Mathematically questionable

Assignment X2.1 (ii) (a) says:

by simplifying (t+dt)W_{t + dt} - tW_t

show that d(tW_t) = tdW_t + W_t dt

The solution begins:

d(tW_t) = (t + dt)W_{t+dt} - tW_t

This is probably one of the most misleading statements I have seen in the actuarial study material.

The stochastic differential symbol d(.) is not a differential operator and you cannot express it as a Newtonian limit as the assignment solution has done.

The limit (t+dt)W_{t+dt} - tW_t as dt tends to zero does not exist. Brownian motion has infinite geometic detail and the limit will never settle at a value.

This is why stochastic calculus is actually an integral calculus and we must remember the d(.) operator is shorthand for an integral.

The given solution only works by chance of notation. It has no mathematical meaning and will only confuse students.

 

Gareth

We have to remember that, at the end of the day, actuarial science is a practical subject -- it is a branch of applied maths, not pure maths. Although pure mathematicians might insist that stochastic calculus is defined only in terms of integrals, the technique of working with stochastic differential equations -- where we interpret the d's intuitively as "small" changes in the variables when second-order effects are ignored -- works well in practice (provided the correct rules are applied) and is very commonly used.

In fact the limit (t+h)W_{t+h} - tW_t as h tends to zero is equal to zero (since Brownian motion has continuous sample paths).

You said that this was the "most" misleading statement you have seen. If you think you have found other mistakes in the ActEd course materials, please send an email to the email address for that subject (eg ST6*bpp.com). The tutor for that subject can then add it to the corrections list (if appropriate). That way, we can minimise any confusion that might be caused to students.

Thank you

David Hopkins (ST6 Tutor)

 

David,

Ok, I can agree that the limit:

W_{t+dt} - W_t

exists, but the implication of writing:

dW_t = W_{t+dt} - W_t

could lead to misunderstanding. It would be rather tempting to divide both sides by dt, and write:

dW_t / dt = (W_{t+dt} - W_t)/dt

and what's wrong with this? The limit on the right hand side does not exist. While you are correct that W_{t+dt} - W_t --> 0 as dt -->0, once you divide by dt it will not converge.

This is because W_t has unbounded variation.

You can think of this intuitively by considering the fractal property of Brownian motion (it's fractual index is 2.0). You fix a point at t and plot another point at t+dt and draw a chord between them.

As dt decreases, the gradient of the chord changes violently, since each time you zoom in on the Brownian motion, more detail appears. The gradient never settles down and you will find that:
1/dt(W_{t_dt}-W_t) follows a Normal distribution, with mean 0 and variance 1/dt.

However, W_t does have bounded quadratic variation, which means we can define an integral operator, the "Ito integral".

Now, coming back to the original assertion:

dW_t = W_{t+dt} - W_t

This simply makes no sense once you remember that dW_t is an integral.

You can use the Riemann representation of the Ito integral to show this is questionable.

Set up t_i such that t_0 = 0, t_n = t and t_i - t_{i-1} = 1/n.

Then consider:



The right hand side is the Riemann representation of dW_t. Correct me if I am wrong, but I find it pretty difficult to believe that these will converge to the same value...

Now, I agree actuarial science is a branch of applied mathematics. But so is physics and so is quantative finance - and neither of these users of stochastic calculus would tolerate this.

If you posted the assertion:

dW_t = W_{t+dt} - W_t

to http://www.wilmott.com/index.cfm?NoCookies=Yes&forumid=1

(which is a quantative finance forum) you would instantly be told that this is nonsense.

If actuaries wish to gain the respect of the financial community then they need to show a proper understanding of stochastic calculus. This is well below the required standard in my opinion.

 

Gareth,

Whilst your analysis may well be theoretically correct, (and i am in no position to judge this), i dont think having a 100% understanding of stochastic calculus will enhance actuarial respect in the financial community as per yast sentence:

one of the responses i read to the morris review was on the lines of the following:
if you want a perfect balance sheet, you employ an accountant, if you want rigorous statistcal analysis you employ a statistician. But the unique skill of actuaries is that they can combine the RELEVANT KNOWLEDGE from the different areas to produce a coherent and practical solution to the financial management of a clients affairs - the aim is to make financial sense of the future, not predict it, so whilst rigour underpins the problem solving approach, some rigour has to be sacrificed in the interests of clarity and practicality for our clients.

so the analysis applied equally to quantitative finance... if one wants rigoorus "stochastic analytics" they will employ a rocket scientist or quant...and probably not an actuary!!

though some of the exam farces of 2005 (admin errors, CT4 imporper question wording highlighted in recent article in actuary, ST6 exam not aligned to syllabus, lost scripts etc) would certainly present a poor image and iot ould be nice to sort those things out urgently.

 

i agree with the point you make, but my issue is this particular example show a poor understanding of the basic theory underpinning derivative pricing.

investment actuaries are trying to expand into new areas - e.g. pricing mortality bonds or insurance risk derivatives, if we cannot demonstrate compentencies in the basics, we lose credibility.

with pensions dying off the profession needs to make a strong move into investment areas, and in theory we should be adaptable enough to work in quant type roles.

the actual core reading for ST6 is theoretically sound and while it does not go into great detail in many areas, there is nothing in it that would be obviously wrong to an expert in this area.