Existence and Uniqueness of Stochastic Differential Equation

• Strong Solution
  • Lemma : Uniqueness
    • proof of Lemma
    • Meaning of Superscript (N)
  • Theorem : Strong Solution
    • proof
  • Conclusion
• Appendix
  • Cauchy Formula
  • Maximal Martingale inequality
  • Doob’s Inequality
  • Borel Cantelli Lemma
  • Corollary
    • Note
• References

This article illustrates the existence and uniqueness of stochastic differential equations. In the procedure of the proof, there are many techniques and ideas for the analysis of the dynamics described by a stochastic differential form. Most of contents in this articles are strongly depending on the reference[1].

Strong Solution

Let the stochastic differential equation as follows:

The integral form of \(\eqref{eq01:strong}​\) is

Lemma : Uniqueness

If assumptions of the measurable and Lipschitz continuous in Important Assumption and Lemma for Stochastic Analysis are hold, then the solution of \(\eqref{eq02:strong}\) corresponding to the same initial value and the same Wiener process are path-wise unique.

proof of Lemma

For \(N>0\) and \(t \in [t_0, T]\), we define the indication function such that

Obviously \(I_t^{(N)}\) is \(\mathcal{A}_t\)-measurable and \(I_t^{(N)} = I_t^{(N)} I_s^{(N)}\) for \(t_0 \leq u \leq t​\).

Let \(Z_t^{(N)} = I_t^{(N)} (X_t - \tilde{X}_t)\)

By the Lipschitz condition, for \(t_0 \leq s \leq t\),

Thus the following inequality is evaluated (and see the Note 1)

Therefore, we can obtain

for \(t \in [t_0, T]\) where \(L = 2(T - t_0 + 1)K^2\).

Apply Grownwall’s Lemma with

and \(\beta(t) \equiv 0​\), then

We conclude that

In addition, for \(t \in [t_0, T]\), \(I_t^{(N)} X_t = I_t^{(N)} \tilde{X}_t\) with probability 1.

Since the sample paths are continuous almost surely, they are bounded almost surely. Thus we can make the probability

arbitrarily small by taking \(N\) sufficiently large. This means that \(P \left( X_t \neq \tilde{X}_t \right) = 0\) for each \(t \in [t_0, T]\), and hence that \(P \left( X_t \neq \tilde{X}_t : t \in D \right) = 0\) for any countable dense subset \(D\) of \([t_0, T]\).

Q.E.D

Meaning of Superscript (N)

See the following diagram.

즉, 위 그림에서 처럼, 어떤 단계를 Stochastic process가 도달 했는지 아닌지를 알리는 것이다.

이는 Lebesgue Integral을 위해 필요한 단계이다.

Theorem : Strong Solution

Under assumptions in the article, Important Assumption and Lemma for Stochastic Analysis, the stochastic differential equation \(\eqref{eq01:strong}\) has a path-wise unique strong solution \(X_t\) on \([t_0, T]\) with

proof

Let \(X_t^{(0)} \equiv X_{t_0}\) and

for \(n = 0, 1, 2, \cdots\).

For assumption of the initial value, it is clear that

Applying the inequality \((a + b +c)^2 \leq 3(a^2 + b^2 + c^2)\), the Cauchy-Schwarz inequality and the linear growth bound to \(\eqref{eq01:strong_pf}​\),

Note 1 By the Cauchy-Schwartz inequality, \(\begin{aligned} \left| \int_{t_0}^T A(s) ds \right|^2 &\leq \int_{t_0}^T \left| A(s) \right| ds \cdot \int_{t_0}^T \left| A(s) \right| ds \\ &\leq \int_{t_0}^T \left| A(s) \right|^2 ds \cdot \int_{t_0}^T ds \;\;\; \because |a|^2 \leq |A|^3 \end{aligned}\)

Therefore,

End of Note 1

we obtain

for \(n=0, 1, 2, \cdots\). By induction we thus have

for \(n=0, 1, 2, \cdots\).

이로써 \(\mathbb{E} \left( {\mid X_t^{(n+1)} \mid }^2 \right)\)의 Finite 특성이 증명되었음. 이를 토대로 Lemma를 적용한다.

Note 2 앞에서 \(\mathbb{E} \left( {\mid X_t^{(n+1)} \mid }^2 \right)\) 가 증명 되었으므로 Lemma 에서의 식 \(\eqref{eq02:pf_lemma}\) 에서 다음의 특성을 얻는다.

for \(t \in [t_0, T]\) and \(n=1, 2, 3 \cdots\) where \(L = 2(T - t_0 + 1)K^2\).

Cauchy Formula \(\eqref{eq01:appd}\)에 의해 \(\mathbb{E} \left( \left| X_t^{(n+1)} - X_t^{(n)} \right|^2 \right)\) 와 \(\mathbb{E} \left( \left| X_s^{(n+1)} - X_s^{(n)} \right|^2 \right)\) 간의 관계식은 다음과 같다. (즉 Level이 \(X_s^{(0)}\) 에서 \(X_s^{(n)}\) 까지 움직여야 한다. Cauchty Formula의 자세한 내용은 Appendix를 참조한다.)

By Cauchy Formula, We obtain

for \(t \in [t_0, T], \; n=0, 1, 2, \cdots ​\).

Herein, we use Growth bound Condition, instead of Lipschitz Condition, (look at the Link of Assumption ) in the derivation of \(\eqref{eq02:strong_pf}\), then, for $n=0$, we find that

On inserting this into \(\eqref{eq02:strong_pf}\), we obtain

for \(t \in [t_0, T], \; n = 0, 1, 2, \cdots\)m and hence

이 자체로 the mean-suqare convergence of the successive approximation uniformly 를 의미한다.

하지만 우리가 원하는 것은 the almost sure convergence of their sample paths uniformly 이다.

이를 위해서는 최종적으로 이것이, 최소한 Chebyshev 부등식을 만족함을 보여야 한다. 그러기 위해서는 다음의 Square 값의 Supremum을 계산하여야 한다.

To show this, we define

For \(n=\mathbf{Z}[0, \infty]\) From \(\eqref{eq01:pf_lemma}​\), we obtain $$ Z_n \leq \int_{t_0}^t \left | a(s, X_s^{n}) - a(s, X_s^{n-1}) \right | ds

• \sup_{t_0 \leq t \leq T} \left | \int_{t_0}^t \left( b(s, X_s^{(n)} - b(s, X_s^{(n-1)}) \right) \right | dW_s. $$

By Lipschitz Condition

By Cauchy-Schawartz

By Doob’s Inequality

Subsequently,

We combined with $\eqref{eq03:strong_pf}$ to conclude that

where $C_2 = C_1 K^2 (T - t_0 + 4)(T - t_0)$ for $n = 1, 2, 3, \cdots$.

After applying the Markov inequality, i.e. \(P(Z_n > \frac{1}{n^2}) \leq \left( n^2 \right) \mathbb{E}|Z_n|^2\) to each term and summing, we have

Checking the internal term of summation in the right-side , we evaluate the ratio of order such that

When \(n > 4\) , \(\frac{e_n}{e_{n-1}} < 1\) and $n^4 < (n -1)!$. Therefore the summation can be converges for $n > n_0$ where $n_0 > 0$ is a sufficient large.

Hence the series on the left side also converges, so by the Borel-Cantelli Lemma, $Z_n$ converges 0, almost surely. that is the successive approximations $X_s^{(n)} $ converges almost surely, uniformly on $[t_0, T]$ to a limit $\tilde{X}_t$ defined by

즉, 여기서 $\sum_{n=0}^{\infty} \left( X_t^{(n+1)} - X_t^{(n)}\right)$ 가 확률적으로 유한하므로 $\left( X_t^{(n+1)} - X_t^{(n)}\right)$ 는 $n \uparrow \infty$ 에 따라 0에 수렴해야 한다.

As the limit of \(\mathcal{A}^*\) -adapted processes, $\tilde{X}$ is \(\mathcal{A}^*\) -adapted and as the uniform limit of continuous, it is continuous

따라서, $n \rightarrow \infty$ 에 따라 Lipschitz Continuous 에 의해

with probability 1, which implies

in probability, as $n \rightarrow \infty​$ for each $t \in [t_0, T] ​$ . Hence the right side of $\eqref{eq01:strong_pf}​$ converges to the right side of $\eqref{eq02:strong}​$, and so the limit process $\tilde{X}​$ satisfies the stochastic integral equation $\eqref{eq02:strong}​$.

This completes the proof of the existence and uniqueness of a strong solution of the stochastic differential equation $\eqref{eq01:strong}$ for an initial value $X_{t_0}$ with \(\mathbb{E}( \mid X_{t_0} \mid^2) < \infty\).

Conclusion

• 다시말해, Assumption A1~A4 를 만족하면, (Measurabilitry, Lipschitz Continuous, Linear Growth, Finite Initial Value) 만 만족하면, pathwise unique strong solution $X_t$ 가 $[t_0, T]$ 에서 존재한다는 것이다.
• $\eqref{eq03:strong_pf}$ 조건이 최근 여러 논문에서 자주 언급된다.

• 생각해 보면 이 조건은 다음 Stochastic integral 에서 $\eqref{eq01:strong_pf}$ 의 solution $X_t^{(n+1)}$ 이 존재한다는 것을 증명하는 것으로 $n \uparrow \infty$ 에서 사실상 $X_t^{(n)} \rightarrow X_t^{(n+1)}$ with the mean-square convergence of the successive approximation uniformly 특성을 통해 다음이 Unique 하게 존재한다는 것을 Strong Sense로 증명한 것이다. \(\begin{aligned} X_t^{(n+1)} &= X_{t_0} + \int_{t_0}^t a(s, X_s^{(n)})ds + \int_{t_0}^t b(S, X_s^{(n)}) dW_s \\ \Rightarrow X_t &= X_{t_0} + \int_{t_0}^t a(s, X_s)ds + \int_{t_0}^t b(S, X_s) dW_s \end{aligned}\)

• 그러므로 알고리즘의 수렴성을 증명하기 위해서는 이러한 Bounded 조건이 만족됨을 보이고 (최소한 Reference를 통해서 … ) 그 다음에 본격적으로 수렴성을 증명해야 한다.
• Therefore, to prove the convergence of any machine learning algorithm, researchers have to verify such a bounded condition of \(\mathbb{E}(\mid X_t \mid )^2 < \infty, \;\; \forall t \in [t_0, T]\) ( at least, by reference). Following this, they have to prove the main convergence property of any proposed algorithm, additionally.

Appendix

Cauchy Formula

Maximal Martingale inequality

A seperable martingale \(X = \{ X_t, t \geq 0 \}\) with finite p-th moment satisfies the maximal martingale inequality

Doob’s Inequality

Borel Cantelli Lemma

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

Corollary

Suppose that A1-A4 (Fundamental Assumptions in the link ) hold and that

for some integer $ n \geq 1$. Then the solution $X_t$ of $\eqref{eq01:strong}$ satisfies

and

for $t \in [t_0, T]$ where $T < \infty, \; C = 2n (2n + 1) K^2$ and $D$ is a positive constant depending only on $K$ and $T - t_0$.

Note

The above Collorlary illustrates that the bounded condition $\eqref{eq03:strong_pf}$ is not suffcient for the convergence. Suppose the case of $n=1$, the bound of $\mathbb{E}\left( \left |X_t - X_{t_0} \right |^{2n} \right)$ is diverge as $t$ is monotone increased to $\infty$. There is not any control parameter the increasing time parameter in $\eqref{eq02:corollary}$.

• The proof is very similar to that of theorem 1.
  • Therefore, it requires an additional analysis of the algorithm.
• Using H\older inequality and Grownwell’s lemma additionally.

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}
}