The Flow Property of Solutions to Stochastic Differential Equations

Summary

The document asks how to justify the flow property for solutions of a stochastic differential equation: evolving from an initial time to an intermediate time and then continuing from the resulting state should agree with evolving from the original initial condition. The equation has drift and diffusion terms depending on the current state, and the question focuses on whether a proof that replaces one process with another is circular.

The text describes the intended uniqueness argument but does not include a full resolution. The central mathematical idea is to compare the two processes after the intermediate time: they start from the same random state and are driven by the same subsequent Brownian increments, so pathwise uniqueness can identify them. The document is a proof question rather than an application to trading, and it leaves technical conditions unstated; the flow property relies on suitable existence and uniqueness assumptions for the SDE.

Key ideas

• The flow property says an SDE solution can be restarted at an intermediate time from its current state.
• The proposed proof concern is whether identifying the restarted process assumes the result it aims to prove.
• Pathwise uniqueness can establish equality when the compared solutions share the same starting state and driving noise after restart.
• The document does not specify the regularity conditions needed for existence and uniqueness.

Tags

# Proving Flow Property of Stochastic Differential Equation


# Proving Flow Property of Stochastic Differential Equation












I am trying to show that $X_t^{s,x} = X_t^{r, X_r^{s,x}}$ for $0 \leq s \leq r \leq t$, $x \in \mathbb{R}^n$ is a given initial condition for time $s$, for some SDE: \begin{equation*} d X(u)=b(X(u))d u+\sigma(X(u))d B(u). \end{equation*}

I have seen a couple of proofs that use the uniqueness of the solution to show this, but I feel like some of the proofs have some circular reasoning in it. I understand the general idea, that since the stochastic process $X_t$ is unique, then at some time $t$, if we can write two expressions for the process's random value, then the expressions must be equal. In Oksendal the proof is basically what I wrote above and is two lines:

But I wanted a more 'formalized' version of this proof, but couldn't find one until I searched online and found this following one:

From the following link: https://www.math.tecnico.ulisboa.pt/~czaja/ISEM/10internetseminar200607.pdf, the proof of this is the following:

I am wondering how to justify the step of 'replacing' $Z_u$ with $X_u^{s,x}$. Isn't this saying that $Z_u = X_u^{s,x}$, where $Z_u = X_u^{r, X_r^{s,x}}$, and so $X_u^{s,x} = X_u^{r, X_r^{s,x}}$. And if we are asserting that this is true, isn't this the same as what we are trying to prove: $X_t^{s,x} = X_t^{r, X_r^{s,x}}$. So I am not sure what I am missing here.

Any help would be greatly appreciated. Thanks!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.