The Itô Integral. Stochastic Differential Equations

Give an example of an adapted process that is not progressively measurable.

Definition 8. The optional $\sigma$-algebra $\mathscr {O}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the Skorokhod space $D[0, \infty )$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {O}$ are called optional.

Prove that, equivalently, the optional $\sigma$-algebra $\mathscr {O}$ can be defined as the $\sigma$-algebra of subsets of $\mathbb {R}_{+} \times \Omega$ generated by the stochastic intervals of the form

where $\tau (\omega )$ are Markov times (with respect to the filtration $\mathbb {F}$ ).

Definition 9. The predictable $\sigma$-algebra $\mathscr {P}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the space $C = C\left(\mathbb {R}_{+}, \mathbb {R}\right)$ of real functions continuous on $\mathbb {R}_{+}$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {P}$ are called predictable.

Prove that, equivalently, the predictable $\sigma$-algebra $\mathscr {P}$ can be defined as the $\sigma$-algebra generated by all adapted processes whose sample paths are merely left-continuous on $(0, \infty )$.

Definition 9. The predictable $\sigma$-algebra $\mathscr {P}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the space $C = C\left(\mathbb {R}_{+}, \mathbb {R}\right)$ of real functions continuous on $\mathbb {R}_{+}$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {P}$ are called predictable.

Verify that the $\sigma$-algebra $\mathscr {P}$ is generated by each of the following classes of subsets:

Definition 7. A subset $A$ of $\mathbb {R}_{+} \times \Omega$ is called progressively measurable (or simply progressive) if the process $X = \mathbf{1}_{A}$ (i.e., $X_{t}(\omega ) = \mathbf{1}_{A}(t, \omega )$, $t \geqslant 0, \omega \in \Omega$) is progressively measurable.

The collection of progressive sets $A$ forms a $\sigma$-algebra, denoted $\operatorname {Prog}$; here $\mathscr {F}_{\infty } = \bigvee_{t \geqslant 0} \mathscr {F}_{t}$, where $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ is the filtration under consideration.

Definition 8. The optional $\sigma$-algebra $\mathscr {O}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the Skorokhod space $D[0, \infty )$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {O}$ are called optional.

Definition 9. The predictable $\sigma$-algebra $\mathscr {P}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the space $C = C\left(\mathbb {R}_{+}, \mathbb {R}\right)$ of real functions continuous on $\mathbb {R}_{+}$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {P}$ are called predictable.

Verify that the following inclusions hold among the $\sigma$-algebras introduced above:

Let the process $X$ be progressively measurable and let $\tau$ be a finite Markov time. Show that the random variable $X_{\tau }$ is $\mathscr {F}_{\tau }$-measurable (this property is very nearly what makes it worthwhile to introduce the notion of progressive measurability at all).

Definition 8. The optional $\sigma$-algebra $\mathscr {O}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the Skorokhod space $D[0, \infty )$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {O}$ are called optional.

Verify that simple (piecewise right-continuous) functions $f = f(t, \omega )$ of the form

where the variables $\xi_{k}(\omega )$ are $\mathscr {F}_{t_{k}}$-measurable and $0 = t_{0} < \cdots < t_{m} = T$, are optional.

Definition 9. The predictable $\sigma$-algebra $\mathscr {P}$ of subsets of $\mathbb {R}_{+} \times \Omega$ is the $\sigma$-algebra generated by all adapted processes $X = \left\{ X_{t}, t \geqslant 0\right\}$, regarded as mappings $(t, \omega ) \mapsto X_{t}(\omega )$ from $\mathbb {R}_{+} \times \Omega$ to $(\mathbb {R}, \mathscr {B}(\mathbb {R}))$, whose sample paths belong to the space $C = C\left(\mathbb {R}_{+}, \mathbb {R}\right)$ of real functions continuous on $\mathbb {R}_{+}$. Processes that are measurable with respect to the $\sigma$-algebra $\mathscr {P}$ are called predictable.

Show that simple (piecewise left-continuous for $t \in (0, T]$ ) functions $f = f(t, \omega )$ of the form

where the variable $\eta (\omega )$ is $\mathscr {F}_{0}$-measurable, the $\eta_{k}(\omega )$ are $\mathscr {F}_{t_{k}}$-measurable, and $0 = t_{0} < \cdots < t_{m} = T$, are predictable.

Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be a Wiener process, $\mathscr {F}_{t}^{W}$ the $\sigma$-algebra generated by the random variables $W_{s}$, $0 \leqslant s \leqslant t$, $\mathbb {F}_{T}^{W} = \left(\mathscr {F}_{t}^{W}\right)_{t \in [0, T]}$ and $\mathscr {F}^{W} = \sigma \left\{ W_{t}, t \geqslant 0\right\}$; finally, $L_{T}^{2} = L^{2}\left(\Omega \times [0, T], \mathscr {F}^{W} \otimes \mathscr {B}([0, T]), \mathbb {P} \otimes \operatorname {mes}\right)$.

Definition 1. We say that a random function $f = \left\{ f_{t}, t \in [0, T]\right\}$ is nonanticipating with respect to the filtration $\mathbb {F}_{T}^{W}$ if it is adapted to $\mathbb {F}_{T}^{W}$ (i.e., for each $t \in [0, T]$ the random variable $f_{t}$ is $\mathscr {F}_{t}^{W}$-measurable) and is $\mathscr {B}([0, T]) \otimes \mathscr {F}^{W}$-measurable in the pair of variables $(t, \omega ) \in [0, T] \times \Omega$. The class of such random functions is denoted by $\mathscr {A}_{T}$.

Carry out the construction of the Itô integral presented in § 2 and § 3, starting from the class of optional simple functions appearing in Exercise 7 (to which the same value of the Itô integral is assigned as to the predictable simple functions of Exercise 8). Can one assert that the closure in $L_{T}^{2}$ of this collection of simple functions belonging to $L_{T}^{2}$ yields the space $\mathscr {A}_{T} \cap L_{T}^{2}$ ?

The idea of the construction of the Itô integral in § 2 and § 3 is as follows: (1) first the integrals are defined (in a quite natural way) for piecewise constant "nonanticipating" random functions, and then (2) for more general functions $f$, by approximating them by simple functions $f^{(n)}$ and relying on isometry ideas, the stochastic integrals $I_{T}(f)$ are defined as the limits (in the mean-square sense) of the variables $I_{T}\left(f^{(n)}\right)$, $n \rightarrow \infty$.

By $I_{t}(f) = \int_{(0, t]} f_{s}(\omega ) d W_{s}(\omega )$, $t \in [0, T]$, we denote the Itô stochastic integral with respect to a Wiener process $W = \left\{ W_{t}, t \geqslant 0\right\}$ of a nonanticipating (i.e., adapted to the filtration and measurable in the pair of variables $(t, \omega )$) random function $f = f_{t}(\omega )$ such that $\mathbb {E}\left[\int_{0}^{T} f_{t}^{2}(\omega ) d t\right] < \infty$.

Let $\tau$ be a Markov time such that $\tau (\omega ) \leqslant T$ for all $\omega \in \Omega$ ($T$ is a positive constant). Prove that $I_{\tau }(f) = I_{T}\left(f \mathbf{1}_{(0, \tau ]}\right)$, where $I_{\tau }(f): = I_{\tau (\omega )}(f)$.

By $I_{t}(f) = \int_{(0, t]} f_{s}(\omega ) d W_{s}(\omega )$, $t \geqslant 0$, we denote the Itô stochastic integral of a function $f$ with respect to a Wiener process $W = \left\{ W_{t}, t \geqslant 0\right\}$.

Prove that if $f:[0, \infty ) \rightarrow \mathbb {R}$ is a nonrandom function from the Skorokhod space $D[0, \infty )$, then the process $I_{t}(f), t \geqslant 0$, is Gaussian. Find its mean and covariance function.

Definition 10. We say that a function $f$ belongs to the class $J_{1}$ if $f:[0, \infty ) \times \Omega \rightarrow \mathbb {R}$ is progressively measurable and

The idea of constructing the integral $I_{t}(f)$ for a function $f \in J_{1}$ is to choose a sequence of functions $f_{n}, n \in \mathbb {N}$, for which the integral $I_{t}\left(f_{n}\right)$ has already been constructed and such that

From this one derives that the sequence $\left\{ I_{t}\left(f_{n}\right)\right\}_{n \in \mathbb {N}}$ is Cauchy in probability. Consequently, there exists a random variable, denoted $I_{t}(f)$, such that $I_{t}\left(f_{n}\right) \stackrel{\mathbb {P}}{\rightarrow } I_{t}(f), n \rightarrow \infty$. The limit $I_{t}(f)$ is still written as $\int_{(0, t]} f(s, \omega ) d W_{s}$ or $(f \cdot W)_{t}$.

Prove that the sequence $\left\{ f_{n}, n \in \mathbb {N}\right\}$ described above in (85) exists. Prove that if $f \in J_{1}$, then there exists an a.s. continuous modification of the process $I_{t}(f), t \geqslant 0$.

Definition 10. We say that a function $f$ belongs to the class $J_{1}$ if $f:[0, \infty ) \times \Omega \rightarrow \mathbb {R}$ is progressively measurable and

Consider, for $f \in J_{1}$, the process

Prove that $d Z_{t} = Z_{t} f(t, \omega ) d W_{t}$ (as before, all differential relations are shorthand for the corresponding integral ones).

Definition 11. A stochastic process $X = \left\{ X_{t}, t \geqslant 0\right\}$ with values in $\mathbb {R}^{m}$ is called automodel (self-similar) if for each $a > 0$ one can find $b > 0$ such that

If in definition (87) the parameter $b = a^{H}$ for every $a > 0$, then the stochastic process $X$ is called a self-similar process with Hurst exponent $H$.

Recall that fractional Brownian motion $B^{(H)} = \left\{ B^{(H)}(t), t \geqslant 0\right\}$ for $0 < H \leqslant 1$ is defined as a centered Gaussian process with covariance function

Prove that $B^{(H)} = \left\{ B^{(H)}(t), t \geqslant 0\right\}$, where $0 < H \leqslant 1$, is a self-similar process (with Hurst exponent $H$ ).

Recall that fractional Brownian motion $B^{(H)} = \left\{ B^{(H)}(t), t \geqslant 0\right\}$ for $0 < H \leqslant 1$ is defined as a centered Gaussian process with covariance function

Prove that for the process $B^{(H)}$, as $n \rightarrow \infty$,

Explain why the quantity $X$ appearing in Theorem 11 is a functional of Brownian motion, i.e. $X(\omega ) = g(W(s, \omega ), 0 \leqslant s \leqslant T)$, where $g: C[0, T] \rightarrow \mathbb {R}$ and $g \in \mathscr {B}(C[0, T]) \mid \mathscr {B}(\mathbb {R})$.

Let $J_{1}([0, T])$ be the space of progressively measurable functions $f:[0, T] \times \Omega \rightarrow \mathbb {R}$ satisfying (84) for $t \in [0, T]$.

Theorem 11 (Itô, Clark). Let $\left(\mathscr {F}_{t}^{W}\right)_{t \in [0, T]}$ be the natural filtration of Brownian motion $W = \left\{ W_{t}, t \in [0, T]\right\}$ and let $X = X(\omega )$ be an $\mathscr {F}_{T}^{W} \mid \mathscr {B}(\mathbb {R})$-measurable random variable. The following assertions hold.

1. If $\mathbb {E}\left[X^{2}\right] < \infty$, then there exists a stochastic process $f = \left\{ f(s), s \in [0, T]\right\} \in \mathscr {A}_{T} \cap L_{T}^{2}$ such that

   $$X = \mathbb {E}\left[X\right]+\int _{0}^{T} f(s, \omega ) d W_{s} \quad \text{ a.s. } \tag {89}$$

2. If $\mathbb {E}\left[\left|X\right|\right] < \infty$, then representation (89) holds with some process $f \in J_{1}([0, T])$.

3. If $X$ is a positive random variable (i.e. $\mathbb {P}\left(X > 0\right) = 1$) and $\mathbb {E}\left[X\right] < \infty$, then there exists a process $f \in J_{1}[0, T]$ such that $X$ admits a representation of the form $X = Z_{T} \mathbb {E}\left[X\right]$, where $Z_{T}$ is defined according to (86).

   $$Z_{t} = \mathrm{exp}\left(\int _{0}^{t} f(s, \omega ) d W_{s}-\frac{1}{2} \int _{0}^{t} f^{2}(s, \omega ) d s\right), \quad t \geqslant 0 \tag {86}$$

Using the Itô formula, prove that for a deterministic function $f = f(t)$ of class $C^{1}$ the "integration by parts" formula holds:

Construct an example of a random function $f = f(t, \omega )$ for which formula (93) does not hold.

Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be a standard (one-dimensional) Brownian motion.

We define, as follows, $[f(W), W]$ — the quadratic covariation of the processes $f(W)$ and $W$:

Here $\mathbb {P}$ - $\lim$ denotes the limit in probability, and $\left\{ t^{(n)}(m), m \in \mathbb {N}\right\}$ for each $n \in \mathbb {N}$ are Riemann sequences of deterministic times $t^{(n)}(m)$, i.e. $t^{(n)}(m) \leqslant t^{(n)}(m+1)$ for $m \in \mathbb {N}$ for each $n \in \mathbb {N}$ and $t^{(n)}(m) \rightarrow \infty$ as $m \rightarrow \infty$; moreover, for any $t > 0$

Let $f \in C^{2}$ be a function. Prove that

Thus, formula (103) turns into the Itô formula.

Let the function $F = F(x)$ be absolutely continuous,

where the function $f = f(y) \in L_{\mathrm{loc}}^{2}(\mathbb {R})$, i.e. for any $c > 0$

Theorem 15 (Protter, Föllmer, Shiryaev; [129]). Under the above assumptions on the function $F = F(x)$, the following formula holds

Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be a standard (one-dimensional) Brownian motion.

We define, as follows, $[f(W), W]$ — the quadratic covariation of the processes $f(W)$ and $W$:

Here $\mathbb {P}$ - $\lim$ denotes the limit in probability, and $\left\{ t^{(n)}(m), m \in \mathbb {N}\right\}$ for each $n \in \mathbb {N}$ are Riemann sequences of deterministic times $t^{(n)}(m)$, i.e. $t^{(n)}(m) \leqslant t^{(n)}(m+1)$ for $m \in \mathbb {N}$ for each $n \in \mathbb {N}$ and $t^{(n)}(m) \rightarrow \infty$ as $m \rightarrow \infty$; moreover, for any $t > 0$

Let $f(x) = \left|x\right|$. Prove that $[f(W), W]_{t} = 2 L_{t}(0)$, where $L_{t}(0)$ is the (P. Lévy) local time of Brownian motion at the point 0 , defined as follows:

Thus, in this case formula (103) turns into Tanaka's formula for Brownian motion.

Let the function $F = F(x)$ be absolutely continuous,

where the function $f = f(y) \in L_{\mathrm{loc}}^{2}(\mathbb {R})$, i.e. for any $c > 0$

Theorem 15 (Protter, Föllmer, Shiryaev; [129]). Under the above assumptions on the function $F = F(x)$, the following formula holds

Let us now turn to a homogeneous diffusion, i.e. consider the solution of the equation

where $W_{t}$ is an $m$-dimensional Wiener process, and the functions $b: \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ and $\sigma : \mathbb {R}^{n} \rightarrow \mathbb {R}^{n \times m}$ satisfy the conditions of Theorem 8 , which in this case reduce to a single requirement: there exists $L > 0$ such that

(for a vector, $\left|\cdot \right|$ denotes the Euclidean norm, and $\left|\sigma \right|^{2} = \sum_{i = 1}^{n} \sum_{k = 1}^{m} \sigma_{i k}^{2}$ ), which implies that for some $c > 0$

Denote by $X_{t}^{s, x}$ the unique strong solution of equation (106) for $t \geqslant s$.

Prove that for each $x \in \mathbb {R}^{n}$ the process $X_{t}^{s, x}, t \geqslant s$, defined above is a homogeneous Markov process.

Prove that, under the conditions of Theorem 8, the uniqueness of the strong solution (on $[0, T]$ ) holds in the wider class of processes for which $\mathbb {E}\left[X_{t}^{2}\right]$ exists for $t \in [0, T]$.

Let a filtration $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ and a Wiener process $W = \left\{ W_{t}, t \geqslant 0\right\}$ with respect to $\mathbb {F}$ be given. Equation (59) is the stochastic differential equation

with initial condition $X_{0} = Z$, which is an $\mathscr {F}_{0}$-measurable random variable. This equation is understood as a formal notation for the stochastic integral equation

where $b$ and $\sigma$ are defined on $[0, T] \times \mathbb {R}$.

Theorem 8. Let the functions $b$ and $\sigma$, measurable on $[0, T] \times \mathbb {R}$, satisfy conditions (61) and (62). Let the random variable $Z$ be $\mathscr {F}_{0}$-measurable, with $\mathbb {E}\left[Z^{2}\right] < \infty$. Then there exists a unique strong solution of equation (60) with $\mathscr {F}_{0}$-measurable initial condition $X_{0} = Z$ such that $X_{t} \in L^{2}(\Omega )$ for every $t \in [0, T]$ and the function $\mathbb {E}\left[X_{t}^{2}\right]$ is bounded on $[0, T]$.

Here (61) is the Lipschitz condition: there exists a constant $L = L(T) > 0$ such that

and (62) is the condition that, for some $c = c(T) > 0$,

Show that if the conditions of Theorem 8 are satisfied on the half-open interval $[0, T)$, then its assertion also holds on this interval, except that in this case one considers the class of processes $X$ for which the function $\mathbb {E}\left[X_{t}^{2}\right]$ is bounded on every closed interval contained in $[0, T)$.

Let a filtration $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ and a Wiener process $W = \left\{ W_{t}, t \geqslant 0\right\}$ with respect to $\mathbb {F}$ be given. Equation (59) is the stochastic differential equation

with initial condition $X_{0} = Z$, which is an $\mathscr {F}_{0}$-measurable random variable. This equation is understood as a formal notation for the stochastic integral equation

where $b$ and $\sigma$ are defined on $[0, T] \times \mathbb {R}$.

Theorem 8. Let the functions $b$ and $\sigma$, measurable on $[0, T] \times \mathbb {R}$, satisfy conditions (61) and (62). Let the random variable $Z$ be $\mathscr {F}_{0}$-measurable, with $\mathbb {E}\left[Z^{2}\right] < \infty$. Then there exists a unique strong solution of equation (60) with $\mathscr {F}_{0}$-measurable initial condition $X_{0} = Z$ such that $X_{t} \in L^{2}(\Omega )$ for every $t \in [0, T]$ and the function $\mathbb {E}\left[X_{t}^{2}\right]$ is bounded on $[0, T]$.

Here (61) is the Lipschitz condition: there exists a constant $L = L(T) > 0$ such that

and (62) is the condition that, for some $c = c(T) > 0$,

Using the Itô formula, prove that the solution of the (one-dimensional) equation

is the process

Let $X_{t}$ be the solution of the (one-dimensional) equation

(continuation of Exercise 23). Prove that $X_{t} \rightarrow \beta$ a.s. as $t \rightarrow T-$. Thus, the solution of equation (109) is a Brownian bridge over the interval $[0, T]$ with pinned endpoints $X_{0} = \alpha$ and $X_{T} = \beta$. The standard Brownian bridge is obtained for $T = 1$ and $\alpha = \beta = 0$.

Prove that the stochastic differential equation

with the "bad" coefficient $\sigma (x) = \operatorname {sgn} x$ has a strong solution, and moreover a unique one.

Prove that the equation

where $\sigma (x) = \operatorname {sgn} x$, has at least two solutions (but "weak" ones). Moreover, on some probability spaces this equation may have no "strong" solution at all.

Let the conditions of Theorem 8 hold. Prove that the solution (weak or strong) of equation (59) is weakly unique.

Let a filtration $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$ and a Wiener process $W = \left\{ W_{t}, t \geqslant 0\right\}$ with respect to $\mathbb {F}$ be given. Equation (59) is the stochastic differential equation

with initial condition $X_{0} = Z$, which is an $\mathscr {F}_{0}$-measurable random variable. This equation is understood as a formal notation for the stochastic integral equation

where $b$ and $\sigma$ are defined on $[0, T] \times \mathbb {R}$.

Theorem 8. Let the functions $b$ and $\sigma$, measurable on $[0, T] \times \mathbb {R}$, satisfy conditions (61) and (62). Let the random variable $Z$ be $\mathscr {F}_{0}$-measurable, with $\mathbb {E}\left[Z^{2}\right] < \infty$. Then there exists a unique strong solution of equation (60) with $\mathscr {F}_{0}$-measurable initial condition $X_{0} = Z$ such that $X_{t} \in L^{2}(\Omega )$ for every $t \in [0, T]$ and the function $\mathbb {E}\left[X_{t}^{2}\right]$ is bounded on $[0, T]$.

Here (61) is the Lipschitz condition: there exists a constant $L = L(T) > 0$ such that

and (62) is the condition that, for some $c = c(T) > 0$,

Definition 15. Weak uniqueness of a (weak or strong) solution of equation (59) means that any two solutions (in the sense under consideration) with the same initial conditions have the same distributions, i.e., the same finite-dimensional distributions.