The Generalized Ito Formula

Recall that Ito’s lemma expresses a twice differentiable function ${f}$ applied to a continuous semimartingale ${X}$ in terms of stochastic integrals, according to the following formula

$\displaystyle f(X) = f(X_0)+\int f^\prime(X)\,dX + \frac{1}{2}\int f^{\prime\prime}(X)\,d[X].$

(1)

In this form, the result only applies to continuous processes but, as I will show in this post, it is possible to generalize to arbitrary noncontinuous semimartingales. The result is also referred to as Ito’s lemma or, to distinguish it from the special case for continuous processes, it is known as the generalized Ito formula or generalized Ito’s lemma.

If equation (1) is to be extended to noncontinuous processes then, there are two immediate points to be considered. The first is that if the process ${X}$ is not continuous then it need not be a predictable process, so ${f^\prime(X),f^{\prime\prime}(X)}$ need not be predictable either. So, the integrands in (1) will not be ${X}$ -integrable. To remedy this, we should instead use the left limits ${X_{t-}}$ in the integrands, which is left-continuous and adapted and therefore is predictable. The second point is that the jumps of the left hand side of (1) are equal to ${\Delta f(X)}$ and, on the right, they are ${f^\prime(X_-)\Delta X+\frac{1}{2}f^{\prime\prime}(X_-)\Delta X^2}$ . There is no reason that these should be equal, and (1) cannot possibly hold in general. To fix this, we can simply add on the correction to the jump terms on the right hand side,

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle f(X_t) =&\displaystyle f(X_0)+\int_0^t f^\prime(X_-)\,dX + \frac{1}{2}\int_0^t f^{\prime\prime}(X_-)\,d[X]\smallskip\\ &\displaystyle +\sum_{s\le t}\left(\Delta f(X_s)-f^\prime(X_{s-})\Delta X_s-\frac{1}{2}f^{\prime\prime}(X_{s-})\Delta X_s^2\right). \end{array}$

(2)

Continue reading “The Generalized Ito Formula” →

Ito’s Lemma

Ito’s lemma, otherwise known as the Ito formula, expresses functions of stochastic processes in terms of stochastic integrals. In standard calculus, the differential of the composition of functions ${f(x), x(t)}$ satisfies ${df(x(t))=f^\prime(x(t))dx(t)}$ . This is just the chain rule for differentiation or, in integral form, it becomes the change of variables formula.

In stochastic calculus, Ito’s lemma should be used instead. For a twice differentiable function ${f}$ applied to a continuous semimartingale ${X}$ , it states the following,

$\displaystyle df(X) = f^\prime(X)\,dX + \frac{1}{2}f^{\prime\prime}(X)\,dX^2.$

This can be understood as a Taylor expansion up to second order in ${dX}$ , where the quadratic term ${dX^2\equiv d[X]}$ is the quadratic variation of the process ${X}$ .

A d-dimensional process ${X=(X^1,\ldots,X^d)}$ is said to be a semimartingale if each of its components, ${X^i}$ , are semimartingales. The first and second order partial derivatives of a function are denoted by ${D_if}$ and ${D_{ij}f}$ , and I make use of the summation convention where indices ${i,j}$ which occur twice in a single term are summed over. Then, the statement of Ito’s lemma is as follows.

Theorem 1 (Ito’s Lemma) Let ${X=(X^1,\ldots,X^d)}$ be a continuous d-dimensional semimartingale taking values in an open subset ${U\subseteq{\mathbb R}^d}$ . Then, for any twice continuously differentiable function ${f\colon U\rightarrow{\mathbb R}}$ , ${f(X)}$ is a semimartingale and,

$\displaystyle df(X) = D_if(X)\,dX^i + \frac{1}{2}D_{ij}f(X)\,d[X^i,X^j].$ (1)

Continue reading “Ito’s Lemma” →

Properties of Quadratic Variations

Being able to handle quadratic variations and covariations of processes is very important in stochastic calculus. Apart from appearing in the integration by parts formula, they are required for the stochastic change of variables formula, known as Ito’s lemma, which will be the subject of the next post. Quadratic covariations satisfy several simple relations which make them easy to handle, especially in conjunction with the stochastic integral.

Recall from the previous post that the covariation ${[X,Y]}$ is a cadlag adapted process, so that its jumps ${\Delta [X,Y]_t\equiv [X,Y]_t-[X,Y]_{t-}}$ are well defined.

Lemma 1 If ${X,Y}$ are semimartingales then

$\displaystyle \Delta [X,Y]=\Delta X\Delta Y.$ (1)

In particular, ${\Delta [X]=\Delta X^2}$ .

Proof: Taking the jumps of the integration by parts formula for ${XY}$ gives

$\displaystyle \Delta XY = X_{-}\Delta Y + Y_{-}\Delta X + \Delta [X,Y],$

and rearranging this gives the result. ⬜

An immediate consequence is that quadratic variations and covariations involving continuous processes are continuous. Another consequence is that the sum of the squares of the jumps of a semimartingale over any bounded interval must be finite.

Corollary 2 Every semimartingale ${X}$ satisfies

$\displaystyle \sum_{s\le t}\Delta X^2_s\le [X]_t<\infty.$

Proof: As ${[X]}$ is increasing, the inequality ${[X]_t\ge \sum_{s\le t}\Delta [X]_s}$ holds. Substituting in ${\Delta[X]=\Delta X^2}$ gives the result. ⬜

Next, the following result shows that covariations involving continuous finite variation processes are zero. As Lebesgue-Stieltjes integration is only defined for finite variation processes, this shows why quadratic variations do not play an important role in standard calculus. For noncontinuous finite variation processes, the covariation must have jumps satisfying (1), so will generally be nonzero. In this case, the covariation is just given by the sum over these jumps. Integration with respect to any FV process ${V}$ can be defined as the Lebesgue-Stieltjes integral on the sample paths, which is well defined for locally bounded measurable integrands and, when the integrand is predictable, agrees with the stochastic integral.

Lemma 3 Let ${X}$ be a semimartingale and ${V}$ be an FV process. Their covariation is

$\displaystyle [X,V]_t = \int_0^t \Delta X\,dV = \sum_{s\le t}\Delta X_s\Delta V_s.$ (2)

In particular, if either of ${X}$ or ${V}$ is continuous then ${[X,V]=0}$ .

Continue reading “Properties of Quadratic Variations” →

Quadratic Variations and Integration by Parts

A major difference between standard integral calculus and stochastic calculus is the existence of quadratic variations and covariations. Such terms show up, for example, in the stochastic version of the integration by parts formula.

For motivation, let us start by considering a standard argument for differentiable processes. The increment of a process ${X}$ over a time step ${\delta t>0}$ can be written as ${\delta X_t\equiv X_{t+\delta t}-X_t}$ . The following identity is easily verified,

$\displaystyle \delta XY = X\delta Y + Y\delta X + \delta X \delta Y.$

(1)

Now, divide the time interval ${[0,t]}$ into ${n}$ equal parts. That is, set ${t_k=kt/n}$ for ${k=0,1,\ldots,n}$ . Then, using ${\delta t=1/n}$ and summing equation (1) over these times,

$\displaystyle X_tY_t -X_0Y_0=\sum_{k=0}^{n-1} X_{t_k}\delta Y_{t_k} +\sum_{k=0}^{n-1}Y_{t_k}\delta X_{t_k}+\sum_{k=0}^{n-1}\delta X_{t_k}\delta Y_{t_k}.$

(2)

If the processes are continuously differentiable, then the final term on the right hand side is a sum of ${n}$ terms, each of order ${1/n^2}$ , and therefore is of order ${1/n}$ . This vanishes in the limit ${n\rightarrow\infty}$ , leading to the integration by parts formula

$\displaystyle X_tY_t-X_0Y_0 = \int_0^t X\,dY + \int_0^t Y\,dX.$

Now, suppose that ${X,Y}$ are standard Brownian motions. Then, ${\delta X,\delta Y}$ are normal random variables with standard deviation ${\sqrt{\delta t}}$ . It follows that the final term on the right hand side of (2) is a sum of ${n}$ terms each of which is, on average, of order ${1/n}$ . So, even in the limit as ${n}$ goes to infinity, it does not vanish. Consequently, in stochastic calculus, the integration by parts formula requires an additional term, which is called the quadratic covariation (or, just covariation) of ${X}$ and ${Y}$ . Continue reading “Quadratic Variations and Integration by Parts” →

Properties of the Stochastic Integral

In the previous two posts I gave a definition of stochastic integration. This was achieved via an explicit expression for elementary integrands, and extended to all bounded predictable integrands by bounded convergence in probability. The extension to unbounded integrands was done using dominated convergence in probability. Similarly, semimartingales were defined as those cadlag adapted processes for which such an integral exists.

The current post will show how the basic properties of stochastic integration follow from this definition. First, if ${V}$ is a cadlag process whose sample paths are almost surely of finite variation over an interval ${[0,t]}$ , then ${\int_0^t\xi\,dV}$ can be interpreted as a Lebesgue-Stieltjes integral on the sample paths. If the process is also adapted, then it will be a semimartingale and the stochastic integral can be used. Fortunately, these two definitions of integration do agree with each other. The term FV process is used to refer to such cadlag adapted processes which are almost surely of finite variation over all bounded time intervals. The notation ${\int_0^t\vert\xi\vert\,\vert dV\vert}$ represents the Lebesgue-Stieltjes integral of ${\vert\xi\vert}$ with respect to the variation of ${V}$ . Then, the condition for ${\xi}$ to be ${V}$ -integrable in the Lebesgue-Stieltjes sense is precisely that this integral is finite.

Lemma 1 Every FV process ${V}$ is a semimartingale. Furthermore, let ${\xi}$ be a predictable process satisfying

$\displaystyle \int_0^t\vert\xi\vert\,\vert dV\vert<\infty$ (1)

almost surely, for each ${t\ge 0}$ . Then, ${\xi\in L^1(V)}$ and the stochastic integral ${\int\xi\,dV}$ agrees with the Lebesgue-Stieltjes integral, with probability one.

Continue reading “Properties of the Stochastic Integral” →

Extending the Stochastic Integral

In the previous post, I used the property of bounded convergence in probability to define stochastic integration for bounded predictable integrands. For most applications, this is rather too restrictive, and in this post the integral will be extended to unbounded integrands. As bounded convergence is not much use in this case, the dominated convergence theorem will be used instead.

The first thing to do is to define a class of integrable processes for which the integral with respect to ${X}$ is well-defined. Suppose that ${\xi^n}$ is a sequence of predictable processes dominated by any such ${X}$ -integrable process ${\alpha}$ , so that ${\vert\xi^n\vert\le\vert\alpha\vert}$ for each ${n}$ . If this sequence converges to a limit ${\xi}$ , then dominated convergence in probability states that the integrals converge in probability,

$\displaystyle \int_0^t\xi^n\,dX\rightarrow\int_0^t\xi\,dX\ \ \text{(in probability)}$

(1)

as ${n\rightarrow\infty}$ . Continue reading “Extending the Stochastic Integral” →

The Stochastic Integral

Having covered the basics of continuous-time processes and filtrations in the previous posts, I now move on to stochastic integration. In standard calculus and ordinary differential equations, a central object of study is the derivative ${df/dt}$ of a function ${f(t)}$ . This does, however, require restricting attention to differentiable functions. By integrating, it is possible to generalize to bounded variation functions. If ${f}$ is such a function and ${g}$ is continuous, then the Riemann-Stieltjes integral ${\int_0^tg\,df}$ is well defined. The Lebesgue-Stieltjes integral further generalizes this to measurable integrands.

However, the kinds of processes studied in stochastic calculus are much less well behaved. For example, with probability one, the sample paths of standard Brownian motion are nowhere differentiable. Furthermore, they have infinite variation over bounded time intervals. Consequently, if ${X}$ is such a process, then the integral ${\int_0^t\xi\,dX}$ is not defined using standard methods.

Stochastic integration with respect to standard Brownian motion was developed by Kiyoshi Ito. This required restricting the class of possible integrands to be adapted processes, and the integral can then be constructed using the Ito isometry. This method was later extended to more general square integrable martingales and, then, to the class of semimartingales. It can then be shown that, as with Lebesgue integration, a version of the bounded and dominated convergence theorems are satisfied.

In these notes, a more direct approach is taken. The idea is that we simply define the stochastic integral such that the required elementary properties are satisfied. That is, it should agree with the explicit expressions for certain simple integrands, and should satisfy the bounded and dominated convergence theorems. Much of the theory of stochastic calculus follows directly from these properties, and detailed constructions of the integral are not required for many practical applications. Continue reading “The Stochastic Integral” →

Local Martingales

Recall from the previous post that a cadlag adapted process ${X}$ is a local martingale if there is a sequence ${\tau_n}$ of stopping times increasing to infinity such that the stopped processes ${1_{\{\tau_n>0\}}X^{\tau_n}}$ are martingales. Local submartingales and local supermartingales are defined similarly.

An example of a local martingale which is not a martingale is given by the `double-loss’ gambling strategy. Interestingly, in 18th century France, such strategies were known as martingales and is the origin of the mathematical term. Suppose that a gambler is betting sums of money, with even odds, on a simple win/lose game. For example, betting that a coin toss comes up heads. He could bet one dollar on the first toss and, if he loses, double his stake to two dollars for the second toss. If he loses again, then he is down three dollars and doubles the stake again to four dollars. If he keeps on doubling the stake after each loss in this way, then he is always gambling one more dollar than the total losses so far. He only needs to continue in this way until the coin eventually does come up heads, and he walks away with net winnings of one dollar. This therefore describes a fair game where, eventually, the gambler is guaranteed to win.

Of course, this is not an effective strategy in practise. The losses grow exponentially and, if he doesn’t win quickly, the gambler must hit his credit limit in which case he loses everything. All that the strategy achieves is to trade a large probability of winning a dollar against a small chance of losing everything. It does, however, give a simple example of a local martingale which is not a martingale.

The gamblers winnings can be defined by a stochastic process ${\{Z_n\}_{n=1,\ldots}}$ representing his net gain (or loss) just before the n’th toss. Let ${\epsilon_1,\epsilon_2,\ldots}$ be a sequence of independent random variables with ${{\mathbb P}(\epsilon_n=1)={\mathbb P}(\epsilon_n=-1)=1/2}$ . Here, ${\epsilon_n}$ represents the outcome of the n’th toss, with 1 referring to a head and -1 referring to a tail. Set ${Z_1=0}$ and

$\displaystyle Z_{n}=\begin{cases} 1,&\text{if }Z_{n-1}=1,\\ Z_{n-1}+\epsilon_n(1-Z_{n-1}),&\text{otherwise}. \end{cases}$

This is a martingale with respect to its natural filtration, starting at zero and, eventually, ending up equal to one. It can be converted into a local martingale by speeding up the time scale to fit infinitely many tosses into a unit time interval

$\displaystyle X_t=\begin{cases} Z_n,&\text{if }1-1/n\le t<1-1/(n+1),\\ 1,&\text{if }t\ge 1. \end{cases}$

This is a martingale with respect to its natural filtration on the time interval ${[0,1)}$ . Letting ${\tau_n=\inf\{t\colon\vert X_t\vert\ge n\}}$ then the optional stopping theorem shows that ${X^{\tau_n}_t}$ is a uniformly bounded martingale on ${t<1}$ , continuous at ${t=1}$ , and constant on ${t\ge 1}$ . This is therefore a martingale, showing that ${X}$ is a local martingale. However, ${{\mathbb E}[X_1]=1\not={\mathbb E}[X_0]=0}$ , so it is not a martingale. Continue reading “Local Martingales” →

Localization

Special classes of processes, such as martingales, are very important to the study of stochastic calculus. In many cases, however, processes under consideration `almost’ satisfy the martingale property, but are not actually martingales. This occurs, for example, when taking limits or stochastic integrals with respect to martingales. It is necessary to generalize the martingale concept to that of local martingales. More generally, localization is a method of extending a given property to a larger class of processes. In this post I mention a few definitions and simple results concerning localization, and look more closely at local martingales in the next post.

Definition 1 Let P be a class of stochastic processes. Then, a process X is locally in P if there exists a sequence of stopping times ${\tau_n\uparrow\infty}$ such that the stopped processes

$\displaystyle 1_{\{\tau_n>0\}}X^{\tau_n}$

are in P. The sequence ${\tau_n}$ is called a localizing sequence for X (w.r.t. P).

I write ${P_{\rm loc}}$ for the processes locally in P. Choosing the sequence ${\tau_n\equiv\infty}$ of stopping times shows that ${P\subseteq P_{\rm loc}}$ . A class of processes is said to be stable if ${1_{\{\tau>0\}}X^\tau}$ is in P whenever X is, for all stopping times ${\tau}$ . For example, the optional stopping theorem shows that the classes of cadlag martingales, cadlag submartingales and cadlag supermartingales are all stable.

Definition 2 A process is a

a local martingale if it is locally in the class of cadlag martingales.

a local submartingale if it is locally in the class of cadlag submartingales.

a local supermartingale if it is locally in the class of cadlag supermartingales.

Continue reading “Localization” →

Class (D) Processes

A stochastic process X is said to be uniformly integrable if the set of random variables ${\{X_t\colon t\in{\mathbb R}_+\}}$ is uniformly integrable. However, even if this is the case, it does not follow that the set of values of the process sampled at arbitrary stopping times is uniformly integrable.

For the case of a cadlag martingale X, optional sampling can be used. If ${t\ge 0}$ is any fixed time then this says that ${X_\tau={\mathbb E}[X_t\mid\mathcal{F}_\tau]}$ for stopping times ${\tau\le t}$ . As sets of conditional expectations of a random variable are uniformly integrable, the following result holds.

Lemma 1 Let X be a cadlag martingale. Then, for each ${t\ge 0}$ , the set

$\displaystyle \{X_\tau\colon\tau\le t\text{\ is\ a\ stopping\ time}\}$

is uniformly integrable.

This suggests the following generalized concepts of uniform integrability for stochastic processes.

Definition 2 Let X be a jointly measurable stochastic process. Then, it is

of class (D) if ${\{X_\tau\colon\tau<\infty\text{ is a stopping time}\}}$ is uniformly integrable.

of class (DL) if, for each ${t\ge 0}$ , ${\{X_\tau\colon\tau\le t\text{ is a stopping time}\}}$ is uniformly integrable.

Continue reading “Class (D) Processes” →

	Anonymous on Spitzer’s Formula
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	Anonymous on Spitzer’s Formula