Important Assumption and Lemma for Stochastic Analysis

This article illustrates fundamental assumptions and very useful lemmas for the stochastic analysis. Mainly, these are from the following reference [1].

Destination Equation

Let the following non-autonomous scalar stochastic differential equation. It is slightly more general form.

\(dX_t = a(t, X_t)dt + b(t, X_t) dW_t \label{eq01:DE} \tag{1}\) where $W_t$ is the Wiener process $W={W_t, t \geq 0 }$, that is $\mathcal{F}^*$ adapted process. each $t \in [t_0, T]$

Suppose that there are two solution $X_t$ and $\tilde{X}_t$ of the equation $\eqref{eq01:DE}$ and the solutions are Path-wise unique almost surely on the same sample path on $[t_0, T]$ such that

\[P\left( \sup_{t_0 \leq t \leq T} \| X_t -\tilde{X}_t \| > 0 \right) = 0,\]

Fundamental Assumption

Measurability

$a = a(t, x)$, and $b = b(t,x)$ are jointly ($\mathcal{L}^2 -$ ) measurable in $(t,x) \in [t_0, T] \times \mathbf{R}$;

  • $\mathcal{L}^2 $ means that the function is the squared Lebsegue measurable.

Lipschitz condition

There exists a constant $K > 0$ such that

\[\begin{aligned} \| a( t,x) - a (t, y) \| \leq K \| x - y\| \\ \| b( t,x) - b (t, y) \| \leq K \| x - y\| \end{aligned}\]

for all $t \in [t_0. T]$ and $x, y \in \mathbf{R}$;

Linear growth bound

There exists a constant $K > 0$ such that

\[\begin{aligned} \| a(t,x) \|^2 \leq K^2 (1 + \| x \|^2) \\ \| b(t,x) \|^2 \leq K^2 (1 + \| x \|^2) \end{aligned}\]

for all $t \in [t_0, T]$ and $x, y \in \mathbf{R}$

Initial value

$X_{t_{0}}$ is \(\mathcal{F}_{t_0}\)measurable with $\mathbb{E}( |X_{t_0} |^2) < \infty$.

Hölder continuous condition

The following is the $\alpha$-Höder continuous \(\| f(x) - f(y) \| \leq C \| x - y\|^{\alpha}\)

If $\alpha=1$ then it is a Lipschitz condition.

Lemma

Grownwall Inequality

Let $\alpha, \beta : [t_0, T] \rightarrow \mathbf{R}$ be integrable with

\[0 \leq \alpha(t) \leq \beta(t) + L \int_{t_0}^t \alpha(s) ds\]

for $t \in [t_0, T]$ where $L > 0$ . Then

\[\alpha(t) \leq \beta(t) + L \int_{t_0}^t e^{L(t-s)}\beta(s) ds \label{eq01:lemma} \tag{2}\]

Simple Version

Let $\alpha, \beta : [t_0, T] \rightarrow \mathbf{R}$ be integrable with

\[\alpha(t) \leq \beta + L \int_{t_0}^t \alpha(s) ds \label{eq02:lemma} \tag{3}\]

for some constant $\beta, L$ then

\[\alpha(t) \leq \beta e^{L(t-t_0)}\]

proof of Simple Version

Define $w(t) = \int_{t_0}^t \alpha(s) ds$. Then, since $w’(t) = \alpha(t)$, by $\eqref{eq02:lemma}$, we obtain

\[\alpha(t) = w'(t) \leq \beta + L w(t)\]

It is important to solve a ordinary differential equation that let $w(t)$ be as follows with $f(t_0) = 0$ \(w(t) \leq e^{L(t - t_0)} f(t). \label{eq03:Lemma} \tag{4}\)

\[w'(t) \leq L \cdot e^{L(t - t_0)} f(t) + e^{L(t - t_0)} f'(t) \label{eq04:Lemma} \tag{5}\]

In $\eqref{eq04:Lemma}$,since the first term is same to $L w(t)$ and the second term shall be $\beta$, we regrad $f’(t)$ as follows:

\[\begin{aligned} f'(t)e^{L(t - t_0)} &= \beta \\ f'(t) &= \beta \cdot e^{-L(t-t_0)} \end{aligned} \label{eq05:lemma} \tag{6}\]

The Integration of the $\eqref{eq05:lemma}$ is

\[f(t) - f(t_0) = \beta \int_{t_0}^t e^{-L(s - t_0)} ds \label{eq06:lemma} \tag{7}\]

Substitute $\eqref{eq06:lemma}$ to $\eqref{eq03:Lemma}$, we obtain

\[\begin{aligned} w(t) &\leq e^{L(t-t_0)}f(t_0) + e^{L(t-t_0)} \int_{t_0}^t e^{-L(s - t_0)}\cdot \beta ds \\ &\leq e^{L(t-t_0)}f(t_0) + \int_{t_0}^t e^{L(t - s)}\cdot \beta ds \end{aligned} \label{eq07:lemma} \tag{8}\]

By assumption $f(t_0) = 0$, the $\eqref{eq07:lemma}$ is

\[w(t) \leq \int_{t_0}^t e^{L(t - s)}\cdot \beta ds \label{eq08:lemma} \tag{9}\]

Therefore,

\[w(t) = \int_{t_0}^t \alpha(s) ds \leq \int_{t_0}^t e^{L(t-s)} \beta ds \Rightarrow \alpha (t) \leq \beta e^{L(t-t_0)}\]

Q.E.D

proof of Grownwell’s lemma

It is very similar to the proof of the sinple version. Suppose that all assumptions are hold, the $\eqref{eq06:lemma}$ is changed to

\[f(t) = \int_{t_0}^t e^{-L(s - t_0)} \beta(s) ds, \;\;\because f(t_0) = 0\]

Thereby,

\[w(t) \leq \int_{t_0}^t e^{L(t - s)}\cdot \beta (s) ds \label{eq09:lemma} \tag{10}\]

since $\alpha(t) = w’(t)$, we can obtain the following

\[\begin{aligned} w'(t) = \alpha(t) &\leq \frac{\partial }{\partial t} \left( \int_{t_0}^t e^{L(t - s)}\cdot \beta (s) ds \right) \\ &\leq \int_{t_0}^t \frac{\partial }{\partial t} e^{L(t - s)}\cdot \beta (s) + e^{L(t - s)}\cdot \frac{\partial }{\partial t} \beta (s) \bigg\rvert_{t=s}ds \\ &\leq L \int_{t_0}^t e^{L(t - s)}\cdot \beta (s) ds + \int_{t_0}^t \frac{\partial }{\partial t} \beta (t) dt \\ &\leq L \int_{t_0}^t e^{L(t - s)}\cdot \beta (s) ds + \beta (t) \end{aligned}\]

Q.E.D

Note

In final integration, correctly

\[\int_{t_0}^t \frac{\partial }{\partial t} \beta (t) dt = \beta(t) - \beta(t_0)\]

However, since $\alpha(t) > 0, \; \forall t \in [t_0, T] $ , $\beta(t) > 0, \; \forall t \in [t_0, T] $. Therefore, it is possible to neglect the $\beta(t_0)$ term.

Triple Squared Inequality

In stochastic Analysis, generally there are 3 terms in a target stochastic differential equation. The first 2 terms are the first order and second order differential with respect to $t$ and the third term is the first order differential to Wiener process.

\[(a + b + c)^2 \leq 3(a^2 + b^2 + c^2)\]

Cauchy Formula

\[\int_{t_0}^t \int_{t_0}^{t_{n-1}} \cdots \int_{t_0}^{t_1} f(s) ds dt_1 \cdots dt_{n-1} = \frac{1}{(n-1)!} \int_{t_0}^t (t -s)^{n-1} f(s) ds\]

이 공식은 다음과 같이 Stochastic Differential 혹은 Difference의 Analysis에서

\[\mathbb{E} \left( \| X_t^{(n+1)} - X_t^{(n)} \|^2 \right) \leq L \int_{t_0}^t \mathbb{E} \left( \| X_s^{(n+1)} - X_s^{(n)} \|^2 \right) ds \label{eq09:CF} \tag{11}\]

방정식 $\eqref{eq09:CF}$를 다음과 같이 변환하기 위하여 사용한다.

\[\mathbb{E} \left( \| X_t^{(n+1)} - X_t^{(n)} \|^2 \right) \leq \frac{L^n}{(n-1)!} \int_{t_0}^t (t -s)^{n-1} \mathbb{E} \left( \| X_s^{(1)} - X_s^{(0)} \|^2 \right) ds\]

이는 Discrete index (n) 에 대한 Continuous index t 가 존재하는 경우에 대한 해석학적 표현에서 매우 중요한 방법론이다.

Borel Cantelli Lemma

For any sequence of events \(A_1, A_2, \cdots A_n, \cdots \in \mathcal{A}\)

\[P \left( \bigcap_{n=1}^{\infty} \bigcup_{k=n}^{\infty} A_k \right) = 0 \;\;\text{if } \sum_{n=1}^{\infty} P(A_n) < \infty\]

Exponential Inequality

Simple but very important. Suppose that $h < 1$ and $r_1 > h$, then

\[(1 + r_1 h)^n \leq e^{r_1}\]

as $n$ is increased.

Gilbarg and Trudinger Inequality [2]

If $\theta:[0,1]^n \rightarrow \mathbf{R}$ is continuosly differentiable, and if

\[\int \theta (x) dx = 0,\]

then

\[\int \theta(x)^2 dx \leq \int \| \nabla \theta (x) \|^2 dx\]

Note

If we obtain an equation of an integral to probability, such that

\[\int (\frac{p}{\pi} - 1) \pi dx = \int p - pi dx = \int p dx - \int \pi dx = 0,\]

we have to the squared error of the internal term of the integral such that

\[\int (\frac{p}{\pi} - 1)^2 \pi dx.\]

In this case, the Gilbarg and Trudinger inequality is very useful to analyze the squared term.

References

[1]

@book{kloeden2011numerical,
  title={Numerical Solution of Stochastic Differential Equations},
  author={Kloeden, P.E. and Platen, E.},
  isbn={9783540540625},
  lccn={92015916},
  series={Stochastic Modelling and Applied Probability},
  url={https://books.google.co.kr/books?id=BCvtssom1CMC},
  year={2011},
  publisher={Springer Berlin Heidelberg}
}

[2]

@book{Gilbarg, 
  title={Elliptic Partial Differential Equations of Second Order},
  author={David Gilbarg, Neil S. Trudinger},
  isbn={978-3-540-41160-4},
  lccn={0072-7830},
  series={Classics in Mathematics},
  url={https://link.springer.com/book/10.1007/978-3-642-61798-0#about},
  year={2001},
  publisher={Springer Berlin Heidelberg}
}

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