Stochastic Calculus

1. Introduction:

In this new chapter, we are going to define- Ito integrals and develop their properties. These are used to model the value of a portfolio that results from trading assets in continuous time. The calculus used to manipulate these integral is based on the Ito-Doeblin formula, that differs from the ordinary calculus.

This difference can be traced to the fact that Brownian motion has non-zero quadratic variation and it is the source of volatility term in the Black-Scholes-Merton partial differential equation.

2. Ito’s integral for simple integrands

Let us first fix a positive number \(T\). We want to see what happens to the following integration- \[\int_{0}^{T}\Delta(t) \mathbb{dW(t)}\] The basic ingredients here are a Brownian motion \(\mathbb{W(t)},\ t \geq 0\), together with filtration \(\mathbb{F(t)},\ t \geq 0\) for this Brownian motion.

We will let the integrand \(\Delta(t)\) be an adaptive stochastic process. Adapted stochastic process is a stochastic process that cannot “see into the future”.

We are doing this because eventually \(\Delta(t)\) will be the position we take in an asset at time \(t\). Tn simple words, the amount of asset owned at time \(t\); and this \(\Delta(t)\) depends on the price path of the asset at time \(t\). Anything that depends on the path of a random process is itself random.

We need \(\Delta(t)\) to be \(\mathbb{F(t)}, \ t \geq 0\) measurable. In simple words, the information available at time \(t\) is sufficient to evaluate \(\Delta(t)\) at time \(t\).

Now remember that, the increments of Brownian motion after time \(t\) are independent of \(\mathbb{F(t)}\) and since \(\Delta(t)\) is \(\mathbb{F(t)}\) measurable, it also must be independent of the future Brownian motion increments. Let \(0 \leq s < t\) \[\begin{aligned} &\mathbb{W(t)} -\mathbb{W(s)} \ \text{is independent of} \ \mathbb{F(s)} \\ &\Delta(s) \ \text{is} \ \mathbb{F(s)} \ \text{measurable} \\ &\text{This implies} \ \mathbb{W(t)} -\mathbb{W(s)} \ \text{is independent of} \ \Delta(s) \\ \end{aligned}\]

Positions (amount of asset) we take in assets may depend on the price history of those assets (i.e. the past prices) but they must be independent of the future increments of the Brownian motion that derives the prices.

The problem we face when trying to assign meaning to the Ito integral is that- Brownian motion paths cannot be differentiated with respect to time.

If \(\mathbb{g(t)}\) is a differentiable function, then we can define- \[\int_{0}^{T}\Delta(t). \mathbb{dg(t)} = \int_{0}^{T}\Delta(t). \mathbb{g'(t)} \ \mathbb{dt}\]

But it will not work in case of Brownian motion.

2.1 Construction of the Ito integral

Ito calculus is named after famous Japanese Mathematician Kiyosi Ito (1915-2008) who made fundamental contribution in Probability Theory, in particular Stochastic processes. He invented the concept of Stochastic integral and stochastic differential equation which is now known as Ito calculus.

To define the integral, Ito devised the following way around the non-differentiability of the Brownian motion paths. we first define the Ito integral for simple integrand \(\Delta(t)\), then extend it to the non-simple integrands as a limit off the integral of simple integrands.

Let \(\Pi = \{t_0, t_1, t_2, \cdots t_n \}\) be a partition of \([0,\mathbb{T}]\), with- \[0 = \mathbb{t_0} \leq \mathbb{t_1} \leq \mathbb{t_2} \leq \cdots \leq \mathbb{t_n} = \mathbb{T}\]

Assume \(\Delta(\mathbb{t})\) is a constant in \(\mathbb{t}\) on each sub-interval \([\mathbb{t_j},\mathbb{t_{j+1}})\). Such a process \(\mathbb{\Delta(t)}\) is a simple process. i.e. if \[\Delta(\mathbb{s}) = \Delta(\mathbb{t}) \quad \text{where,} \ \mathbb{s} \ \text{&} \ \mathbb{t} \ \in \ [\mathbb{t_j},\mathbb{t_{j+1}})\]

Form this picture it is not straight forward the path shown depends on the same \(\mathbb{\omega}\) on which the path of the Brownian motion \(\mathbb{W(t)}\) depends. If we had different \(\mathbb{\omega}\) there would be a different path of Brownian motion \(\mathbb{W(t)}\) and possibly a different path of \(\Delta(\mathbb{t})\)

Here, the value of \(\Delta(\mathbb{t})\) only depends on the information available at time \(\mathbb{t}\). Now since there is no information at time \(\mathbb{t = 0}\), \(\Delta(\mathbb{0})\) must be same for all possible paths. So in the partition \(\mathbb{\Pi}\) the value of \(\Delta(\mathbb{t})\) does not depend on \(\omega\) for \(0 \leq \mathbb{t} < \mathbb{t_1}\). But the value of \(\Delta(\mathbb{t})\) for the second interval i.e. \([\mathbb{t_1,t_2})\) can and will depend on the observation made on the interval \([\mathbb{0,t_1})\).

Now we will compare all these notations to the financial terms and what that implies. \[\begin{aligned} &\mathbb{W(t)} \ = \ \text{The price per share of an asset at time} \ \mathbb{t} \\ \end{aligned}\] Suppose you have fixed \(\mathbb{T}\) time up to which we want to keep track of the asset price or the horizon of the portfolio. The partition \(\mathbb{\Pi := \{0 = t_0,t_1, \cdots, t_n = T \}}\) is nothing but the transaction dates of assets between \(0\) and \(\mathbb{T}\).

And consider, \[\begin{aligned} \mathbb{\Delta(t_j)} = \ &\text{Position / number of shares taken in asset }& \\ &\text{at each trading dates held to the next trading dates} \\ &\text{For-} \quad \mathbb{j = 0,1, \cdots, n-1} \\ \end{aligned}\]

In this process, the gain from trading at each time \(\mathbb{t}\) is given by- \[\begin{aligned} \mathbb{I(t)} &= \Delta(\mathbb{t_0})[\mathbb{W(t)}-\mathbb{W(t_0)}] \qquad \text{where} \quad \mathbb{0} \leq \mathbb{t} < \mathbb{t_1} \\ \mathbb{I(t)} &= \Delta(\mathbb{t_0})[\mathbb{W(t_1)}-\mathbb{W(t_0)}]+\Delta(\mathbb{t_1})[\mathbb{W(t)}-\mathbb{W(t_1)}] \qquad \text{where} \quad \mathbb{t_1} \leq \mathbb{t} < \mathbb{t_2} \\ \mathbb{I(t)} &= \Delta(\mathbb{t_0})[\mathbb{W(t_1)}-\mathbb{W(t_0)}]+\Delta(\mathbb{t_1})[\mathbb{W(t)}-\mathbb{W(t_1)}]+\Delta(\mathbb{t_2})[\mathbb{W(t)}-\mathbb{W(t_2)}] \qquad \text{where} \quad \mathbb{t_2} \leq \mathbb{t} < \mathbb{t_3} \\ &\text{and so on} \ \cdots \end{aligned}\]

In general if- \(\mathbb{t_k \leq t < t_{k+1}}\)- \[\mathbb{I(t) = \sum_{j = 0}^{k-1}\Delta(t_j)[W(t_{j+1})-W(t_j)]+ \Delta(t_k)[W(t)-W(t_k)]}\]

Now this \(\mathbb{I(t)}\) is the Ito integral of the simple process \(\Delta(\mathbb{t})\). Ito integral can be represented as- \[\mathbb{I(t)} = \int_{0}^{\mathbb{t}} \Delta(\mathbb{u}) \mathbb{dW(u)}\] In particular, we take \(\mathbb{t = t_n = T}\) in the summation definition.

2.2 Properties of the Ito integral

The Ito integral- \[\mathbb{I(t) = \sum_{j = 0}^{k-1}\Delta(t_j)[W(t_{j+1})-W(t_j)]+ \Delta(t_k)[W(t)-W(t_k)]} \qquad \text{For} \ \mathbb{t_k} \leq \mathbb{t} < \mathbb{t_{k+1}}\] is nothing but the gain from trading in the martingale \(\mathbb{W(t)}\). Now, a martingale has no tendency to rise or to fall.

Let us see what happens in case of \(\mathbb{I(t)}\), which is integration of \(\mathbb{\Delta(u)}\) with respect to \(\mathbb{W(u)}\); if \(\mathbb{I(t)}\) is a martingale or not?

Let \(\mathbb{0 \leq s \leq t \leq T}\) be given. We assume that \(\mathbb{s}\) and \(\mathbb{t}\) are in different sub-intervals of partition \(\Pi\), i.e. \[\begin{aligned} &\text{There exists partition points-} \quad \mathbb{t_l} \ \text{&} \ \mathbb{t_k} \ \text{such that-} \quad \mathbb{t_l} \leq \mathbb{t_k} \ \text{and} \\ &\mathbb{s} \in \mathbb{[t_l,t_{l+1})} \ \text{and} \ \mathbb{t} \in \mathbb{[t_k,t_{k+1})} \end{aligned}\]

In that case, we can write the Ito integral \(\mathbb{I(t)}\) as follows- \[\begin{aligned} \mathbb{I(t)} = &\mathbb{\sum_{j=1}^{k-1} \Delta(t_j)[W(t_{j+1})- W(t_{j})] + \Delta(t_k)[W(t)- W(t_{k})]} \\ = &\mathbb{\sum_{j=1}^{l-1} \Delta(t_j)[W(t_{j+1})- W(t_{j})] + \Delta(t_l)[W(t_{l+1})- W(t_{l})]} \\ & + \mathbb{\sum_{j=l+1}^{k-1} \Delta(t_j)[W(t_{j+1})- W(t_{j})] + \Delta(t_k)[W(t)- W(t_{k})]} \end{aligned}\]

Now, to see if \(\mathbb{I(t)}\) is a martingale or not, we need to calculate- \(\mathbb{E[I(t)|F(s)]}\).

\[\begin{aligned} & \ \mathbb{E[I(t)|F(s)]} \\ = & \ \mathbb{E[\mathbb{\sum_{j=1}^{l-1} \Delta(t_j)[W(t_{j+1})- W(t_{j})] + \Delta(t_l)[W(t_{l+1})- W(t_{l})]} \\ + \mathbb{\sum_{j=l+1}^{k-1} \Delta(t_j)[W(t_{j+1})- W(t_{j})] + \Delta(t_k)[W(t)- W(t_{k})}]|F(s)]} \\ = & \ \mathbb{\mathbb{\sum_{j=1}^{l-1} E[\Delta(t_j)[W(t_{j+1})- W(t_{j})]| F(s)] + E[\Delta(t_l)[W(t_{l+1})- W(t_{l})]|F(s)]} \\ + \mathbb{\sum_{j=l+1}^{k-1}E[\Delta(t_j)[W(t_{j+1})- W(t_{j})]|F(s)] + E[\Delta(t_k)[W(t)- W(t_{k})}]|F(s)]} \\ \end{aligned}\]

So now we have got 4 parts.

The first part is- \[\mathbb{\mathbb{\sum_{j=1}^{l-1} E[\Delta(t_j)[W(t_{j+1})- W(t_{j})]| F(s)]}}\] Now we have previously assumed that- \(\mathbb{t_l \leq s}\)- \[\text{Therefore} \quad \mathbb{\sum_{j=1}^{l-1} \Delta(t_j)[W(t_{j+1})- W(t_{j})]} \ \text{is} \quad \mathbb{F(s)}- \text{measurable}\] Therefore we can write- \[\mathbb{\mathbb{\sum_{j=1}^{l-1} E[\Delta(t_j)[W(t_{j+1})- W(t_{j})]| F(s)]}} = \mathbb{\sum_{j=1}^{l-1}\Delta(t_j)[W(t_{j+1})- W(t_{j})]}\]

The second part is- \[\begin{aligned} &\mathbb{E[\Delta(t_l)[W(t_{l+1})- W(t_{l})]|F(s)]} \qquad \text{with} \quad \mathbb{t_l \leq s < t_{l+1}} \\ \end{aligned}\] Hence we can write- \[\begin{aligned} &\mathbb{E[\Delta(t_l)[W(t_{l+1})- W(t_{l})]|F(s)]} \\ = \ & \mathbb{\Delta(t_l)[E[W(t_{l+1})|F(s)]- W(t_{l})]} \\ = \ & \mathbb{\Delta(t_l)[W(s)- W(t_{l})]} \\ \end{aligned}\]

Now, let us go to the third part, which is- \[\begin{aligned} \mathbb{\sum_{j=l+1}^{k-1}E[\Delta(t_j)[W(t_{j+1})- W(t_{j})]|F(s)]} \qquad \text{where} \quad \mathbb{t_j \geq t_{l+1} > s} \end{aligned}\] Here we will use a very useful result- \(\mathbb{E[E[X|Y]] = E[X]}\)

Since \(\mathbb{t_j \geq t_{l+1} > s}\), we can write- \[\begin{aligned} & \mathbb{E\{\Delta(t_j)[W(t_{j+1})- W(t_{j})]|F(s)\}} \\ = \ & \mathbb{E\{E[\Delta(t_j)[W(t_{j+1})- W(t_{j})]|F(t_j)]| F(s) \}} \\ = \ & \mathbb{E\{\Delta(t_j).\{E[W(t_{j+1})|F(t_j)]- W(t_{j})\}| F(s) \}} \\ = \ & \mathbb{E\{\Delta(t_j).\{W(t_{j})- W(t_{j})\}| F(s) \}} \\ = \ & 0 \\ \end{aligned}\]

Hence we can write- \[\begin{aligned} \mathbb{\sum_{j=l+1}^{k-1}E\{\Delta(t_j)[W(t_{j+1})- W(t_{j})]|F(s)\}} = 0 \end{aligned}\]

The fourth part is- \[\mathbb{E\{\Delta(t_k)[W(t)- W(t_{k})]|F(s)\}}\]

We are going to apply the same trick used in part-3, i.e. \(\mathbb{E[E[X|Y]] = E[X]}\).

\[\begin{aligned} & \ \mathbb{E\{\Delta(t_k)[W(t)- W(t_{k})]|F(s)\}} \\ = & \ \mathbb{E\{E[\Delta(t_k)[W(t)- W(t_{k})]|F(t_k)]| F(s)\}} \\ = & \ \mathbb{E\{\Delta(t_k)(E(W(t)|F(t_k))- W(t_k))|F(s)\}} \\ = & \ \mathbb{E\{\Delta(t_k)(W(t_k)- W(t_k))|F(s)\}} \\ = & \ 0 \\ \end{aligned}\]

Combining the above 4 parts, we have-

\[\begin{aligned} & \ \mathbb{E[I(t)|F(s)]} \\ = & \ \mathbb{\mathbb{\sum_{j=1}^{l-1} E[\Delta(t_j)[W(t_{j+1})- W(t_{j})]| F(s)] + E[\Delta(t_l)[W(t_{l+1})- W(t_{l})]|F(s)]} \\ + \mathbb{\sum_{j=l+1}^{k-1}E[\Delta(t_j)[W(t_{j+1})- W(t_{j})]|F(s)] + E[\Delta(t_k)[W(t)- W(t_{k})}]|F(s)]} \\ = & \ \mathbb{\sum_{j=1}^{l-1}\Delta(t_j)[W(t_{j+1})- W(t_{j})]} + \mathbb{\Delta(t_l)[W(s)- W(t_{l})]} + 0 + 0 \\ = & \ \mathbb{I(s)} \end{aligned}\]

So we have proved that \(\mathbb{E[I(t)|F(s)]} = \mathbb{I(s)}\), i.e. \(\mathbb{I(t)}\) is a martingale.

Now, because \(\mathbb{I(t)}\) is a martingale and \(\mathbb{I(0)=0}\) we can write- \[\begin{aligned} &\mathbb{E[I(t)] = 0} \qquad \text{and} \\ &\mathbb{Var[I(t)] = E[I^2(t)]- E^2[I(t)]} = \mathbb{E[I^2(t)]} \\ \end{aligned}\]

In the next blog, we are going to see Ito isometry and quadratic variation of Ito integral. Happy reading!!