Stochastic differential of a particular martingale

In summary, the conversation discusses a problem from Oksendal's book involving finding the differential form of a given function involving a Brownian Motion. The person is seeking guidance on how to approach the problem and someone suggests using Ito's formula for the first term and applying the stochastic version of the fundamental law of calculus for the second term.
  • #1
steve1985
1
0
Hello everyone,
I'm studying from Oksendal's book, and I'm stuck at an excercise which asks you to find the differential form of:

X(t) = (W(t)[itex]^{2}[/itex]-t)[itex]^{2}[/itex] - 4[itex]\int[/itex] (W(s))[itex]^{2}[/itex]ds
where W(t) is a Brownian Motion.

I tried several possible functions g(t,W(t)) which could have led to a potential solution (by finding d(g(t,W(t))) with Ito), but none led me any closer to a solution.

Can you please put me in the right direction?

thanks!
Steve

PS. I posted this in the wrong place, it should have been in "Homeworks and coursework questions", but I don't know how to move it...
 
Last edited:
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  • #2
Yes Ito's formula only applies to the first term in the sum. For the second term, if the integral is from 0 to t, just apply the stochastic version of the fundamental law of calculus.
 

What is a stochastic differential of a particular martingale?

A stochastic differential of a particular martingale is a mathematical concept used in probability theory and stochastic processes. It refers to the change in value of a martingale, which is a type of random process that satisfies a certain property.

How is a stochastic differential of a particular martingale calculated?

A stochastic differential of a particular martingale is typically calculated using Ito's Lemma, which is a formula that relates the change in a function of a stochastic process to the changes in the process itself.

What is the significance of a stochastic differential of a particular martingale in finance?

In finance, a stochastic differential of a particular martingale is used to model the dynamics of stock prices and other financial assets. It is an important tool in option pricing and risk management.

How does a stochastic differential of a particular martingale differ from a regular differential?

A stochastic differential takes into account the random fluctuations of a process, while a regular differential only considers the deterministic part of the process. This makes stochastic differentials more appropriate for modeling real-world processes.

Can stochastic differentials be used for non-martingale processes?

Yes, stochastic differentials can be used for non-martingale processes, but they may not always be the most appropriate tool. In such cases, other methods such as stochastic integration may be more suitable.

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