Taking Expectations of a Stochastic Differential Equation

Summary

The document derives an expression for the expected value of a process described by a stochastic differential equation with drift and diffusion terms. Rewriting the equation in integral form separates accumulated drift from the stochastic integral. Under the stated zero mean assumption for the random increments, the stochastic integral has zero expectation, leaving the initial value plus the time integral of the expected drift.

This result does not generally yield a closed form for the mean of the process: the drift depends on the process itself, so evaluating its expectation requires additional information about the drift and the distribution or dynamics of the state. The answer gives the general expectation relation but no worked example or conditions for existence and interchange of expectation and integration. It is a useful starting point for stochastic modeling in finance, not a complete solution for every nonlinear process.

Key ideas

• Writing the stochastic differential equation in integral form clarifies how to take expectations.
• With appropriate integrability conditions and mean zero increments, the stochastic integral contributes zero to the expectation.
• The expected process equals its initial value plus the accumulated expected drift.
• Computing the drift expectation explicitly depends on the drift function and process dynamics.

Tags

# Expected Value of Stochastic Process


# Expected Value of Stochastic Process












Given the following stochastic process:

$$ dX = a(X,t)dt + b(X,t)dz $$

where:

$$ dz = A \sqrt{dt}$$

and $A$ is a random variable with mean zero and variance $1$.

Is there a way to calculate the expected value of $X$ at some time $t$? My suspection is that it is simply the integral of the expected value of $dX$. However, the function impacts its own expected value, so I could also imagine that the answer is very different. Any help here? Thanks :)

## Answer by clarkmaio (score 3)

https://quant.stackexchange.com/a/36940

If you write the SDE in the integral form everything should be straightforward: $$ X_t = X_0 + \int_0^t a(X_s, s) ds + \int_0^t b(X_s, s) dz_s $$

If you now take the expected value the third term disappear since $A$ has $0$ mean (and by definition of the integral of stochastic process).

At the end you obtain:

$$ \mathbb{E} \left[ X_t\right] = X_0 + \int_0^t \mathbb{E} \left[ a(X_s, s) \right] ds. $$

Now the explicit computation depends on the structure of the term $a$

Ciao!

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