_k-t_{k-1}|$ the size of Π. Then we have ${\lim }_{\Vert \mathrm{\Pi }\Vert \to 0}{V}_t^{(2)}(\mathrm{\Pi })={\langle \langle X\rangle \rangle }_t$ in probability, i.e. for all $\epsilon >0$, $\eta >0$ there is $\delta >0$, such that we have that

$\mathbb{P}(|V_t^{(2)}(\mathrm{\Pi })-{\langle X\rangle }_t|>\epsilon )<\eta ,$

for $\Vert \mathrm{\Pi }\Vert <\delta$.

We define ${\mathcal{L}}^{⁎}$ as the space of equivalence classes of progressively ${({\mathcal{F}}_t)}_{t⩾0}$-measurable processes X with ${[X]}_T<\infty$ for all $T>0$.

By setting $[X-Y]:={\sum }_{n=0}^{\infty }{2}^{-n}(1\wedge {[X-Y]}_n)$ we can define a metric on ${\mathcal{L}}^{⁎}$.

In the following we do not distinguish between X and the equivalence class $X^{⁎}$ of X.

A process X is called step process if there is a strictly increasing sequence ${(t_n)}_{n\in \mathbb{N}}\subseteq \mathbb{R}$ with $t_0=0$ and $\underset{n\to \infty }{\lim }{t}_n=\infty$, a sequence of random variables ${({\xi }_n)}_{n\in {\mathbb{N}}_0}$ and $C<\infty$ with ${\sup }_{n\in {\mathbb{N}}_0}|{\xi }_n(\omega )|⩽C$ such that the following holds: for every $\omega \in \mathrm{\Omega }$ we have that ${\xi }_n$ is ${\mathcal{F}}_{t_n}$-measurable for every $n\in {\mathbb{N}}_0$ and we have the representation

$X_t(\omega )={\xi }_0(\omega ){\mathbb{I}}_{\{0\}}(t)+\sum _{i=0}^{\infty }{\xi }_i(\omega ){\mathbb{I}}_{(t_i,t_i+1]}(t),$

for all $0⩽t<\infty$. The space of step processes is denoted by ${\mathcal{L}}_0$.

(i) Step processes are progressively measurable and bounded. (ii) There holds ${\mathcal{L}}_0\subseteq {\mathcal{L}}^{⁎}$.

Let $X\in {\mathcal{L}}_0$ and $M\in {\mathcal{M}}_2^c$. The stochastic integral of X with respect to M is the martingale transformation

$I_t(X):=\sum _{i=0}^{n-1}{\xi }_i(M_{t_{i+1}}-M_{t_i})+{\xi }_n(M_t-M_{t_n})=\sum _{i=0}^{\infty }{\xi }_i(M_{t\wedge t_{i+1}}-M_{t\wedge t_i}),$

for $0⩽t<\infty$. Here $n\in \mathbb{N}$ is the unique natural number such that $t_n⩽t<t_{n+1}$.

The space of step processes ${\mathcal{L}}_0$ is dense in ${\mathcal{L}}^{⁎}$ with respect to the metric defined in Remark 8.2.16.

Let $X\in {\mathcal{L}}^{⁎}$ and $M\in {\mathcal{M}}_2^c$. The stochastic integral of X with respect to M is the unique quadratically integrable martingale

$I(X)=\{I_t(X),{\mathcal{F}}_t,0⩽t<\infty \},$

which satisfies $\underset{n\to \infty }{\lim }\Vert I(X^{(n)})-I(X)\Vert =0$, for every sequence ${(X^{(n)})}_{n\in \mathbb{N}}$ $\subseteq {\mathcal{L}}_0$ with $\underset{n\to \infty }{\lim }[X^{(n)}-X]=0$. We write

$I_t(X)={\int }_0^t{X}_s\mathrm{d}{M}_s;0⩽t<\infty .$

Let $X,Y\in {\mathcal{L}}^{⁎}$ and $0⩽s<t<\infty$. For the stochastic integrals $I(X),I(Y)$ we have

a) $I_0(X)=0$ $\mathbb{P}$-a.s.,

b) $\mathbb{E}[I_t(X)|{\mathcal{F}}_s]=I_s(X)$, $\mathbb{P}$-a.s. (martingale property),

c) $\mathbb{E}\left[{(I_t(X))}^2\right]=\mathbb{E}\left[{\int }_0^t{X}_u^2\mathrm{d}{\langle \langle M\rangle \rangle }_u\right]$ (Itô-isometry),

d) $\Vert I(X)\Vert =[X]$,

e) $\mathbb{E}[{(I_t(X)-I_s(X))}^2|{\mathcal{F}}_s]=\mathbb{E}[{\int }_s^t{X}_u^2\mathrm{d}{\langle \langle M\rangle \rangle }_u|{\mathcal{F}}_s]$ $\mathbb{P}$-a.s.,

f) $I(\alpha X+\beta Y)=\alpha I(X)+\beta I(Y)$, for $\alpha ,\beta \in \mathbb{R}$.

Let ${(X_t)}_{t⩾0}$ be a continuous process adapted to ${({\mathcal{F}}_t)}_{t⩾0}$. Assume there is a sequence of stopping times ${({\mathcal{T}}_n)}_{n\in \mathbb{N}}$ of the filtration ${({\mathcal{F}}_t)}_{t⩾0}$, such that ${(X_t^{(n)}:=X_{t\wedge {\mathcal{T}}_n})}_{t⩾0}$ is a ${({\mathcal{F}}_t)}_{t⩾0}$-martingale for all $n\in \mathbb{N}$ and $\mathbb{P}({\lim }_{n\to \infty }{\mathcal{T}}_n=\infty )=1$. In this case we call X a continuous local martingale. If, in addition, $X_0=0$ $\mathbb{P}$-a.s., we write $X\in {\mathcal{M}}_2^{c,loc}$.

A continuous semi-martingale ${(X_t)}_{t⩾0}$ is a ${({\mathcal{F}}_t)}_{t⩾0}$-adapted process such that the following (unique) decomposition holds:

$X_t=X_0+M_t+B_t,0⩽t<\infty .$

In the above $M={(M_t)}_{t⩾0}\in {\mathcal{M}}_2^{c,loc}$ and $B={(B_t)}_{t⩾0}$ is the difference of two continuous increasing and ${({\mathcal{F}}_t)}_{t⩾0}$-adapted processes $A^{\pm }={(A_t^{\pm })}_{t⩾0}$, i.e. there holds

$B_t=A_t^{+}-A_t^{-},0⩽t<\infty ,$

with $A_0^{\pm }=0$ $\mathbb{P}$-a.s.

Let $f:\mathbb{R}\to \mathbb{R}$ be a $C^2$-function and ${(X_t)}_{t⩾0}$ be a continuous ${({\mathcal{F}}_t)}_{t⩾0}$ semi-martingale with the decomposition (8.3.8). The following holds $\mathbb{P}$-a.s. for $0⩽t<\infty$

$\begin{array}{cc}\hfill f(X_t)& =f(X_0)+{\int }_0^t{f}^{\prime }(X_s)\mathrm{d}{X}_s+\frac{1}{2}{\int }_0^t{f}^{″}(X_s)\mathrm{d}{\langle \langle M\rangle \rangle }_s\hfill \\ \hfill & =f(X_0)+{\int }_0^t{f}^{\prime }(X_s)\mathrm{d}{M}_s+{\int }_0^t{f}^{\prime }(X_s)\mathrm{d}{B}_s+\frac{1}{2}{\int }_0^t{f}^{″}(X_s)\mathrm{d}{\langle \langle M\rangle \rangle }_s.\hfill \end{array}$

The stochastic integral ${\int }_0^t{f}^{\prime }(X_s)\mathrm{d}{M}_s$ in (8.3.10) is a continuous local martingale. The other two integrals in (8.3.10) are Lebesgue–Stieltjes integrals. They are of bounded variation as a function of t. Due to this ${(f(X_t))}_{t⩾0}$ is a continuous ${({\mathcal{F}}_t)}_{t⩾0}$ semi-martingale.

Equation (8.3.10) is often written in differential form

$\begin{array}{cc}\hfill \mathrm{d}f(X_t)& =f^{\prime }(X_t)\mathrm{d}{X}_t+\frac{1}{2}{f}^{″}(X_t)\mathrm{d}{\langle \langle M\rangle \rangle }_t\hfill \\ \hfill & =f^{\prime }(X_t)\mathrm{d}{M}_t+f^{\prime }(X_t)\mathrm{d}{B}_t+\frac{1}{2}{f}^{″}(X_t)\mathrm{d}{\langle \langle M\rangle \rangle }_t.\hfill \end{array}$

Note that this does not have a rigorous meaning. It only serves as an abbreviation of (8.3.10).

Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a probability space with filtration ${({\mathcal{F}}_t)}_{t⩾0}$ and $X_0\in L^2(\mathrm{\Omega },{\mathcal{F}}_0,\mathbb{P})$. Assume that μ and Σ are continuous on $[0,T]\times \mathbb{R}$ and globally Lipschitz continuous with respect to the second variable. Then there is a unique ${({\mathcal{F}}_t)}_{t⩾0}$-adapted process X such that (8.4.12) holds $\mathbb{P}$-a.s. for every $0⩽t⩽T$ and we have $X(0)=X_0$ a.s. The trajectories of X are a.s. continuous and we have

$\mathbb{E}[\underset{t\in (0,T)}{\sup }|X_t{|}^2]<\infty .$

Let ${\mathrm{\Lambda }}_0$ be a Borel probability measure on $\mathbb{R}$. Assume that μ and Σ are continuous on $[0,T]\times \mathbb{R}$ and have linear growth, i.e. there is $K⩾0$ such that

$|\mu (t,X)|+|\mathrm{\Sigma }(t,X)|⩽K(1+|X|)\forall (t,X)\in [0,T]\times \mathbb{R}.$

There is a quantity $((\mathrm{\Omega },\mathcal{F},{({\mathcal{F}}_t)}_{t⩾0},\mathbb{P}),W,X)$ with the following properties.

i) X is a ${({\mathcal{F}}_t)}_{t⩾0}$-adapted stochastic process with a.s. continuous trajectories such that

$\mathbb{E}[\underset{t\in (0,T)}{\sup }|X_t{|}^2]<\infty .$

ii) Equation (8.4.12) holds $\mathbb{P}$-a.s. for every $0⩽t⩽T$.

iii) We have $\mathbb{P}\circ X{(0)}^{-1}={\mathrm{\Lambda }}_0$.

Let μ and Σ satisfy (A1)–(A4). Assume we have a given probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ with filtration ${({\mathcal{F}}_t)}_{t⩾0}$, an initial datum ${\mathbf{X}}_0\in L^2(\mathrm{\Omega },{\mathcal{F}}_0,\mathbb{P})$ and a Brownian motion W with respect to ${({\mathcal{F}}_t)}_{t⩾0}$. Then there is a unique ${({\mathcal{F}}_t)}_{t⩾0}$-adapted process satisfying

$\mathbf{X}(t)={\mathbf{X}}_0+{\int }_0^t\mathit{\mu }(\sigma ,\mathbf{X}(\sigma ))\mathrm{d}\sigma +{\int }_0^t\mathbf{\Sigma }(\sigma ,\mathbf{X}(\sigma ))\mathrm{d}{\mathbf{W}}_{\sigma },\mathbb{P}\mathit{\text{-a.s.}},$

for every $t\in [0,T]$. The trajectories of X are $\mathbb{P}$-a.s. continuous and we have

$\mathbb{E}[\underset{t\in (0,T)}{\sup }|{\mathbf{X}}_t{|}^2]<\infty .$

Let the assumptions of Theorem 8.4.37 hold. Assume that ${\mathbf{X}}_0\in L^{\beta }(\mathrm{\Omega },{\mathcal{F}}_0,\mathbb{P})$ and $\tilde{\mu }\in L^{\beta }(\mathrm{\Omega },\mathcal{F},\mathbb{P};L^{\beta }(0,T))$ for some $\beta >2$. Then we have

$\mathbb{E}[\underset{t\in (0,T)}{\sup }|{\mathbf{X}}_t{|}^{\beta }]<\infty .$