15.2.1 Riemann sums

Intuitively, if we fix a scenario ω ∈ Ω, then the corresponding paths H_t(ω) and W_t(ω) are both functions of t and then we can write down the non-random Riemann sum:

For each n, this is the total area of n rectangles, where the height of the i-th rectangle is given by and the width of its base is given by .

Example 15.2.2 Realization of a Riemann sum

Let us fix n = 4 and , yielding t_i = i/12, and let us choose H_t = |W_t|. Consider a scenario ω in which we have:

Note that W₀(ω) = 0 by definition, no matter which scenario ω occurs. Since H_t = |W_t| at any time t, we have

One should remark that in this scenario, the area of the first (non-null) rectangle is negative: |0.382927566| × ( − 0.141739491 − 0.382927566) = −0.2009095.

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As we can repeat this for every scenario ω ∈ Ω, then for each n we have a doubly random Riemann sum that we denote by

(15.2.2)

It is a random variable that we will call the n-th (random) Riemann sum (of H).

Example 15.2.3 Riemann sums of geometric Brownian motion

Let us compute the expectation of the n-th Riemann sum of the process H_t = exp (W_t), i.e.

Recall that since increments of Brownian motion are independent, we have that is independent of , for each i. Consequently,

as increments of Brownian motion are zero-mean random variables.

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15.2.2 Elementary stochastic processes

Depending on the integrand, the limiting procedure involved in defining the stochastic integral as given in (15.2.1) is not always necessary. We have seen such a trivial illustration in example 15.2.1. In this section, we introduce a class of stochastic processes called elementary stochastic processes whose stochastic integrals do not require the limiting procedure. Hence, each such stochastic integral is equal to a Riemann sum.

The simplest stochastic process one can imagine² is a one-step elementary stochastic process, i.e. a process of the form

(15.2.3)

where φ is a random variable³ and where 0 ⩽ t₁ < t₂ ⩽ T. Here, φ, t₁ and t₂ specify the elementary process: they are the parameters of H.

Figure 15.2 shows four different trajectories of this process H associated to four different scenarios.

As we can see in Figure 15.2, the trajectories of the process H given in (15.2.3) consist of a single step of height φ(ω) when the scenario is ω. Consequently, for such a process, its stochastic integral does not require the limiting procedure: it is directly equal to the Riemann sum . In other words, the stochastic integral of the process H given in (15.2.3), over the time interval [0, T], is given by the following random variable:

(15.2.4)

Example 15.2.4

If , with 0 ⩽ t₁ < t₂ ⩽ T, then it is a one-step elementary process as given in (15.2.3) with a constant. Consequently, the stochastic integral of H is given by

In this case, the stochastic integral is normally distributed with mean 0 and variance 3 × (t₂ − t₁).

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Example 15.2.5

If , with 0 ⩽ t₁ < t₂ ⩽ T, then it is a one-step elementary process as given in (15.2.3) with . Consequently, the stochastic integral of H is given by

In this case, the stochastic integral is the product of two independent zero-mean normal random variables. In particular, we have

and consequently

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More generally, a stochastic process H = {H_t, 0 ⩽ t ⩽ T} is said to be an elementary stochastic process if it is such that

(15.2.5)

where 0 = t₀ < t₁ < … < t_{n − 1} < t_n = T generate non-overlapping time intervals and where, for each i, the random variable φ_i is completely determined by the Brownian motion up to time t_i; this is a mathematical technicality we have to deal with. Note that we have H₀ = 0 for any scenario ω.

For a given ω ∈ Ω, the trajectory H_t(ω) is made of steps with various heights determined by the draws of φ₁(ω), φ₂(ω), …, φ_n(ω). This is illustrated in Figure 15.3 showing trajectories of H_t(ω) for two scenarios.

Again, as we can see in Figure 15.3, since the trajectories of the process H given in (15.2.5) consist of steps of height φ_i(ω) when the scenario is ω, then its stochastic integral does not require the limiting procedure. The stochastic integral of H given in (15.2.5) over the time interval [0, T] is the extension of equation (15.2.4): it is given by the following random variable:

(15.2.6)

Example 15.2.6

For the elementary process , the corresponding stochastic integral is simply

Since increments of Brownian motion over non-overlapping intervals are independent, this is a sum of independent normally distributed random variables. Hence, the stochastic integral is normally distributed with mean

and variance

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As observed in the last example, if H is an elementary stochastic process as in (15.2.5) with non-random φ_i, then its stochastic integral is normally distributed with mean zero and variance equal to

Note that in other/most cases, we cannot say much about the distribution of a stochastic integral.

Example 15.2.7

For the process , its stochastic integral is the random variable given by

Using the properties of Brownian motion, we can compute its expectation, which is equal to zero. However, the variance is more challenging to compute because W₄ − W₂ and W₆ are not independent. First, since the mean of the integral is 0, we have

We then need to expand the square within the expectation, which yields three expectations. The first expectation is

The third expectation is

since we are taking the expectation of independent random variables (Brownian increments over non-overlapping intervals). The second expectation is more complicated. We have

which is the expectation of the product of three random variables, some of them being dependent. However, since we can write

then

and its expectation is equal to zero. Combining everything, we obtain

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15.2.3 Ito-integrable stochastic processes

We would like to analyze the stochastic integral of a more general class of stochastic processes, which we will call Ito-integrable stochastic processes. Such processes need to have the following properties.

First, the process H = {H_t, 0 ⩽ t ⩽ T} should be an adapted stochastic process, that is a stochastic process such that for each time t, the random variable H_t is completely determined by the Brownian trajectory up to time t, i.e. by the random variables W_s, for all 0 ⩽ s ⩽ t. In some sense, an adapted process follows the flow of information provided by the Brownian motion W: it does not look into the future. Second, the process H also needs to be sufficiently integrable for the variance of its stochastic integral to be finite; more on this below. These are mathematical technicalities we have to deal with, but we will not say more than we already have on this matter. Elementary stochastic processes defined in (15.2.5) are Ito-integrable processes by definition.

The stochastic integral of an Ito-integrable process H, as defined in (15.2.1), requires the limiting procedure, meaning that

is obtained as the limit of the Riemann sums where t_i = it/n, for each i = 0, 1, …, n. We can also say that each n-th Riemann sum is the stochastic integral of an elementary stochastic process given by

implying that

Said differently, the stochastic integral of a process H is obtained by approximating H with elementary stochastic processes having an increasing number of time steps n. This is illustrated in Figure 15.4.

Example 15.2.8 Stochastic integrals of functions of Brownian motion

First, let us have a look at the following stochastic integral: ∫^t₀W_sdW_s. In this case, we have H = W and then, by definition,

Similarly, the stochastic integral ∫^t₀exp (W_s)dW_s is obtained by the following limiting procedure:

Recall that these last Riemann sums have been studied in example 15.2.3.

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Convergence in quadratic mean

The type of convergence involved in the definition of the stochastic integral given in (15.2.1) is the convergence in quadratic mean. Mathematically, it means that

In other words, as n increases, the mean-square error between the Riemann sums and the stochastic integral gets closer to zero.

15.2.4 Properties

Let us now analyze some of the properties of Ito’s stochastic integral. It must be noted that since a stochastic integral is a random variable, there will be two types of properties:

1. properties related to its random nature, i.e. its mean and variance, and whenever possible its probability distribution;
2. properties related to its integral nature, i.e. its approximation by Riemann sums and its linearity.

As mentioned before, the properties of stochastic integrals are mostly inherited from those of the approximating Riemann sums.

Let H = {H_t, 0 ⩽ t ⩽ T} and G = {G_t, 0 ⩽ t ⩽ T} be Ito-integrable stochastic processes and let α₁, α₂ be real numbers. Then, for any time T > 0, we have:

1. Stochastic integrals are zero-mean random variables:
   
2. Stochastic integrals have a variance given by:
   (15.2.7)
   which is also known as Ito’s isometry.
3. If H is a non-random function of time, i.e. if it is not affected by the scenario ω, then
   
4. Stochastic integrals are linear:
   

Let us show all these properties and how they are inherited from the Riemann sums. For any Ito-integrable process H = {H_t, 0 ⩽ t ⩽ T}, its n-th Riemann sum

where t_i = iT/n for each i, is a zero-mean random variable. Indeed, we have

where in the second last equality we used the fact that, for each i, the random variable is independent of the random variable ; this is because is completely determined by the Brownian trajectory up to time t_i.

Given that the mean of the Riemann sum is equal to zero, its variance is such that

Distributing the square and using the linearity property of the expectation, we get

When i = j, again by independence, we have

When i < j, using the formula of iterated expectations, we get

where, once more, we used the independence properties of Brownian increments. Of course, changing the role played by i and j, we get a symmetric result when j < i. In conclusion,

(15.2.8)

Informally speaking, since by definition

we can expect that⁴

Therefore, using the above discussion, we can conclude that

where the last equality comes from the definition of the (deterministic) Riemann integral of the function , as defined in (15.1.1).

Finally, if H = {H_t, 0 ⩽ t ⩽ T} is a non-random function of time, then the s are no longer genuine random variables: they are simple real numbers. Therefore, the n-th Riemann sum

is a linear combination of independent normal random variables given by the increments . This means that the Riemann sum itself is normally distributed. We have already checked that the mean is equal to zero and now the variance becomes

because the expectation of a constant is equal to the constant itself. If we summarize, we have: if H is deterministic, then

The normality of the Riemann sums is not affected by the limiting procedure, so if H is deterministic, then ∫^T₀H_tdW_t is indeed normally distributed with mean 0 and variance given by:

Finally, since Riemann sums are clearly linear (they are sums) and since limits are themselves linear, it should be clear that stochastic integrals are linear with respect to their integrands. The details are left to the reader.

Example 15.2.9 Stochastic integral of a Brownian motion

In Example 15.2.8, we considered the stochastic integral of a standard Brownian motion, namely the random variable ∫^T₀W_tdW_t.

By the above properties, we have that . Note that, in Example 15.2.3, we had already verified that the corresponding n-th Riemann sums were also zero-mean random variables. By Ito’s isometry given in (15.2.7), the variance is given by

Note that since the integrand is W, i.e. a stochastic process (as opposed to a function), we do not know the distribution of ∫^T₀W_tdW_t.

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Example 15.2.10. Linearity of stochastic integrals

In example 15.2.8, we also considered the stochastic integral ∫^T₀exp (W_t)dW_t.

Let us now consider the following stochastic integral:

We know that it has a zero mean and that its variance can be computed using Ito’s isometry.

Moreover, by the linearity property of stochastic integrals, we can write

where the two stochastic integrals on the right-hand-side have been defined in example 15.2.8.

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15.4.1 Stochastic differential equations

In many actuarial and financial models, the dynamics of a financial variable over time (e.g. the stock price) is expressed with a stochastic differential equation. Such an equation specifies the desirable behavior of this quantity over the next infinitesimal period of time.

Symbolically, a stochastic differential equation (SDE) is a stochastic equation: for given functions a( · ) and b( · ), we are looking for a process X such that

(15.4.2)

with initial condition X₀ = x₀. More rigorously, we are looking for a diffusion process X such that

as already defined in (15.4.1). The stochastic differential equation in (15.4.2) is also called the differential form whereas the expression in (15.4.1) is known as the integral form of a diffusion process.

Intuitively, a SDE can be interpreted as follows: given that we know the value of X_t at time t, the future value of the process one millisecond after is explained by two components:

• a component a(X_t)dt: a change of magnitude a(X_t) in the direction of the infinitesimal increase in time dt;
• a component b(X_t)dW_t: a change of magnitude b(X_t) in the direction of the infinitesimal increment of the Brownian motion dW_t.

As a SDE describes the evolution of a quantity X_t as time passes, we are interested in finding an explicit expression for X_t: this is what we will mean by solving the SDE. But recall that a SDE of the form given in (15.4.2) has no meaning by itself: it is rather a symbolic representation for the integral equation given in (15.4.1). Hence, solving a SDE and studying the underlying stochastic process means analyzing its corresponding integral form and its existence. Most of the time, this is a difficult task.

Example 15.4.1 Examples of diffusion processes

Let us consider the following SDEs: for given numbers μ and σ,

1. , with initial value X₀ = −1;
2. , with initial value X₀ = 2.

For the first SDE, according to the general form given in (15.4.2), we have a constant drift coefficient a(x) = μ and a constant volatility coefficient b(x) = σ. Therefore, the integral form is given by

and we have recovered a linear Brownian motion issued from − 1. Note also that for this SDE, there is clearly a solution (we found it).

For the second SDE, according to the general form given in (15.4.2), we have a drift coefficient given by a(x) = μx and a volatility coefficient given by b(x) = σx. Therefore, the integral form is given by

For this SDE, it is not clear whether there exists a solution X that verifies this equality.

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15.4.2 Ito’s lemma for diffusion processes

In the last example, we were not able to obtain an explicit expression for the diffusion process solving

with initial value X₀ = 2. In order to solve such SDEs, we will need a generalized version of Ito’s lemma for diffusion processes. Indeed, in Section 15.3, we saw the simplest form of Ito’s lemma; the most general form, still known as Ito’s lemma, applies to deterministic transformations f(t, X_t) of a diffusion process X.

Let f(t, x) be a function of two variables, i.e. of time and space respectively, such that ∂f/∂t and ∂²f/∂x², i.e. the first partial derivative in time and the second partial derivative in space,⁶ are both continuous functions. If X = {X_t, 0 ⩽ t ⩽ T} is a diffusion process solving the general SDE given in (15.4.2), then the process Y_t = f(t, X_t) solves the following SDE:

(15.4.3)

Before going any further, we need to make two clarifications. First, the terms and must be understood as follows:

1. Compute the partial derivatives , and of the function f(t, x).
2. Evaluate each partial derivative at (t, X_t).

Second, the expression (dX_t)² must be understood as follows: it can be shown that

This is an application of the product rule; see the box below for more details. Hence, substituting dX_t by its expression given in (15.4.2) and reorganizing equation (15.4.3), we get

(15.4.4)

This last SDE is equivalent to the one in (15.4.3). They are both referred to as Ito’s lemma for diffusion processes.

While the last expression in (15.4.4) can be tedious, the one in (15.4.3) is more intuitive: it is more or less a Taylor expansion, of order one in time and order two in space. Then, one only needs to be familiar with the product rule to further simplify the expressions.

Product rule

Whenever we need to multiply two SDEs, there exists a convenient and symbolic tool (backed by rigorous mathematics) known as the product rule. It is described in the following table:

×	dWt	dt
dWt	dt	0
dt	0	0

More precisely, if we have two SDEs

then

In particular,

as used in Ito’s lemma (15.4.3) above.

Example 15.4.2 Simple geometric Brownian motion

Using Ito’s lemma, let us verify that the following simple geometric Brownian motion, i.e. the process

is a diffusion process. We must show that S satisfies an SDE as given in (15.4.2), that is we must find the SDE coefficients.

Clearly, we have S_t = f(t, W_t), with f(t, x) = e^{x − t/2}, and S₀ = 1. First, we compute the partial derivatives:

Second, we replace x by W_t and we use Ito’s lemma (15.4.3) to f(t, W_t) = exp (W_t − t/2), noticing that . We get

since . Substituting f(t, W_t) by S_t, we see that exp (W_t − t/2) is a solution to the SDE given by

with S₀ = 1. In conclusion, we have indeed verified that the simple geometric Brownian motion

is a diffusion process with drift a(S_t) = 0 and diffusion b(S_t) = S_t.

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Note that if we consider a transformation f(X_t) in space only, all partial derivatives simplify to ordinary derivatives and then Ito’s lemma of (15.4.3) and (15.4.4) simplify respectively to:

(15.4.5)

and, equivalently,

(15.4.6)

15.4.3 Geometric Brownian motion

We saw in example 15.4.1 that an SDE of the form

is that of a linear Brownian motion with drift μ and diffusion/volatility σ. In other words, X_t = x₀ + μt + σW_t is a diffusion process.

In this section, we are interested in obtaining a similar characterization, as a diffusion process, for geometric Brownian motions. We want to extend what we did in example 15.4.2 and thus solve the second SDE in example 15.4.1.

We know from Chapter 14 that a geometric Brownian motion S is obtained as the exponential of a linear Brownian motion X. As this is a simple deterministic transformation of a diffusion process, then, using Ito’s lemma, we should be able to find its corresponding SDE.

Let us apply Ito’s lemma to S_t = exp (X_t), where X_t = x₀ + μt + σW_t. Here, as f(x) = e^x, we will use the version of Ito’s lemma in (15.4.5). First, we have

Consequently, as S_t = f′(X_t) = f′′(S_t), using Ito’s lemma we can write

where . Since, by the product rule,

we further have

with . By analogy with Example 15.4.1, this is an SDE with drift coefficient given by a(x) = (μ + σ²/2)x and volatility coefficient given by b(x) = σx. In other words, a geometric Brownian motion is indeed a diffusion process.

It is important to realize that if we consider instead the GBM S given by

then it will be the solution to the following SDE:

This representation corresponds to the SDE in Equation (17.1.8) of Chapter 17.

In summary, we have obtained Table 15.1, for converting a geometric Brownian motion from its SDE representation to its exponential representation, and vice versa.

SDE representation		Exponential representation
	⇔	St = exp ((μ − σ2/2)t + σWt)
	⇔	St = exp (μt + σWt)

15.4.4 Ornstein-Uhlenbeck process

In this section, we analyze the Ornstein-Uhlenbeck (OU) process, which is widely used in actuarial finance, especially to model mortality/longevity risk and interest rates in continuous-time short rate models.

First, let us look at a simple version of the Ornstein-Uhlenbeck SDE, namely

(15.4.7)

with an arbitrary initial condition X₀ = x₀. We know this SDE means that

for all t ⩾ 0. Unfortunately, this cannot be considered as an explicit expression: the process X itself appears on both sides of the equation.

To find the solution to the SDE in (15.4.7), we can use Ito’s lemma along with the function f(t, x) = xe^t. The partial derivatives of this function are

Thus, using the version of Ito’s lemma in (15.4.3), we can write

with f(0, x₀) = x₀, which is equivalent to

If we move things around, we get

(15.4.8)

In other words, is a solution to . This is now an explicit expression for the OU process X.

For a fixed time t, since the integrand is deterministic, the stochastic integral in (15.4.8) is normally distributed (see Section 15.2.4) with

and

In conclusion, for each fixed time t, we have

More generally, the simple OU SDE in (15.4.7) can be extended to get the following general OU SDE:

for parameters α, β, σ > 0, and initial value X₀ = x₀.

This OU dynamic is said to be mean reverting:

• if X_t < β, then the drift is positive, pushing the process upward (toward β);
• if X_t > β, then the drift is negative, pushing the process downward (toward β).

Note that this is a desirable feature for interest rate modeling. We interpret β as an equilibrium level whereas α is the speed of mean reversion, thus amplifying or mitigating the effect of the gap between X_t and β. This version of the mean-reverting OU process is used in the Vasicek model for the short rate.

It is also possible to find an explicit expression for the general OU process. Again, using Ito’s lemma with f(t, x) = xe^αt, we obtain

As a consequence, we have, for each fixed time t > 0, that

15.4.5 Square-root diffusion process

A general OU process can take both positive and negative values. Indeed, since the random variable X_t is normally distributed, it has a non-zero probability of being negative. This can be a major impediment for some actuarial and financial modeling.

This is not the case for the square-root diffusion process. A simple version of this other process can be obtained as the square of a specific OU process. Let X be given by the SDE in (15.4.7), i.e.

Set Y_t = X²_t. If we use Ito’s lemma with the function f(x) = x², for which the first two derivatives are

then, after a few manipulations, we get

The general mean-reverting square-root process is defined as the solution of

for parameters α > 0, β > 0, σ > 0, and initial condition X₀ = x₀. In general, there is no closed-form expression for the solution of this SDE.

This version of the square-root dynamic is used for interest rate modeling, e.g. in the Cox-Ingersoll-Ross (CIR) model, and for volatility modeling, e.g. in the Heston model.