Why Integrating a Wiener Process Differential Gives Its Endpoint

Summary

The document asks whether integrating the differential of a Wiener process from zero to a terminal time equals the process value at that time, assuming the process starts at zero. It answers yes: this is the integral of the constant integrand one with respect to the process.

The explanation uses the defining sums for a stochastic integral over partitions. With the integrand equal to one, each term is an increment of the Wiener process, and the increments telescope to the difference between the endpoint values. Since the initial value is zero, every such sum equals the terminal value, so taking the limit preserves the equality. The response adds that choosing between the integral notation and the endpoint notation is a matter of taste in this case. The result is a basic identity, not a claim about integrals with nonconstant integrands or broader stochastic calculus conventions.

Key ideas

• With a Wiener process starting at zero, its integral differential over the interval equals its terminal value.
• The defining partition sums reduce to sums of process increments.
• Those increments telescope to the terminal value minus the initial value.
• For this constant-integrand case, the integral and endpoint notations express the same quantity.

Tags

# Wiener process integral


# Wiener process integral












Suppose that $W_{t}$ is a Wiener process.

Assume $W_{0}=0.$ Is it true that $\int_{t=0}^{T}dW_{t}=W_{T}$? If so, why?

Is one preferred to the other?

## Answer by Sergio Almada (score 6, accepted)

https://quant.stackexchange.com/a/15670

yes, by definition. The integral is by definition the limit $$ \int_0^T f_s dW_s = lim_{ n \to \infty } \sum_{ s_i \in \mathcal{P}_n } f_{s_i} ( W_{ s_{i+1} } - W_{s_i} ), $$ where $\mathcal{P}_n$ is a partition of $n$ points of the interval $[0,T]$. In your case, for any partition, the sum above is telescopic and always evaluates to $W_T$. so the limit does too and the equality follows.

Once this idea is establish to use one or the other is a matter of taste and notation.

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