Stochastic Integration

Introduction

Suppose we wish to model a process described by a differential equation and initial condition

\displaystyle   \begin{aligned}  \dot{x}(t) &= a(x, t) \\  x(0) &= a_0  \end{aligned}

But we wish to do this in the presence of noise. It’s not clear how do to this but maybe we can model the process discretely, add noise and somehow take limits.

Let \pi = \{0 = t_0 \leq t_1 \leq \ldots \leq t_n = t\} be a partition of [0, t] then we can discretise the above, allow the state to be random and add in some noise which we model as samples of Brownian motion at the selected times multiplied by b so that we can vary the amount noise depending on the state. We change the notation from x to X(\omega) to indicate that the variable is now random over some probability space.

\displaystyle   \begin{aligned}  {X}(t_{i+1}, \omega) - {X}(t_i, \omega)  &= a({X}(t_i, \omega))(t_{i+1} - t_i) +                                              b({X}(t_i, \omega))(W(t_{i+1}, \omega) - W(t_i, \omega)) \\  X(t_0, \omega) &= A_{0}(\omega)  \end{aligned}

We can suppress explicit mention of \omega and use subscripts to avoid clutter.

\displaystyle   \begin{aligned}  {X}_{t_{i+1}} - {X}_{t_i}  &= a({X}_{t_i})(t_{i+1} - t_i) +                                b({X}_{t_i})(W_{t_{i+1}} - W_{t_i}) \\  X(t_0) &= A_{0}(\omega)  \end{aligned}

We can make this depend continuously on time specifying that

\displaystyle   X_t = X_{t_i} \quad \mathrm{for} \, t \in (t_i, t_{i+1}]

and then telescoping to obtain

\displaystyle   \begin{aligned}  {X}_{t} &= X_{t_0} + \sum_{i=0}^{k-1} a({X}_{t_i})(t_{i+1} - t_i) +                       \sum_{i=0}^{k-1} b({X}_{t_i})(W_{t_{i+1}} - W_{t_i})                       \quad \mathrm{for} \, t \in (t_k, t_{k+1}]  \end{aligned}

In the limit, the second term on the right looks like an ordinary integral with respect to time albeit the integrand is stochastic but what are we to make of the the third term? We know that Brownian motion is nowhere differentiable so it would seem the task is impossible. However, let us see what progress we can make with so-called simple proceses.

Simple Processes

Let

\displaystyle   X(t,\omega) = \sum_{i=0}^{k-1} B_i(\omega)\mathbb{I}_{(t_i, t_{i+1}]}(t)

where B_i is {\cal{F}}(t_i)-measurable. We call such a process simple. We can then define

\displaystyle   \int_0^\infty X_s \mathrm{d}W_s \triangleq \sum_{i=0}^{k-1} B_i{(W_{t_{i+1}} - W_{t_{i+1}})}

So if we can produce a sequence of simple processes, X_n that converge in some norm to X then we can define

\displaystyle   \int_0^\infty X(s)\mathrm{d}W(s) \triangleq \lim_{n \to \infty}\int_0^\infty X_n(s)\mathrm{d}W(s)

Of course we need to put some conditions of the particular class of stochastic processes for which this is possible and check that the limit exists and is unique.

We consider the {\cal{L}}^2(\mu \times \mathbb{P}), the space of square integrable functions with respect to the product measure \mu \otimes \mathbb{P} where \mu is Lesbegue measure on {\mathbb{R}^+} and \mathbb{P} is some given probability measure. We further restrict ourselves to progressively measurable functions. More explicitly, we consider the latter class of stochastic processes such that

\displaystyle   \mathbb{E}\int_0^\infty X^2_s\,\mathrm{d}s < \infty

Less Simple Processes

Bounded, Almost Surely Continuous and Progressively Adapted

Let X be a bounded, almost surely continuous and progressively measurable process which is (almost surely) 0 for t > T for some positive constant T. Define

\displaystyle   X_n(t, \omega) \triangleq X\bigg(T\frac{i}{n}, \omega\bigg) \quad \mathrm{for} \quad T\frac{i}{n} \leq t < T\frac{i + 1}{n}

These processes are cleary progressively measurable and by bounded convergence (X is bounded by hypothesis and \{X_n\}_{n=0,\ldots} is uniformly bounded by the same bound).

\displaystyle   \lim_{n \to \infty}\|X - X_n\|_2 = 0

Bounded and Progressively Measurable

Let X be a bounded and progressively measurable process which is (almost surely) 0 for t > T for some positive constant T. Define

\displaystyle   X_n(t, \omega) \triangleq \frac{1}{1/n}\int_{t-1/n}^t X(s, \omega) \,\mathrm{d}s

Then X^n(s, \omega) is bounded, continuous and progressively measurable and it is well known that X^n(t, \omega) \rightarrow X(t, \omega) as n \rightarrow 0. Again by bounded convergence

\displaystyle   \lim_{n \to \infty}\|X - X_n\|_2 = 0

Progressively Measurable

Firstly, let X be a progressively measurable process which is (almost surely) 0 for t > T for some positive constant T. Define X_n(t, \omega) = X(t, \omega) \land n. Then X_n is bounded and by dominated convergence

\displaystyle   \lim_{n \to \infty}\|X - X_n\|_2 = 0

Finally let X be a progressively measurable process. Define

\displaystyle   X_n(t, \omega) \triangleq  \begin{cases}  X(t, \omega) & \text{if } t \leq n \\  0            & \text{if } \mathrm{otherwise}  \end{cases}

Clearly

\displaystyle   \lim_{n \to \infty}\|X - X_n\|_2 = 0

The Itô Isometry

Let X be a simple process such that

\displaystyle   \mathbb{E}\int_0^\infty X^2_s\,\mathrm{d}s < \infty

then

\displaystyle   \mathbb{E}\bigg(\int_0^\infty X_s\,\mathrm{d}W_s\bigg)^2 =  \mathbb{E}\bigg(\sum_{i=0}^{k-1} B_i{(W_{t_{i+1}} - W_{t_{i}})}\bigg)^2 =  \sum_{i=0}^{k-1} \mathbb{E}(B_i)^2({t_{i+1}} - {t_{i}}) =  \mathbb{E}\int_0^\infty X^2_s\,\mathrm{d}s

Now suppose that \{H_n\}_{n \in \mathbb{N}} is a Cauchy sequence of progressively measurable simple functions in {\cal{L}}^2(\mu \times \mathbb{P}) then since the difference of two simple processes is again a simple process we can apply the Itô Isometry to deduce that

\displaystyle   \lim_{m,n \to \infty}\mathbb{E}\bigg(\int_0^\infty (H_n(s) - H_m(s))\,\mathrm{d}W(s)\bigg)^2 =  \lim_{m,n \to \infty}\mathbb{E}\int_0^\infty (H_n(s) - H_m(s))^2\,\mathrm{d}s =  0

In other words, \int_0^\infty H_n(s)\,\mathrm{d}W(s) is also Cauchy in {\cal{L}}^2(\mathbb{P}) and since this is complete, we can conclude that

\displaystyle   \int_0^\infty X(s)\mathrm{d}W(s) \triangleq \lim_{n \to \infty}\int_0^\infty X_n(s)\mathrm{d}W(s)

exists (in {\cal{L}}^2(\mathbb{P})). Uniqueness follows using the triangle inequality and the Itô isometry.

Notes

  1. We defer proving the definition also makes sense almost surely to another blog post.

  2. This approach seems fairly standard see for example Handel (2007) and Mörters et al. (2010).

  3. Rogers and Williams (2000) takes a more general approach.

  4. Protter (2004) takes a different approach by defining stochastic processes which are good integrators, a more abstract motivation than the one we give here.

  5. The requirement of progressive measurability can be relaxed.

Bibliography

Handel, Ramon von. 2007. “Stochastic Calculus, Filtering, and Stochastic Control (Lecture Notes).”

Mörters, P, Y Peres, O Schramm, and W Werner. 2010. Brownian motion. Cambridge Series on Statistical and Probabilistic Mathematics. Cambridge University Press. http://books.google.co.uk/books?id=e-TbA-dSrzYC.

Protter, P.E. 2004. Stochastic Integration and Differential Equations: Version 2.1. Applications of Mathematics. Springer. http://books.google.co.uk/books?id=mJkFuqwr5xgC.

Rogers, L.C.G., and D. Williams. 2000. Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus. Cambridge Mathematical Library. Cambridge University Press. https://books.google.co.uk/books?id=bDQy-zoHWfcC.

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