Modelling Real World Using Stochastic Processes and Filtration

First we give an implementation in Mizar [2] basic important definitions of stochastic finance, i.e. filtration ([9], pp. 183 and 185), adapted stochastic process ([9], p. 185) and predictable stochastic process ([6], p. 224). Second we give some concrete formalization and verification to real world examples.

In article [8] we started to define random variables for a similar presentation to the book [6]. Here we continue this study. Next we define the stochastic process. For further definitions based on stochastic process we implement the definition of filtration.

To get a better understanding we give a real world example and connect the statements to the theorems. Other similar examples are given in [10], pp. 143-159 and in [12], pp. 110-124. First we introduce sets which give informations referring to today (Ω_now, Def.6), tomorrow (Ω_fut1 , Def.7) and the day after tomorrow (Ω_fut2 , Def.8). We give an overview for some events in the σ-algebras Ω_now, Ω_fut1 and Ω_fut2, see theorems (22) and (23).

The given events are necessary for creating our next functions. The implementations take the form of: Ω_now ⊂ Ω_fut1 ⊂ Ω_fut2 see theorem (24). This tells us growing informations from now to the future 1=now, 2=tomorrow, 3=the day after tomorrow.

We install functions f : {1, 2, 3, 4} → ℝ as following:

f₁ : x → 100, ∀x ∈ dom f, see theorem (36),

f₂ : x → 80, for x = 1 or x = 2 and

f₂ : x → 120, for x = 3 or x = 4, see theorem (37),

f₃ : x → 60, for x = 1, f₃ : x → 80, for x = 2 and

f₃ : x → 100, for x = 3, f₃ : x → 120, for x = 4 see theorem (38).

These functions are real random variable: f₁ over Ω_now, f₂ over Ω_fut1, f₃ over Ω_fut2, see theorems (46), (43) and (40). We can prove that these functions can be used for giving an example for an adapted stochastic process. See theorem (49).

We want to give an interpretation to these functions: suppose you have an equity A which has now (= w₁) the value 100. Tomorrow A changes depending which scenario occurs − e.g. another marketing strategy. In scenario 1 (= w₁₁) it has the value 80, in scenario 2 (= w₁₂) it has the value 120. The day after tomorrow A changes again. In scenario 1 (= w₁₁₁) it has the value 60, in scenario 2 (= w₁₁₂) the value 80, in scenario 3 (= w₁₂₁) the value 100 and in scenario 4 (= w₁₂₂) it has the value 120. For a visualization refer to the tree:

The sets w₁,w₁₁,w₁₂,w₁₁₁,w₁₁₂,w₁₂₁,w₁₂₂ which are subsets of {1, 2, 3, 4}, see (22), tell us which market scenario occurs. The functions tell us the values to the relevant market scenario:

For a better understanding of the definition of the random variable and the relation to the functions refer to [7], p. 20. For the proof of certain sets as σ-fields refer to [7], pp. 10-11 and [9], pp. 1-2.

This article is the next step to the arbitrage opportunity. If you use for example a simple probability measure, refer, for example to literature [3], pp. 28-34, [6], p. 6 and p. 232 you can calculate whether an arbitrage exists or not. Note, that the example given in literature [3] needs 8 instead of 4 informations as in our model. If we want to code the first 3 given time points into our model we would have the following graph, see theorems (47), (44) and (41):

The function for the “Call-Option” is given in literature [3], p. 28. The function is realized in Def.5. As a background, more examples for using the definition of filtration are given in [9], pp. 185-188.