Tag: financial-mathematics

Studies in Chaos and Finance: Chapter III

Pasha ZavariLeave a comment

The nonlinear properties of f(W(\boldsymbol{z})) described in Chapter II, which can be thought of a type of memory function, arise from its intrinsic dependence on past values of W_t. This means that for some integer-time stochastic process \{Z_n; n \geqslant 1\} our model satisfies one of two relations: a Submartingale defined by

{\displaystyle \begin{aligned} \mathsf{E} [|Z_n|]<\infty; \quad \mathsf{E}[Z_n|Z_{n-1},Z_{n-2},...,Z_1] \geq Z_{n-1} ;\quad n \geq 1 \nonumber \end{aligned} }     (29)

or a Supermartinagle defined by

{\displaystyle \begin{aligned} \mathsf{E}[|Z_n|] <\infty; \quad \mathsf{E}[Z_n|Z_{n-1},Z_{n-1},...,Z_1]\leq Z_{n-1};\quad n \geq 1 \nonumber \end{aligned} }     (30)

The focus of this chapter will be to explore the fundamental stochastic characteristics of (28) using the above defintions. To begin, it will help to re-write the integral using summation nation:

{\displaystyle \int_0^t \boldsymbol{W}_s^T (e^{\textbf{A} s} \circ \Sigma^{-1}) \boldsymbol{W}_s \, ds \, \equiv \sum_ {i = 1}^{n} \sum_ {j = 1}^{n} \Sigma^{-1}_{ij} \int_0^t e^{\alpha_{ij} s} W_s^{\{i\}} W_s^{\{j\}} \, ds }     (31)

Note that when i = j we have W_s^{\{j\}, 2}.

In order to address these non-deterministic integrals in (31) we will need to apply Itô’s Lemma expressed as

{\displaystyle f(t,W_t) - f(0,0) = \int_0^t \frac{\partial f}{\partial t}(s,W_s) \, ds + \int_0^t \frac{\partial f}{\partial x}(s,W_s) \, dW_s + \frac{1}{2}\int_0^t \frac{\partial^2 f}{\partial x^2}(s,W_s) \, ds }     (32)

For i = j we have

{\displaystyle \begin{aligned} f(t, W_t^{i}) - f(0, 0) & = \frac {1}{\alpha_{ii}} [W_s^{\{i\}, 2} e^{\alpha_{ii} s}]_{0}^{t} \\ \int_0^t \frac{\partial f}{\partial t}(s,W_s^{\{i\}}) \, ds & = \int_0^t W_s^{\{i\}, 2} e^{\alpha_{ii} s} \, ds \\ \int_0^t \frac{\partial f}{\partial x}(s,W_s^{\{i\}}) \, dW_s^{\{i\}} & = \frac {2}{\alpha_{ii}} \int_0^t W_s^{\{i\}} \, e^{\alpha_{ii} s} \, dW_s^{\{i\}} \\ \frac{1}{2}\int_0^t \frac{\partial^2 f}{\partial x^2}(s,W_s^{\{i\}}) \, ds & = \frac {1}{\alpha_{ii}} \int_0^t e^{\alpha_{ii} s} \, d_s \end{aligned} }     (33)

which yields

{\displaystyle \begin{aligned} \int_0^t W_s^{\{i\}, 2} e^{\alpha_{ii} s} \, ds & = W_t^{\{i\}, 2} \frac {e^{\alpha_{ii} t}}{\alpha_{ii}} - \frac {2}{\alpha_{ii}} \int_0^t W_s^{\{i\}} e^{\alpha_{ii} s} \, dW_s^{\{i\}} - \frac {1}{\alpha_{ii}} \int_0^t e^{\alpha_{ii} s} \, d_s \\ & = W_t^{\{i\}, 2} \frac {e^{\alpha_{ii} t}}{\alpha_{ii}} - \frac {2}{\alpha} \int_0^t W_s^{\{i\}} e^{\alpha_{ii} s} \, dW_s^{\{i\}} + \frac {1 - e^{\alpha_{ii} t}}{\alpha_{ii}^2} \end{aligned} }     (34)

The case i \neq j will require a little more mathematical machinery. The first step is to re-write the integral using Itô’s Differential:

{\displaystyle d(xy)_{t} = x_{t}dy_{t} + y_{t}dx_{t} + dx_{t}dy_{t}, \quad dx_{t}dy_{t} = d[x,y]_{t} }     (35)

The quadratic covariation denoted by [x,y]_{t} underpins a critical difference between classical calculus and stochastic calculus. This topic is quite involved and, for the sake of brevity, will be skipped in this chapter. I plan to return to it in the future with a comprehensive overview. Continuing on, we know that using Itô’s multiplication table we can deduce that dx_{t}dy_{t} = \delta_{ij} s, where \delta_{ij} is the Kronecker delta and where \delta_{ij} = 1 when i = j. The expression can then be re-written as

{\displaystyle d(W_s^{\{i\}} W_s^{\{j\}}) = W_s^{\{i\}}\, dW_s^{\{j\}} + W_s^{\{j\}}\, dW_s^{\{i\}} + \delta_{ij}\, ds }     (36)

Applying Itô’s product rule to:

{\displaystyle Y_s = e^{\alpha_{ij} s} W_s^{\{i\}} W_s^{\{j\}} }     (37)

we can derive:

{\displaystyle \begin{aligned} dY_s & = \alpha_{ij} e^{\alpha_{ij} s} W_s^{\{i\}} W_s^{\{j\}}\, ds + e^{\alpha_{ij} s}\, d(W_s^{\{i\}} W_s^{\{j\}}) \\ & = \alpha_{ij} e^{\alpha_{ij} s} W_s^{\{i\}} W_s^{\{j\}}\, ds + e^{\alpha_{ij} s} \left[ W_s^{\{i\}}\, dW_s^{\{j\}} + W_s^{\{j\}}\, dW_s^{\{i\}} + \delta_{ij}\, ds \right] \end{aligned} }     (38)

Integrating both sides in (36):

{\displaystyle Y_t = Y_0 + \int_0^t \alpha_{ij} e^{\alpha_{ij} s} W_s^{\{i\}} W_s^{\{j\}}\, ds + \int_0^t e^{\alpha_{ij} s} \left[ W_s^{\{i\}}\, dW_s^{\{j\}} + W_s^{\{j\}}\, dW_s^{\{i\}} \right] + \delta_{ij} \int_0^t e^{\alpha_{ij} s}\, ds }     (39)

Since {W_0^{\{i\}} = W_0^{\{j\}} = 0 }, we have {Y_0 = 0 }, so:

{\displaystyle e^{\alpha_{ij} t} W_t^{\{i\}} W_t^{\{j\}} = \alpha_{ij} \int_0^t e^{\alpha_{ij} s} W_s^{\{i\}} W_s^{\{j\}}\, ds + \int_0^t e^{\alpha_{ij} s} \left[ W_s^{\{i\}}\, dW_s^{\{j\}} + W_s^{\{j\}}\, dW_s^{\{i\}} \right] + \delta_{ij} \int_0^t e^{\alpha_{ij} s}\, ds }     (40)

Rearranging (38) and solving for I_t = \int_0^t e^{\alpha_{ij} s} W_s^{\{i\}} W_s^{\{j\}}\, ds:

{\displaystyle \alpha I_t = e^{\alpha_{ij} t} W_t^{\{i\}} W_t^{\{j\}} - \int_0^t e^{\alpha_{ij} s} \left[ W_s^{\{i\}}\, dW_s^{\{j\}} + W_s^{\{j\}}\, dW_s^{\{i\}} \right] - \delta_{ij} \int_0^t e^{\alpha_{ij} s}\, ds }     (41)

Provided that \alpha_{ij} \neq 0, we can isolate I_t:

{\displaystyle I_t = \frac{1}{\alpha_{ij}} e^{\alpha_{ij} t} W_t^{\{i\}} W_t^{\{j\}} - \frac{1}{\alpha_{ij}} \int_0^t e^{\alpha_{ij}s} \left[ W_s^{\{i\}}\, dW_s^{\{j\}} + W_s^{\{j\}}\, dW_s^{\{i\}} \right] - \frac{\delta_{ij}}{\alpha_{ij}} \int_0^t e^{\alpha_{ij} s}\, ds }     (42)

Given that the closed form solution to \int_0^t e^{\alpha_{ij} s}\, ds = \frac{e^{\alpha_{ij} t} - 1}{\alpha_{ij}}, we can also write:

{\displaystyle \frac{\delta_{ij}}{\alpha_{ij}} \int_0^t e^{\alpha_{ij} s}\, ds = \delta_{ij} \frac{e^{\alpha_{ij} t} - 1}{\alpha_{ij}^2} }     (43)

Finally we have:

{\displaystyle \int_0^t e^{\alpha_{ij} s} W_s^{\{i\}} W_s^{\{j\}}\, ds = \frac{1}{\alpha_{ij}} e^{\alpha_{ij} t} W_t^{\{i\}} W_t^{\{j\}} - \frac{1}{\alpha_{ij}} \int_0^t e^{\alpha_{ij} s} \left[ W_s^{\{i\}}\, dW_s^{\{j\}} + W_s^{\{j\}}\, dW_s^{\{i\}} \right] - \delta_{ij} \frac{e^{\alpha_{ij} t} - 1}{\alpha_{ij}^2} }     (44)