The stochastic process is based upon set functions whose domain is a \(\sigma\) -algebra formed from a collection of subsets of a random experimental space, \(\Omega\).
As the process unfolds in time, t, more information is obtained about the outcomes to date of the random experiment. As t increases, the \(\sigma\) -algebra \(f(X_t)\) , has an increasing number of subsets, each with a decreasing number of elements i.e. outcomes.
As an illustration 1 suppose the experiment is tossing a fair coin three times. The atomic outcomes are the \(2^3\) combinations of heads and tails. Each outcome defines a discrete sequence or path in time. All possible unions of these atoms are also in the \(\sigma\) -algebra (and all complements).
At time zero f0=\(\sigma (\)t0) is composed of two subsets: \(\emptyset\), and \(\Omega\). The subset \(\Omega\) contains all possible events of the three coin tosses and their unions. In this discrete example we have, including complements and finite unions, \(2^{2^3}\)= 256 subsets, as below.
As time progresses, say at t=1, we can see which sets are resolved as true or false conditional on the first coin toss. f1 = {\(\emptyset, \Omega\), Ah, At}. Ah is the subset containing all sequences of three coin tosses with the first toss equal to a head: {(HTT),(HHH),(HTH), (HHT)}, plus all unions of these four atomic outcomes. At is the subset containing events with a tail as the first outcome: {(TTT),(TTH),(THT),(THH)} plus all possible unions of these four atomic outcomes. These sets are atomic because they are indivisible and all other sets in the \(\sigma\)-algebra f1 are (countable)unions of these atoms. These include the empty set, \(\emptyset\), and the universal event, \(\Omega\) which are carried into f1 from f0.
This preservation of earlier sets to a later \(\sigma\) -algebra is a property of a filtration in which the stochastic process accumulates more information over time. If at each t the process is measurable , according to a countably additive function which maps each of the subsets in the increasing \(\sigma\) -algebras to Borel sets on {R}, the process is said to be adapted. As the number of subsets in the \(\sigma\) -algebra increases, the number of elements in the new sets added compared to the previous value of t decreases.
For instance after coin toss 2 we have new atomic (i.e. indivisible)sets Ahh= {(HHH),(HHT)}, Aht={(HTT),(HTH)},Att={(TTT),(TTH)}, Ath={(THH),(THT)}.Each of the four subsets contains 2 elements. In contrast, at f1 we had only two subsets each containing 4 elements. We have ignored the unions between the subsets in each \(\sigma\) -algebra but the union of Ahh and Aht in f2 preserves the set Ah from f1; and the union of Att and Ath preserves set At from \(\sigma\) -algebra f1. f2 has more subsets than are in f1, indicating a less coarse collection. Moreover these additional sets carry more information as only two possible paths belong to each set compared to four in f1.
A sub-\(\sigma\) algebra is a coarser collection of subsets. Each \(\sigma\)-algebra is nested: f0 \(\subset\) f1 and so on as information accumulates with time while preserving previous information i.e. the sets in the sub- \(\sigma\)-algebra are carried over as unions in the next period’s \(\sigma\)-algebra.
For a sub-\(\sigma\)-algebra g and a random variable X on a triple (\(\Omega\) ,f,P)2
\(\tau _m\) is a random variable because it depends upon a path of a BM process, and records the first time period in which state m is reached.
\(EQ_\pi\)= \(lim _{n \to \infty}\) \(\sum_{j=0}^{n}[W(j+1)-W(j)]^2 = t\) 5. This follows because each interval within the n+1 partitions of the domain is an independent random variable, with each W(tj) ~ N(0,\(t_{j+1}\) - \(t_{j}\)). The expectation is over all paths \(\omega \varepsilon \Omega\) which smooths out any differences, although since the variance of BM Quadratic Variation is zero then any outlier paths must be countable (with measure zero) and the Expectation holds with probability 1 (‘almost surely’)6
Brownian Motion accumulates Quadratic Variation at the rate one per unit of time7 In fact, Brownian Motion is the limiting distribution of a scaled symmetrical random walk as the time between trails decreases towards zero and the length of up and down ticks consequent of each trial is \(n^{-\frac{1}{2}}\).
For Brownian Motion Quadratic Variation and Variance are both t. However the former is for a particular realised pathof random variables W(t) whilst the latter is an average over all paths. Some paths will have QV <> t but these are countable and hence of null measure so t still holds almost surely.
Usually, for a sufficiently smooth continuous function, ,
f(),quadratic variation is zero8:
\(\sum_{t=0}^{n}|f_{t+1}- f_{t}|^2\)=
\(\sum_{t=0}^{n}|f^{'}_t|^2((t+1) -
t)^2)\) \(\le\sum_{t=0}^{n}||t|||f^{'}_t|^2((t+1) -
t)\)
The first equality follows from the definition of a slope.
The mesh approaches \(\lim_{n \to \infty}||t||\to 0\) as the number of partition points increases so the final inequality becomes an integral…
\(\le\lim{n\to \infty}\sum_{t=0}^{n}||t|||f^{'}_t|^2((t+1)-t)\) \(\to 0.\int_{t=0}^{T}f{'}^{2}dt\)
but multipled by the zero mesh length. In contrast, although BM is a continuous function it is not smooth at all, in fact it is fractal-like at all scales (‘self-similar’) and jagged and thus non-differentiable at all points.
An integral having a BM integrator is called an Ito Integral. Ito Integrals, I(t), are no where differentiable i.e. no tangents exist to points on its graph, and ,exceptionally, have non-zero quadratic variation because of the distributional assumption that its variance is proportion to the length of t intervals.
Ito Integral I(t)= \(\int _{0}^{t}w(t)\; dw(t)\) = \(\sum_{j=1}^{n}w(j)(w(j+1)-w(j))\) = \(\frac{W^2}{2}\)- \(\frac{t^2}{2}\) where w(t) is a function, here Brownian Motion.
I(t) has the properties
The above result can be inferred from the simple functions11: \(\sum_{j=0}^{n-1}W_{j}(W_{j+1}-W_{j})\) = \(\frac{1}{2}W^{2}_{n}\) - \(\frac{1}{2}\sum_{j=0}^{n-1}( W_{j+1}-W_{j})^{2}\) As the max interval size decreases towards zero as the number of intervals, n , in the partition increases, the last term approaches t (as it is just the quadratic variation) while the LHS approaches the integral of a continuous function:
\(\int_{-\infty}^{t}w(t)dw(t) = \frac{1}{2}W^{2}_{n} -\frac{1}{2}t\)
Also consider12: \(\frac{1}{2}\sum_{j=0}^{n-1}(W_{j+1}-W{j})^2\). This multiplies out to \(\frac{1}{2}\sum_{j=0}^{n-1}W_{j+1}^{2} -\frac{1}{2}\sum_{j=0}^{n-1}W_{j+1}W_{j} +\frac{1}{2}\sum_{j=0}^{n-1}W_j^{2}\) = \(\frac{1}{2}W^{2}_{n}+\sum_{j=0}^{n-1}W_{j}(W_{j}-W_{j+1})\)
Re-arranging, \(\sum_{j=0}^{n-1}W_{j}(W_{j}-W_{j+1}) = \frac{1}{2}W^{2} -\frac{1}{2}\sum_{j=0}^{n-1}(W_{j+1}-W{j})^2\).
As the number of intervals n increases and the max interval length approaches zero, we get the continous integrals \(\int_{-\infty}^{t}W(t)dW(t)= \int_{-\infty}^{t}=\frac{1}{2}W^{2}(t)-\frac{1}{2}[W,W](t)\). But the second term on the rhs is just the quadratic variation and equal to \(\frac{t}{2}\)
Note that we taking the left most point in each interval of this partition to evaluate the function, here w(j). If we took the midpoint of each interval, w(j+1/2), we would get the Stratonovich integral and the normal calculus rules would apply 13. However in cases where the function to be integrated, ( ‘integrand’ ),has non-zero quadratic variation then the above Ito integral is the one to choose. Heuristically it applies where a function takes a value at the start of each interval and this value is constant until a moment before the start of the next interval. A process such as buying/selling shares at set times based on previous information fulfills this criterion.
The term \(-\frac{t}{2}\) is a correction to the Ito integral and relates to the integrand having non-zero quadratic variation.
I(t)= \(\int _{0}^{t}w(t)\; dw(t)\) for BM (or any integrand having non-zero Quadratic Variation) is a Martingale. Since all Martingales have a constant unconditional expectation for all t14, E(M)=c. It follows that c= M(t) = M(0). Since BM always starts at zero, W(0)= 0, and EW(t)= 0.
For a BM, \(E(W^2(t))\) is the variance (as EW=0 for BM) so equals t. Therefore \(\frac{1}{2}EW^2(t)\)= \(\frac{t}{2}\) so the correction term, -\(\frac{t}{2}\), ensures that E(I)=\(E(W^2(t))\) - E(\(-\frac{t^2}{2}\)) = 0 to make I, the Ito integral, have an unconditional expectation of zero 15.
Stochastic integration involves taking the integral of functions with time integrators, which are normal integrals, and integrals of Ito integrals which are functions with Brownian Motion integrators. All that is happening is that the multivariate function to integrate is decomposed into its Taylor series up to the second order (higher orders are zero) using , as a rule of thumb, the stochastic Multiplication table17.
\(f(t,w(t))= f(0,w(0))+
\int_{-\infty}^{t}f_{t}(t,w(t))dt\) + \(\int_{-\infty}^{t}f_{w}(t,w(t))dw(t)\) +
\(\frac{1}{2}\int_{-\infty}^{t}f_{ww}(t,w(t))dt\)
where in the last term on the RHS dt is the result of dW(t)dW(t) as per
the multiplication table and the subscripts indicate orders of partial
differentiation wrt to the variable subscripted.
If X = g(t,W) then the formula for the Ito process f(t,X) is slightly more complicated. Remember that df(t,X)= ft(t,W)dt +fw(t,W)dW+ fww(t,W)dWdW where dt = gt(t,W)dt and dW= gw(t,W)dW (where W is Brownian Motion and ft is a partial differential \(\frac{\partial f}{\partial t}\) and fxx is the second partial differential \(\frac{\partial f^2}{\partial x^2}\)
is a process of the form:
\(X(t) = X(0)+\int_{0}^{t}\Delta(t)dW(t) +\int_{0}^{t}\Theta(t)dt\), see P184 [equ.4.8.4] for the integral form.
where \(\Delta(u)\) and \(\Theta(u)\) are adapted stochastic processes and X(0) is non-random. Note that a \(\Theta(t)\) is still random and a stochastic process (which is why it is a function of t which indexes the sequence of realised \(\Theta_t\)) albeit not one with BM as an integrator. An example of a random \(\Theta(t)\) is a variable interest rate process which might be random if it can go up or down between time intervals.
If \(\Delta(u)\) is a non-random,deterministic function of u=time then X(t) is normal ~N(0,\(\int_{0}^{t}\Delta(U)^2du)\). However if \(\Delta(u)\) is a function of BM then X cannot be assumed to be Normal.
The Ito Product for ITo Process X(t)and Y(t): d(X(t)Y(t))= X(t)dY(t) + dX(t)Y(t) + dX(t)dY(t)20
X(t)= \(e^{\int_{o}^{t} \Delta(t)dW(t)-\frac{1}{2}\int_{0}^{t} \Delta(t)^2dt}\) X is a martingale.
It is usually easier to work with GMB in its differential form:
\(dS(t)=\alpha(t)S(t)dt + \sigma S(t)dW(t)\) 22
Allow \(\sigma\) and \(\alpha\) to either vary with time or be random. Then S(t)=\(e^{\int_{0}^{t}\sigma (t)dW(t)+\int_{0}^{t}(\alpha(t)-\frac{1}{2}\sigma ^2)dt}\)
does not have a Lognormal distribution even though the process depends on a single BM, W(t).
However if \(\sigma\) and \(\alpha\) are both constants then S(t)= S(0)\(e^{\sigma (t)dW(t) + (\alpha - \frac{1}{2}\sigma ^2)t}\) is a Geometric Brownian Motion and S(t) has a lognormal distribution.
If M(t) is a martingale then the stochastic integral \(\int_{0}^{t}f_{x}(t,M(t))dM(s)\) is also a Martingale. If at t=0 this integral is zero then its expectation is always zero.
The Radon-Nikodym random variable \(\frac{d\hat p}{dp}\) is a Martingale. 25
These are partial derivatives with respect to a function S(t,W(t)) which represents a portfolio valuation.
A market is complete if a derivative of a security can always be hedged by an appropriate portfolio of cash and stocks.28
A portfolio process is X(t) =V(t) for all t from 0 to T where V(t)is the pay-off of a derivative ( (S-K)^{+}) where K is the strike price at termination T.29
The discounted portfolio process under the risk-neutral measure is a Martingale 30
Stochastic Calculus for finance II , S Shreve, Springer Finance 2009↩︎
Stochastic Calculus for finance II , S Shreve, Springer Finance 2009, P76↩︎
Ibid p109↩︎
Shreve, ibid,p111↩︎
Shreve, ibid, p102↩︎
Shreve, ibid, p103↩︎
Shreve, ibid,p105↩︎
Ibid, p101 & p215↩︎
ibid ,p153↩︎
ibid p213↩︎
ibid,p136↩︎
Ibid,pp135-136↩︎
Shreve , ibid, p136↩︎
Shreve, ibid,p137↩︎
Shreve,ibid,p137↩︎
ibid, P147↩︎
ibid, p147↩︎
Ibid, p141↩︎
ibid,p143↩︎
ibid,p168 Corollary 4.6.3↩︎
ibid,p149↩︎
ibid, p238 equ.(5.5.14)↩︎
ibid, p148↩︎
Ibid p169↩︎
Ibid p211↩︎
Ibid pp159-161↩︎
Ibid p 162↩︎
Ibid, p223↩︎
Ibid, p223↩︎
Ibid, p231 and p234↩︎