Solution of Fuzzy Stochastic Differential Equations

Let us consider an SDE with fuzzy parameters, then Equation 11.1 may be written as

Equation 11.7 is solved here by exact as well as numerical methods.

Using the limit method (Chakraverty and Nayak 2013), the FSDE (11.7) may be represented in modified limit form as

where

and

Initially, for the exact (or crisp) case, we take the crisp representation of X (a), a (a), b (a) and use the Ito integral to solve the problem.

Now, if we apply the fuzzy concept for the EMM, then we get

where

By applying lim and lim on the solution, we get the left and right bounds, whereas we obtain various®solutior®s ets by considering different values of a e [0, 1]. It is noticed that sometimes we get weak solutions (the left- and right-bound solutions overlap or intersect each other), and this occurs due to the randomness of the system. This may easily be observed from the following example problems.

Example Problems

In this section, we have considered two example problems and taken the parameters as fuzzy. Initially, the problem is studied for crisp parameters for both the exact and numerical methods and then the fuzzy parameters are incorporated.

Example 11.2

Consider the Black-Scholes SDE (Black and Scholes 1973).

The crisp EM approximation for this SDE is as follows:

where the values of the involved parameters are given in Table 11.1.

TABLE 11.1

Crisp and Fuzzy Values of the Involved Parameters

Parameters

Crisp

TFN

M

a

0.75

0.30

[0.65, 0.75, 0.85] [0.25, 0.30, 0.35]

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Initially, the Black-Scholes SDE has been solved for the crisp parameter and then the fuzzy parameters are considered for investigation. Here, we compute a discretized Brownian path over [0, 1] with 5t = 2^{-8} and the obtained solution is plotted with a solid light gray line in Figure 11.1. We then apply the EMM using a step size At = R5t, where R is a constant (with R = 4) and the solution is presented in Figure 11.1 with a dark gray line.

Now, drift (p) and diffusion (o) coefficients are taken as TFN, which are given in Table 11.1. The exact method (Ito process) is used to obtain the solution that is depicted in Figure 11.2. Here, the black and light gray solid lines represent the left and right bounds of the uncertainties. Next, the left and right values of the uncertainties are plotted with the exact solutions in Figures 11.3 and 11.4, respectively. Then the EMM is used to solve the uncertain SDE and the results are graphically depicted in Figure 11.5, where the black and light gray lines represent the left and right bounds of the uncertain solutions. The region covered in between the left and right bounds is the uncertain solution of the Black-Scholes SDE. In Figure 11.6, we have given the left and right bounds, which represent the uncertain solution of the Black-Scholes SDE along with the crisp solution, and we found that the exact solution lies within the region covered by the left and right solutions. Furthermore, the EMM is used for the next example (Figure 11.7).

Example 11.3

The SDE of the Langevin equation is
where p and o are positive constants.

FIGURE 11.1

Solution of the Black-Scholes SDE when parameters are crisp.

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FIGURE 11.2

Ito solution of the Black-Scholes SDE when parameters are fuzzy.

FIGURE 11.3

Crisp EM solution of the Black-Scholes SDE and the left uncertain bound solutions.

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FIGURE 11.4

Crisp EM solution of the Black-Scholes SDE and the right uncertain bound solutions.

FIGURE 11.5

EM solution of the Black-Scholes SDE when parameters are fuzzy.

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FIGURE 11.6

EM solution of the Black-Scholes SDE when parameters are fuzzy with the Ito solution.

FIGURE 11.7

EM solution of the Langevin SDE when parameters are fuzzy.

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The EM approximation for Equation 11.11 is

The same parameter values that are used in Equation 11.8 are considered here and those are given in Table 11.2.

In Figure 11.8, we have given a plot for the solution of the Langevin SDE when parameters are crisp, whereas the solution for the Langevin SDE is presented in Figure 11.9, where the parameters are taken as fuzzy. The left- and right-bound solutions are shown in blue and magenta colours, respectively.

For better visualization of uncertain distribution of EM approximation results, fuzzy plots are represented in Figures 11.9 and 11.10 for Examples 11.2 and 11.3, respectively.

One may also see that the uncertain widths are randomly varying. It may be noted that if the uncertainty of the parameter changes, the uncertain width of the solution sets varies accordingly.

TABLE 11.2

Crisp and Fuzzy Values of the Used Parameters (for Section 11.1.4)

Parameters

Crisp

TFN

P

10

[8, 10, 12]

a

1

[0.5, 1, 1.5]

FIGURE 11.8

EM solution of the Langevin SDE when parameters are fuzzy with the crisp solution.

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FIGURE 11.9

Fuzzy plot of EM solution of the Black-Scholes SDE when parameters are TFN (Example 11.2).

FIGURE 11.10

Fuzzy plot of EM solution of the Langevin SDE when parameters are TFN (Example 11.3).