Research and Lecture notes

In this lecture, we study the regularity of the solution of a stochastic differential equation with respect to its initial condition. The key tool is a multimensional parameter extension of the Kolmogorov continuity theorem whose proof is almost identical to the one-dimensional case.

Theorem. Let be a -dimensional stochastic process such that there exist positive constants such that for every

There exists a modification of the process such that for every there exists a finite random variable such that for every

As above, we consider two functions and and we assume that there exists such that

As we already know, for every , there exists a continuous and adapted process such that for ,

Proposition. Let and be a compact set in . For every , there exists a constant such that for every and ,
.
As a consequence, there exists a modification of the process such that for , ,

and such that is continuous for almost every .

Proof. As before, we can find such that
, ;
and ), .

We fix and . Let

By using the inequality , we obtain

We now have

and from Burkholder-Davis-Gundy inequality



As a conclusion we obtain

Gronwall’s inequality yields then

where is a continuous function.

We have for ,
,
and



The conclusion then easily follows by combining the two previous estimates

In the sequel, of course, we shall always work with this bicontinuous version of the solution.

Definition.The continuous process of continuous maps is called the stochastic flow associated to the equation.

If the maps and are moreover , then the stochastic flow is itself differentiable and the equation for the derivative can be obtained by formally differentiating the equation with respect to its initial condition. We willl admit this result without proof:

Theorem. Let us assume that and are Lipschitz functions, then for every , the flow associated to the equation is a flow of differentiable maps. Moreover, the first variation process which is defined as the Jacobian matrix is the unique solution of the matrix stochastic differential equation:

We now turn to the theory of stochastic differential equations. Stochastic differential equations are the differential equations corresponding to the theory of the stochastic integration.

As usual, we consider a filtered probability space which satisfies the usual conditions and on which is defined a -dimensional Brownian motion . Let , and be functions.

Theorem. Let us assume that there exists such that

Then, for every , there exists a unique continuous and adapted process such that for

Moreover, for every ,

Proof.
Let us first observe that from our assumptions, there exists such that

• , ;
• ), .

The idea is to apply a fixed point theorem in a convenient Banach space.
For , let us consider the space of continuous and adapted processes such that
.
We endow that space with the norm

It is easily seen that is a Banach space.

Step one: We first prove that if a continuous and adapted process is a solution of the equation then, for every , .

Let us fix and consider for the stopping times For ,

Therefore, by using the inequality , we get

Thus, we have


By using our assumptions, we first estimate

By using our assumptions and Doob’s inequality, we now estimate

Therefore, from the inequality , we get


We may now apply Gronwall’s lemma to the function and deduce

where is a constant that does not depend on . Fatou’s lemma implies by passing to the limit when that

We conclude, as expected, that
More generally, by using the same arguments we can observe that if a continuous and adapted process satisfies

with , then .

Step 2: We now show existence and uniqueness of solutions for the equation on a time interval where is small enough.

Let us consider the application that sends a continuous and adapted process to the process By using successively the inequalities , Cauchy-Schwarz inequality and Doob’s inequality, we get . Moreover, arguing the same way as above, we can prove
Therefore, if is small enough is a Lipschitz map whose Lipshitz constant is strictly less than 1. Consequently, it has a unique fixed point. This fixed point is, of course the unique solution of the equation on the time interval . Here again, we can observe that the same reasoning applies if is replaced by a random variable that satisfies .

Step 3.
In order to get a solution of the equation on , we may apply the previous step to get a solution on intervals , where is small enough and . This will provide a solution of the equation on . This solution is unique, from the uniqueness on each interval

Definition: An equation like in the previous theorem is called a stochastic differential equation.

Exercise: (Ornstein-Uhlenbeck process) Let . We consider the following stochastic differential equation,

Exercise.(Brownian bridge) We consider for the following stochastic differential equation

Exercise. Let and . We consider the following stochastic differential equation,

Show that

is the unique solution.

The next proposition shows that solutions of stochastic differential equations are intrinsically related to a second order differential operator. This connection will later be investigated in more details.

Proposition. Let be the solution of a stochastic differential equation

where and are Borel functions. Let now be a function. The process

is a local martingale, where is the second order differential operator

and .

In this section, we describe a theorem which has far reaching consequences in mathematical finance: The Girsanov theorem. It describes the impact of a probability change on stochastic calculus.

Let be a filtered probability space. We assume that is the usual completion of the filtration of a Brownian motion . Let be a probability measure on which is equivalent to . We denote by the density of with respect to .

Theorem (Girsanov theorem) There exists a progressively measurable process such that for every , and Moreover, the process is a Brownian motion on the filtered probability space . As a consequence, a continuous and adapted process is a -semimartingale if and only if it is a -semimartingale.

Proof. Since and are equivalent on , there are of course also equivalent on for every . The density of with respect to is given by . As a consequence, the process is a positive martingale. From Itō’s representation theorem, we therefore deduce that there exists a progressively measurable process such that Let now . We have then,

By using Itō’s formula to the process , we see that it implies

We now want to prove that the process is a -Brownian motion. It is clear the -quadratic variation of this process is . From the Levy’s characterization result, we therefore just need to prove that it is a local martingale. For this, we are going to prove that that the process

is a -local martingale. Indeed, from the integration by parts formula, it is immediate that

Since is the density of with respect to , it is then easy to deduce that is a -local martingale and thus a Brownian motion

Exercise. Let be a filtered probability space that satisfies the usual conditions. As before, let be a probability measure on which is equivalent to . We denote by the density of with respect to and . Let be a local martingale. Show that the process

is a local martingale. As a consequence, a continuous and adapted process is a -semimartingale if and only if it is a -semimartingale.

Exercise Let be a Brownian motion. We denote by the Wiener measure, by the coordinate process and by its natural filtration.

• Let and be the distribution of the process . Show that for every , and that
  
• Is it true that
• For , we denote Compute the density function of (You may use the previous question).
• More generally, let be a measurable function such that for every , . We denote by the distribution of the process . Show that for every ,
  
  and that
  

Let be a filtered probability space that satisfies the usual conditions and let be a Brownian motion on it. Let now be a progressively measurable process such that for every ,   . We denote

As a consequence of Itō’s formula, it is clear that is a local martingale. In general is not a martingale, but the following two lemmas provide simple sufficient conditions that it is.

Lemma. If for every , then is a martingale.

Proof. The process is a non negative local martingale and thus a super martingale

Lemma. (Novikov’s condition) If , then is a uniformly integrable martingale.

Proof. We denote . As a consequence of , the random variable has moments of all order. So from Burkholder-Davis-Gundy inequalities, has moments of all orders, which implies that is a uniformly integrable martingale. We have then

The Cauchy-Schwarz inequality implies then that .
We deduce from the Doob’s convergence theorem that the process is a uniformly integrable submartingale. Let now and . We have

Holder’s inequality shows then that




If we can prove that , then by letting in the above inequality, we would get

and thus .

Let such that . Consider and so that . Using

and then Holder’s inequality, shows that there is a constant (depending only on ) such that for any stopping time

By the Doob’s maximal inequality, it implies that the local martingale is actually a true martingale. This implies and the desired conclusion

We now assume that is a uniformly integrable martingale. In that case, it is easy to see that on the -field , there is a unique probability measure equivalent to such that for every , The same argument as before shows then that with respect to , the process
is a Brownian motion.

In this section, we study some of the most important martingale inequalities: The Burkholder–Davis–Gundy inequalities. Interestingly, the range of application of these inequalities is very large and they play an important role in harmonic analysis and the study of singular integrals (see for instance the nice survey by my colleague Pr. Bañuelos). These inequalities admit several proofs. We present here a proof using Itō’s formula and an interesting domination inequality which is due to Lenglart. For an alternative proof, you may refer to the original approach by Burkholder-Davis-Gundy.

We admit without proof, the following domination inequality which is is due to Lenglart.

Proposition.(Lenglart) Let be a positive adapted right-continuous process and be an increasing process. Assume that for every bounded stopping time , . Then, for every ,

We shall use this lemma to prove the following

Theorem. (Burkholder-Davis-Gundy inequalities) Let and be a continuous local martingale such that . For every , there exist universal constants and , independent of and such that

Proof. By stopping it is enough to prove the result for bounded . Let . From Itō’s formula we have


As a consequence of the Doob’s stopping theorem, we get that for every bounded stopping time ,

From the Lenglart’s domination inequality, we deduce then that for every ,

We now bound



As a consequence, we obtain:

Letting yields the claimed result, that is

We proceed now to the proof of the left hand side inequality. We have,

Therefore, we get

By using the previous argument, we now have



As a conclusion, we obtained

This is an inequality of the form , which easily implies
, thanks to the inequality , with a conveniently chosen

In the previous Lecture, we proved that any martingale which is adapted to a Brownian filtration can be written as a stochastic integral. In this section, we prove that any martingale can also be represented as a time changed Brownian motion. To prove this fact, we give first first a characterization of the Brownian motion which is interesting in itself. In this section, we denote by a filtration that satisfies the usual conditions.

Proposition: (Levy’s characterization theorem) Let be a continuous local martingale such that and such that for every , . The process is a standard Brownian motion.

Proof. Let . By using Itō’s formula, we obtain that for ,
As a consequence, the process is a martingale and, from the above equality we get

The process is therefore a continuous process with stationary and independent increments such that is normally distributed with mean 0 and variance . It is thus a Brownian motion

The next proposition shows that continuous martingales behave in a nice way with respect to time changes.

Proposition Let be a continuous and increasing process such that for every , is a finite stopping time of the filtration . Let be a continuous martingale with respect to . The process is a local martingale with respect to the filtration . Moreover .

Proof. . By using localization, we may assume to be bounded. According to the Doob’s stopping theorem, we need to prove that for every bounded stopping time of the filtration , we have . But is obviously a bounded stopping time of the filtration and thus from Doob’s stopping theorem we have . The same argument shows that

Exercise. Let be an increasing and right continuous process such that for every , is a finite stopping time of the filtration . Let be a continuous martingale with respect to such that is constant on each interval . Show that the process is a continuous local martingale with respect to the filtration and that .

We can now prove the following nice representation result for martingales.

Theorem. ( Dambis, Dubins-Schwarz) Let be a continuous martingale such that and . There exists a Brownian motion , such that for every ,

Proof. Let . is a right continuous and increasing process such that for every , is a finite stopping time of the filtration and is obviously constant on each interval . From the previous exercise the process is a local martingale whose quadratic variation is equal to . From Levy’s characterization theorem, it is thus a Brownian motion

Exercise. Show that if is a continuous local martingale such that and , there exists a Brownian motion , such that for every ,

Exercise.
Let be a continuous adapted process and let be a Brownian motion. Show that for every , the process has Holder paths, where .

The study of the planar Brownian is deeply connected to the theory of analytic functions. The fundamental property of the Brownian curve is that it is a conformal invariant. The following proposition is easily proved as a consequence of Itō’s formula and of the Dambins-Dubins-Schwarz theorem. By definition, a complex Brownian motion is a process in the complex plane that can be decomposed as where and are independent Brownian motions.

Proposition.(Conformal invariance of the planar Brownian motion) Let be a complex Brownian motion and be an analytic function. Then

As a consequence, there exists a complex Brownian motion such that

To study the complex Brownian motion, it is useful to look at it in polar coordinates. It leads to the so-called skew-product decomposition of the complex Brownian motion.

Proposition. Let be a complex Brownian motion started at . There exists a complex Brownian motion such that
where .

Proof. The proof is let as an exercise to the reader. The main idea is to prove, by using Itō’s formula, that and then to used the Dambins-Dubins-Schwarz theorem

Exercise. In the previous proposition, show that the process is independent from the process .

You will find below a video of a talk by Pr. Marc Yor concerning quadratic functionals of the planar Brownian motion. The talk was given at the University of Bristol in December 2008 for a special event.

In this Lecture we show that, remarkably, any square integrable integrable random variable which is measurable with respect to a Brownian motion, can be expressed as a stochastic integral with respect to this Brownian motion. A striking consequence of this result, which is known as Itō’s representation theorem, is that any square integrable martingale of a Brownian filtration has a continuous version.

Let be a Brownian motion. In the sequel, we consider the filtration which is the usual completion of the natural filtration of (such a filtration is called a Brownian filtration).

The following lemma is a straightforward consequence of Itō’s formula.

Lemma. Let be a locally square integrable function. The process is a square integrable martingale.

Proof. From Itō’s formula we have

The random variable is a Gaussian random variable with mean 0 and variance . As a consequence

and the process

is a martingale.

Lemma. Let be the set of compactly supported and piecewise constant functions , i.e. the set of functions that can be written as for some and . The family is total in .

Proof.
Let such that for every ,

Let . We have for every ,

By analytic continuation, we see that

actually also holds for every . By using the Fourier transform, it implies that

Since were arbitrary, we conclude that . As a conclusion

We are now in position to state the representation theorem.

Theorem. For every , there is a unique progressively measurable process such that and

Proof. The uniqueness is immediate as a consequence of the Itō’s isometry for stochastic integrals. Let be the set of random variables such that there exists a progressively measurable process such that and From the above lemma, it is clear that contains the set of set of random variables

Since this set is total in , we just need to prove that is closed in . So, let be a sequence of random variables such that and in . There is a progressively measurable process such that and By using Itō’s isometry, it is seen that the sequence is a Cauchy sequence and therefore converges to a process which is seen to satisfy

As a consequence of the representation theorem, we obtain the following description of the square integrable martingales of the filtration .

Corollary. Let be a square integrable martingale of the filtration . There is a unique progressively measurable process such that for every , and In particular, admits a continuous version.

Exercise. Show that if is a local martingale of the filtration , then there is a unique progressively measurable process such that for every , and

In the next few Lectures we will illustrate through several examples of application the power of the stochastic integration theory.

We start with a study of the multidimensional Brownian motion. As already pointed out, a multidimensional stochastic process , is called a Brownian motion if the processes , , are independent Brownian motions. In the sequel we denote by the Laplace operator on , that is

The following result is an easy consequence of the Itō’s formula.

Proposition. Let be a function that is once continuously differentiable with respect to its first variable and twice continuously differentiable with respect to its second variable and let be a -dimensional Brownian motion. The process

is a local martingale. If moreover is such that

for some continuous function and some constant , then is a martingale.

In particular, if is a harmonic function, i.e. , and if is a multidimensional Brownian motion, then the process is a local martingale. As we will see it later, this nice fact has many consequences. A first nice application is the study of recurrence or transience of the multidimensional Brownian motion paths. As we have seen before, the Brownian motion recurrent: It reaches any value with probability 1. In higher dimensions, the situation is more subtle.

Let be a -dimensional Brownian motion with . For and , we consider the stopping time

Proposition. For ,

Proof. For , we consider the function

A straightforward computation shows that . The process is therefore a martingale, which implies . This yields

Since

we deduce that

By letting , we get
Corolllary. For ,

As a consequence, for the Brownian motion is recurrent, that is, for every non empty set ,

Though the two-dimensional Brownian motion is recurrent, points are always polar.

Proposition.
For every ,

Proof. It suffices to prove that for every , , . We have

Since , we get

As we have just seen, the two-dimensional Brownian motion will hit every non empty open set with probability one. The situation is different in dimension higher than 3: Brownian motion paths will eventually leave any bounded set with probability one.

Proposition. Let be a -dimensional Brownian motion. If then almost surely

Proof. Let us assume . Let where . Since will never hit the point , we can consider the process which is seen to be a positive local martingale from Itō’s formula. A positive local martingale is always a supermartingale. Therefore from the Doob’s convergence theorem, the process converges almost surely when to an integrable and non negative random variable . From Fatou’s lemma, we have . By the scaling property of the Brownian motion, it is clear that . We conclude

Exercise (Probabilistic proof of Liouville theorem) By using martingale methods, prove that if is a bounded harmonic function, then is constant.

Itō’s formula is certainly the most important and useful formula of stochastic calculus. It is the change of variable formula for stochastic integrals. It is a very simple formula whose specificity is the appearance of a quadratic variation term, that reflects the fact that semimartingales have a finite quadratic variation.

Due to its importance, we first provide a heuristic argument on how to derive Itō ‘s formula. Let be a smooth function and be a path . We have the following heuristic computation:



This suggests, by summation, the following correct formula:

Let us now try to consider a Brownian motion instead of the smooth path and let us try to adapt the previous computation to this case. Since Brownian motion has quadratic variation which is not zero, , we need to go at the order 2 in the Taylor expansion of . This leads to the following heuristic computation:



By summation, we are therefore led to the formula

which is, as we will see it later perfectly correct.

In what follows, we consider a filtered probability space that satisfies the usual conditions. Our starting point to prove Itō’s formula is the following formula which is known as the integration by parts formula for semimartingales:

Proposition. (Integration by parts formula)
Let and be two continuous semimartingales, then the process is a continuous semimartingale and we have:

Proof. By bilinearity of the multiplication, we may assume . Also by considering, if needed, instead of , we may assume that . Let . For every sequence such that
we have

By letting , we therefore obtain the following identity which yields the expected result:

We are now in position to prove Itō’s formula in its simpler form.

Theorem. (Itō’s formula I) Let be a continuous and adapted semimartingale and let be a function which is twice continuously differentiable. The process is a semimartingale and the following change of variable formula holds:

Proof. We assume that the semimartingale is bounded. If it is not, we may apply the following arguments to the semimartingale , where and then let . Let be the set of two times continuously differentiable functions for which the formula given in the statement of the theorem holds holds. It is straightforward that is a vector space. Let us show that is also an algebra, that is also let stable by multiplication. Let . By using the integration by parts formula with the semimartingales and , we obtain

The terms of the previous sum may be separately treated in the following way. Since , we get:



Therefore,


We deduce that .

As a conclusion, is an algebra of functions. Since contains the function , we deduce that actually contains every polynomial function. Now in order to show that every function which is twice continuously differentiable is actually in , we first observe that since is assumed to be bounded, it take its values in a compact set. It is then possible to find a sequence of polynomials such that, on this compact set, uniformly converges toward , uniformly converges toward and uniformly converges toward

As a particular case of the previous formula, if we apply this formula with as a Brownian motion, we get the formula that was already pointed out at the beginning of the section: If is twice continuously differentiable function, then

It is easy to derive the following variations of Itō’s formula:

Theorem: (Itō’s formula II) Let be a continuous and adapted semimartingale, and let be an adapted bounded variation process. If is a function that is once continuously differentiable with respect to its first variable and that is twice continuously differentiable with respect to its second variable, then for :

Theorem. (Itō’s formula III) Let ,…, be adapted and continuous semimartingales and let be a twice continuously differentiable function. We have:

Exercise. Let be a function that is once continuously differentiable with respect to its first variable and twice continuously differentiable with respect to its second variable that satisfies
Show that if is a continuous local martingale, then is a continuous local martingale. Deduce that for , the process is a local martingale.

Exercise. The Hermite polynomial of order is defined as

The goal of this Lecture is to extend the domain of definition of the Itō integral with respect to Brownian motion. The idea is to use the fruitful concept of localization. We will then be interested in the wider class of processes for which it is possible to define a stochastic integral satisfying natural probabilistic properties. This will lead to the natural notion of semimartingales.

As before, we consider here a Brownian motion that is defined on a filtered probability space that satisfies the usual conditions.

Definition. We define the space , as the set of the processes that are progressively measurable with respect to the filtration and such that for every ,

We first have the following fact:

Lemma. Let . There exists an increasing family of stopping times for the filtration such that:

Thanks to this Lemma, it is now easy to naturally define for . Indeed, let and let . According to the previous lemma, let us now consider an increasing sequence of stopping times such that:

• Almost surely,
• 

Since the stochastic integral

exists. We may therefore define in a unique way a stochastic process such that:

• is a continuous stochastic process adapted to the filtration ;
• The stochastic process is a uniformly integrable martingale with respect to the filtration (because it is bounded in ).

This leads to the following definition:

Definition. A stochastic process is called a local martingale (with respect to the filtration ) if there is a sequence of stopping times such that:

Thus, as an example, if then the process is a local martingale. Of course, any martingale turns out to be a local martingale. But, as we will see it later, in general the converse is not true. The following Exercise gives a useful criterion to prove that a given local martingale is actually martingale.

Exercise.
Let be a continuous local martingale such that for ,

Show that is a martingale. As a consequence, bounded local martingales necessarily are martingales.

It is interesting to observe that if is a local martingale, then the sequence of stopping times may explicitly be chosen so that the resulting stopped martingales enjoy nice properties.

Lemma. Let be a continuous local martingale on such that . Let

Then, for , the process is a bounded martingale.

Proof. Let be a sequence of stopping times such that:

• The sequence is increasing and almost surely ;
• For every , the process is a uniformly integrable martingale with respect to the filtration .

For and , we have:

Letting leads then to the expected result

Since bounded martingales are of course square integrable, we easily deduce from the previous Lemma that the following result holds:

Theorem. Let be a continuous local martingale on such that . Then, there is a unique continuous increasing process such that:

Furthermore, for every and every sequence of subdivisions such that , the following limit holds in probability:

Moreover, if is a progressively measurable process such that for every , then we may define a stochastic integral such that the stochastic process is a continuous local martingale.

At that point, we already almost found the widest class of stochastic processes with respect to which it was possible to naturally construct a stochastic integral. To go further in that direction, let us first observe that if we add a bounded variation process to a local martingale, then we obtain a process with respect to which a stochastic integral is naturally defined.

More precisely, if may be written under the form:
, where is a bounded variation process and where
is a continuous local martingale on such that , then if is a progressively measurable process such that for ,
we may define a stochastic integral as

where is simply understood as the Riemann-Stieltjes integral with respect to the process .

The class of stochastic processes that we obtained is called the class of semimartingales and, as we will see it later, is the most relevant one:
Definition.
Let be an adapted continuous stochastic process on the filtered probability space . We say that is a semimartingale with respect to the filtration if may be written as:

where is a bounded variation process and is a continuous local martingale such that . If it exists, the previous decomposition is unique.

Exercise. Let be a continuous local martingale on the filtered probability space . Show that is a semimartingale.

Since a bounded variation process has a zero quadratic variation, it is easy to prove the following theorem:
Proposition. Let

be a continuous adapted semimartingale. For every and every sequence of subdivisions such that the following limit holds in probability:

We therefore call the quadratic variation of and denote .

Exercise. Let be a continuous semimartingale on the filtered probability space . If is a subdivision of the time interval $[0,T]$, we denote , where is such that . Let be a sequence of subdivisions of such that Show that the following limit holds in probability,

Exercise. Let be a continuous semimartingale on . Let be a sequence of locally bounded and adapted processes almost surely converging toward 0 such that , where is a locally bounded process. Show that for , the following limit holds in probability

It already has been observed that in the Brownian case, though the stochastic integral is not an almost sure limit of Riemann sums, it is however a limit in probability of such sums. This may extended to semimartingales in the following way.

Proposition. Let be a continuous and adapted process, let be a continuous and adapted semimartingale and let . For every sequence of subdivisions such that

the following limit holds in probability:

As we already suggested it, the class of semimartingales is actually the wider class of stochastic processes with respect to which we may define a stochastic integral that enjoys natural properties. Let us more precisely explain what the previous statement means.

Let us denote by the set of processes such that:
where are bounded stopping times and where the ‘s are random variable that are bounded and measurable with respect to . If is a continuous and adapted process and if , then we naturally define

We have the following theorem that we shall admit without proof:

Proposition. Let be a continuous and adapted process. The process is a semimartingale if and only if for every sequence in that almost surely converges to 0, we have for every and ,

In the same way that a stochastic integral with respect to Brownian motion was constructed, a stochastic integral with respect to square integrable martingales may be defined. We shall not repeat this construction, since it was done in the Brownian motion case, but we point out the main results without proofs.

Let be a continuous square integrable martingale on a filtered probability space that satisfies the usual conditions. We assume that and . Let us denote by the set of processes that are progressively measurable with respect to the filtration and such that

We still denote by the set of simple and predictable processes, that is the set of processes that may be written as:

where and where is a random variable that is measurable with respect to and such that . We define an equivalence relation on the set as follows:

and denote by

the set of equivalence classes. It is easy to check that endowed with the norm

is a Hilbert space.

Theorem. There exists is a unique linear map

such that:

The map is called the Itō integral with respect to the continuous and square integrable martingale . We denote for ,

Proposition. Let be a stochastic process which is progressively measurable with respect to the filtration and such that for every , . The process

is a square integrable martingale with respect to the filtration that admits a continuous modification.

Proposition. Let be a stochastic process which is progressively measurable with respect to the filtration and such that for every , . We have

Proposition. Let be a stochastic process whose paths are left continuous. Let . For every sequence of subdivisions such that

the following convergence holds in probability:

Proposition. Let us assume that where is a Brownian motion on and where . For ,

Exercise. Let and be two square integrable martingales such that for every ,

For ,