Use Ito's Lemma to solve the stochastic differential equatio

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In summary, the conversation discusses using Ito's Lemma to solve a stochastic differential equation and finding the expected value of the solution. The equation is given in standard form and its solution is described using the function $\varphi_{t}$. The solution involves integrals and the final result is expressed as (3).
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I've been at this for ages but I can't make sense of it. Can somebody help me out?Use Ito's Lemma to solve the stochastic differential equation:

[tex]X_t=2+\int_{0}^{t}(15-9X_s)ds+7\int_{0}^{t}dB_s[/tex]

and find:

[tex]E(X_t)[/tex]
 
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  • #2
Jason said:
I've been at this for ages but I can't make sense of it. Can somebody help me out?Use Ito's Lemma to solve the stochastic differential equation:

[tex]X_t=2+\int_{0}^{t}(15-9X_s)ds+7\int_{0}^{t}dB_s[/tex]

and find:

[tex]E(X_t)[/tex]

In 'standard form' the SDE is written as...

$\displaystyle d X_{t}= (15-9\ X_{t})\ dt + 7\ dW_{t}\ ,\ x_{0}=2$ (1)

The (1) is a linear in narrow sense SDE andits solving procedure has been described in...

http://www.mathhelpboards.com/f23/unsolved-statistic-questions-other-sites-part-ii-1566/index2.html#post8411

... and its solution is...

$\displaystyle X_{t}= \varphi_{t}\ \{ x_{0} +\int_{0}^{t} \varphi_{s}^{-1}\ u_{s}\ ds + \int_{0}^{t} \varphi_{s}^{-1}\ v_{s}\ dW_{s} \}$ (2)


Here is $a_{t}=-9$ , so that is $\varphi_{t}=e^{-9 t}$,$u_{t}=15$, $v_{t}=7$ and $x_{0}=2$ so that (2) becomes...

$\displaystyle X_{t}= e^{- 9 t}\ \{ 2 + 15\ \int_{0}^{t} e^{9 s} ds + 7\ \int_{0}^{t} e^{9 s}\ dW_{s} \}= e^{-9 t}\ \{2 + \frac{5}{3}\ (e^{9 t}-1) + \frac{7}{9}\ (e^{9 W_{t}}-1) - \frac{7}{2}\ (e^{9t}-1) \} = $

$\displaystyle = - \frac{11}{6} + \frac {55}{18}\ e^{-9 t} + \frac{7}{9}\ e^{9\ (W_{t}-t)}$ (3)

Kind regards

$\chi$ $\sigma$
 

FAQ: Use Ito's Lemma to solve the stochastic differential equatio

What is Ito's Lemma?

Ito's Lemma is a mathematical formula used in stochastic calculus to describe the evolution of a stochastic process. It is named after the Japanese mathematician Kiyoshi Ito.

What is the significance of Ito's Lemma in mathematics?

Ito's Lemma is a fundamental tool in the study of stochastic processes and has applications in fields such as finance, physics, and engineering. It allows for the analysis of complex systems with random variables and helps in the development of mathematical models.

What are the assumptions made in Ito's Lemma?

Ito's Lemma assumes that the random variables in the stochastic process are normally distributed and have finite variances. It also assumes that the process is continuous and differentiable.

How is Ito's Lemma used in finance?

In finance, Ito's Lemma is used to model the changes in the value of financial assets over time. It is an essential tool in the development of financial models, such as the Black-Scholes model for option pricing.

Are there any limitations to Ito's Lemma?

Ito's Lemma is limited in its applicability to certain types of stochastic processes, such as those with jumps or discontinuities. It also assumes that the process is continuously differentiable, which may not always be the case in real-world applications.

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