Construction of Ito Integral - simple integrands

Dear experts,

I am trying to understand the application of the below concept:

Question 1 how can [W(t1) - W(t0)] = [W(t1) - W(0)] =[W(t1) - 0] = gain? In this calculation the purchase price is not taken into consideration. The above positive number make sense if it called portfolio value at the end of time t1.

Below is the text extract from Stochastic Calculus Bk2: Continuous Time Models Also below is construction of the topic leading to the above question.

Ito integral for simple integrands

Assume that Δ(t) is constant in t on each subinterval [tj,tj+1). Such a process Δ(t) is a simple process.

Here we choose a single path of a simple process Δ(t)

Values of Δ(t) depends up on a particular path ω belonging sample space Ω. If we choose a different ω then we will have a different Δ(t) for each time interval.

Δ(t) depends only upon the information available till time t.

Since there is no information available at time t = 0, Δ(0) will be the same for all the paths. Hence the first piece of Δ(t) for 0≦t≦t1 does not really depend on ω

The value of the Δ(t) on the second interval [t1,t2) can depend on observations made during the first time interval [0,t1)

We shall think of the interplay between the simple process Δ and the Brownian Motion W(t) in the following way.

Regard W(t) as the price per share of an asset at time t. For here we assume W(t) can take only positive values.

Think of t0, t1,. . . .tn-1 as the trading dates in the asset, and think of ∆(t0), ∆(t1), . . . .,∆(tn-1) as the position (number of shares) taken in the asset at each trading date and held to the next trading date.

The gain from trading at each time t is given by:

I(t) = ∆(t0)[W(t) - W(t0)] = ∆(0)W(t), 0 ≤ t ≤ t1,

∆(t0) is similar to ∆(0), this is some positive quantity. Here W(t0) is similar to W(0), W(0)=0

I wanted to understand the process for first time interval/first trading date with the help of an example. Below are the details: ∆(t0): Quantity at the beginning say 10 numbers
W(t0) Purchase price is $6 at time t0

We apply the above two possible paths to calculate the gain using the below formula for first trading date/first time interval I(t) = ∆(t0)[ W(t up) - W(t0)] = ∆(0)W(t), 0 ≤ t ≤ t1, For Brownian motion

W(t) Up move will result in the price going up to USD 7

W(t) Down move will result in the price going down to USD5

  • If W(0) taking the value of purchase price
Gain on uptick\ I(t) = 10 ( 7-6) =10 * 1 = 10, This I can understand is a gain and not portfolio value\ Gain on downtick\ I(t) = 10 ( 5-6) =10 * -1 = -10, This I can understand is a negative gain and not portfolio value\

Parking the above understanding aside, since W(t) is mentioned/given as Brownian motion, the starting value has to be equal to zero in order to meet Levy’s condition.

In which case the above two would change to the following:

  • If W(t) is a Brownian motion W(0) taking the initial value (W(0) = 0)
Gain on uptick\ I(t) = 10 ( 7-0) =10 * 7 = 70, This I can understand is portfolio and not gain \

Gain on downtick\ I(t) = 10 ( 5-0) =10 * 5 = 50, This I can understand is portfolio and not gain\

So does the below calculation represent a gain during the time period or is it portfolio value at the end of the time period 1 I(t) = ∆(t0)[ W(t up) - W(t0)] = ∆(0)W(t), 0 ≤ t ≤ t1,

Sorry for these lengthy message/or not presented my challenge in pure mathematical terms.

Kindly clarify/guide.

Thank you
 
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