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Equation 5.3 in The Concepts and Practice of Mathematical Finance by Mark Joshi

I would like to thank Dr. Paul Davis for clarifying this to me.

In Section 5.3 (Stochastic Processes), the following equation appears on page 94:

LaTex: X_{t+h} - X_t = h \mu(t) + \int_{t}^{t+h} e(r) dr + h^{1/2} \sigma(t) N(0, 1) + g(t, h) N(0, 1)

Unfortunately it isn't immediately apparent from the explanation how this equation is derived (at least it wasn't immediately apparent to me, which certainly doesn't mean much).

Here

LaTex: g(t, h) = (h \sigma(t)^2 + \int_{t}^{t+h} f(r) dr)^{1/2} - h^{1/2} \sigma(t)

What is happening? Nothing extraordinary. The algebra is trivial once the logic behind the derivation is understood. I shall derive (5.3) carefully (perhaps too carefully!), step by step.

We have written

LaTex: \mu(r) = \mu(t) + e(r)

and

LaTex: \sigma^2(r) = \sigma^2(t) + f(r)

Thus we have expressed the mean (variance) at LaTex: r in terms of the mean (variance) at LaTex: t plus the error term.

In the first equation on page 94 we were assuming that LaTex: \sigma is identically zero:

LaTex: X_t - X_s = \int_s^t \mu(r) dr

We are no longer assuming that. Let us reinstate our stochastic component, as in page 90 (remember LaTex: \mu + \sigma N(0, 1) ):

LaTex: X_{t+h} - X_t = \int_t^{t+h} \mu(r) dr + \sqrt{\int_t^{t+h} \sigma^2(r) dr} \cdot N(0, 1)

Now let us substitute our expressions for LaTex: \mu(r) and LaTex: \sigma^2(r) :

LaTex: X_{t+h} - X_t = \int_t^{t+h} (\mu(t) + e(r)) dr + \sqrt{\int_t^{t+h} (\sigma^2(t) + f(r)) dr} \cdot N(0, 1)

Hence

LaTex: X_{t+h} - X_t = \mu(t) \int_t^{t+h} dr + \int_t^{t+h} e(r) dr + \sqrt{\sigma^2(t) \int_t^{t+h} dr + \int_t^{t+h} f(r) dr} \cdot N(0, 1)
LaTex: X_{t+h} - X_t = h \mu(t) + \int_t^{t+h} e(r) dr + \sqrt{h \sigma^2(t) + \int_t^{t+h} f(r) dr} \cdot N(0, 1)
LaTex: X_{t+h} - X_t = h \mu(t) + \int_t^{t+h} e(r) dr + \sqrt{h \sigma^2(t) (1 + h^{-1} \sigma^{-2}(t) \int_t^{t+h} f(r) dr)} \cdot N(0, 1)
LaTex: X_{t+h} - X_t = h \mu(t) + \int_t^{t+h} e(r) dr + h^{1/2} \sigma(t) \sqrt{1 + h^{-1} \sigma^{-2}(t) \int_t^{t+h} f(r) dr} \cdot N(0, 1)
LaTex: X_{t+h} - X_t = h \mu(t) + \int_t^{t+h} e(r) dr + h^{1/2} \sigma(t) \left( 1 + \sqrt{1 + h^{-1} \sigma^{-2}(t) \int_t^{t+h} f(r) dr} - 1 \right) \cdot N(0, 1)
LaTex: X_{t+h} - X_t = h \mu(t) + \int_t^{t+h} e(r) dr + h^{1/2} \sigma(t) N(0, 1) + \left[ h^{1/2} \sigma(t) \left( \left( 1 + h^{-1} \sigma^{-2}(t) \int_t^{t+h} f(r) dr \right)^{\frac{1}{2}} - 1 \right) \right] \cdot N(0, 1)

The factor in square brackets is, of course, LaTex: g(t, h) , and we are home.

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