Conditional Expectations (Stochastic Calculus)

In summary, conditional expectation in stochastic calculus is a way to incorporate information about one random variable into the expected value of another random variable. It is calculated by finding the joint probability distribution of the two variables and using a specific formula. It is important in predicting and analyzing the behavior of random variables and has various real-world applications in fields such as finance, economics, and engineering. It can also be negative and is used in option pricing models to estimate future asset values.
  • #1
irony of truth
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Homework Statement



Let (X_n; for all counting number n) be a sequence of independent random
variables. We focus on the random walk S_n := X_1 + . . . + X_n and set
F_n = 'sigma-algebra' of (S_1, . . . , S_n).

1. Compute E[S_(n+1) \ F_n]
2. For any z belonging to the complex plane C, show that:

E[ z^S_(n+1) \ F_n] = z^S_n E(z^X_(n+1))


Homework Equations



-N.A.- (Hoping for your consideration)

The Attempt at a Solution



Here's what I've done:

1. E[S_(n+1) \ F_n] = E[S_n + X_(n+1) \ F_n]

= E[S_n \ F_n] + E[X_(n+1) \ F_n]

= S_n + E[X_(n+1) \ F_n] <- S_n is F_n - measurable

The last line is my final answer. Should I leave it as is? Is there
something I can do with the second term?

2. Note that X_(n+1) is independent of X_1, X_2, ..., X_n. Also,
S_(n+1) = Sn + X_(n+1). Thus,

z^S_(n+1) = z^{Sn + X_(n+1)}

= (z^Sn)(z^X_(n+1))

Hence, E[ z^S_(n+1) \ F_n] = E[(z^Sn)(z^X_(n+1)) \ F_n]
Because S_n is F_n - measurable,

E[ z^S_(n+1) \ F_n] = (z^S_n)E[z^X_(n+1)\ F_n]

= z^S_n E(z^X_(n+1))

...because X_(n+1) is independent of X_1, X_2, ..., X_n.

Please inform me if there are some mistakes in my solution. Hoping for your
consideration.
 
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  • #2
Thank you.



Thank you for your response and for sharing your solution. It looks like you have correctly computed the conditional expectation in part 1 and have used the independence of X_(n+1) in part 2 to show that E[ z^S_(n+1) \ F_n] = z^S_n E(z^X_(n+1)). This is a correct and concise solution.

As for the second term in part 1, you could potentially simplify it further by using the fact that E[X_(n+1) \ F_n] = E[X_(n+1)] since X_(n+1) is independent of F_n. This would result in a final answer of E[S_(n+1) \ F_n] = S_n + E[X_(n+1)]. However, leaving it as E[X_(n+1) \ F_n] is also a valid and correct answer.

Overall, your solution is well-written and clear. Keep up the good work in your studies!
 

1. What is conditional expectation in stochastic calculus?

Conditional expectation in stochastic calculus is a mathematical concept that represents the expected value of a random variable given the knowledge of another random variable. It is a way to incorporate information about one variable into the expected value of another variable.

2. How is conditional expectation calculated?

Conditional expectation is calculated by first finding the joint probability distribution of the two random variables and then applying the formula for conditional expectation, which involves integrating the product of the joint probability distribution and the conditional probability density function.

3. What is the importance of conditional expectation in stochastic calculus?

Conditional expectation is important in stochastic calculus because it allows us to make predictions and analyze the behavior of random variables based on the knowledge of other related variables. It is also a fundamental concept in the study of stochastic processes and their applications.

4. Can conditional expectation be negative?

Yes, conditional expectation can be negative. It represents the expected value of a random variable given the knowledge of another random variable, so it can take on any real value depending on the specific variables and their relationship.

5. How is conditional expectation used in real-world applications?

Conditional expectation has various applications in fields such as finance, economics, and engineering. It is used to model and analyze stochastic processes, make predictions, and estimate risk in various scenarios. For example, it can be used in option pricing models in finance to estimate the future value of an asset based on current market conditions.

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