# Tagged Questions

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### Markov property question

In every book I can find, the Markov property for ito diffusions, $E[f(X_{t+h})\mid F_s] = E^{X_t}f(X_h)$ is stated for $\textbf{bounded}$ Borel functions. However, I have the following statement ...
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### Brownian motion-Holder [on hold]

there exists a positive constant $c$, such that almost surely, for h small enough , for all $0< t < 1- h$ \begin{align} |B(t+h)-B(t)| < c\sqrt{h\log(1/h)} \end{align} As a result ...
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Consider a stochastic process $X_t$ with continuous paths. I'd like to apply the mean value theorem inside the expectation, i.e. write something like $$\operatorname{E} \left[ \int_0^t X_s \, ... 1answer 70 views ### Derivation of Black-Scholes equation by riskless portfolio The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the ... 1answer 81 views ### In stochastic calculus, why do we have (dt)^2=0 and other results? I'm doing actuarial problems of Exam MFE and it covers some of the stochastic calculus (like Ito's Lemma). One of the frequently used results are the so-called "multiplication rules": (dt)^2=0 ... 1answer 15 views ### Multi-dimensional Feynman Kac Theorem I am trying to understand how to prove the multi-dimensional version of the Feynman-Kac formula. The single-dimensional version is proved on this page: en.wikipedia.org/wiki/Feynman–Kac_formula ... 0answers 19 views ### Change of variable in stochastic integral Let B be a standard Bronwian motion. Can we do a change of variable in the sense s=\theta+h$$\int_{0}^{t+h}X_sdB_s=\int_{-h}^{t}X_{\theta+h}dY_\theta.$$In this case what is the process ... 0answers 50 views ### Brownian Motion and stochastic integration on the complete real line I'm struggling to understand stochastic integration over intervals containing zero, i.e. integrals of the form \int_{a}^{b} X_s \, d B_s where -\infty \leq a < b \leq \infty, (X_t)_{t \in ... 1answer 38 views ### Invariant mesure of a reflected random walk Let (X_n), n \geq 0 be a Reflected Random Walk defined by: X_0 = 0 and:  X_{n+1}=\max( 0 , X_n + \xi ) \xi  is a random variable such that P(\xi=a)=\theta and P(\xi=-b)=1-\theta for a ... 0answers 53 views ### Ito formula proof for bounded functions using stopping time I'm self studying with the Oksendal book "Stochastic differential equations" and trying to do some exercises by myself. P.57 the exercise asks for the following (a screenshot will save us typing ... 1answer 48 views ### The Lévy-Khintchine formula and integrability conditions of a random measure I am trying to see the connection between the Lévy-Khintchine and the integrability conditions of a Lévy measure. The literature seems to always connect both, but I cannot make sense of this relation ... 0answers 34 views ### Reflected random walk Suppose that X_n is a reflected (in 0) random walk with parameter \theta. So X_{n+1}-X_n = 1 with probability \theta , and -1 with probability 1-\theta when X_n \geq 1, if X_n=0 then ... 1answer 47 views ### Stochastic integral wrt the compensated Poisson random measure I am solving the exercises in a book I have about Lévy processes ("Lévy Processes and Stochastic Calculus", Applebaum, 2003), and I cannot get my head around an exercise that seems rather simple. I ... 1answer 22 views ### Integration with respect to two different Brownian motions Let B be the standard Brownian motion. The process W_s=B_{s+a}-B_a is also a Brownian motion. I just want an example of a process X_s such that$$E\int_0^tX_sdB_s\neq E\int_0^tX_sdW_s.$$0answers 10 views ### Time homogeneous asset dynamics model I'm studying asset process. As i know, Black scholes model and CEV model is time homogeneous diffusion model. Are there time homogeneous model ??? 0answers 24 views ### Differential of stochastic process How do I find the dynamics of X_t=\int_0 ^t \sigma (s,t) dW_s? It seems that the simple solution of dX_t = \sigma(t,t)dW_t is not correct since I get X_t = \int _0 ^t \sigma(s,s) dW_s if I ... 1answer 24 views ### How to find the dynamics of stochastic process? We have Y_t=e^{\int_0^t W_sds}. How do I obtain the dynamics of Y_t (i.e. dY_t)? It seems that we can't use Ito Lemma because \int_0^t W_sds is not in the form X_t = \int_0 ^t \sigma_s dW_s ... 1answer 31 views ### Dynamics of short rate in HJM According to a simplified HJM framework, we have: Forward Rate: f(t,T)=\sigma W_t +f(0,T) +\int_0^t{\alpha(s,T)}ds, where W_t is brownian motion. Dynamics of forward rate: ... 1answer 22 views ### Comparison between these Ito Lemma versions According to wikipedia : I found another version : Please explain the difference for me. 1answer 37 views ### upper bound for Ito integral of deterministic integrand It is well known that Ito integrals with respect to a Brownian motion cannot be defined pathwise because the Brownian motion has infinite 1st order variation. These integrals are defined as limits of ... 0answers 31 views ### Matlab code for higher order scheme Can somebody help me how to generate the code for the increment \DeltaZ in the document I have attached? I know how to generate the rest of the increments but struggling in how to generate ... 2answers 35 views ### Resolvent operators and inverses proof I am trying to prove for myself that A(R_{\alpha}g)=\alpha R_{\alpha}g-g which is proving problematic. The definition of A, the generator, is \displaystyle Af(x)= \lim_{t \rightarrow 0} ... 1answer 36 views ### Question on generators in the proof of Kolmogorov's Backward Equation Here is a part of the proof of the Kolmogorov's Backward Equation. I cannot see why Y_t has been picked as it has. In particular, I cannot see why you would want to subtract t in the first bit of ... 0answers 18 views ### Milestein Scheme Im struggling in the following schemes. I cant understand how the first scheme is equivalent to the second one. Can somebody help me? Thanks in advance. Moreover there is a typo error in the ... 1answer 42 views ### More on the Existence and Uniqueness of the solutions of an SDE Proof An extract from the proof of the existence and uniqueness of the solution of a SDE from Oksendal. I cannot see how holders inequality and the ito isometry are applied. 1answer 22 views ### Part of Proof of the Uniqueness of the Solution of SDE's This is an extract from Oksendal's SDE of the proof of the uniqueness of the solution of a SDE. I cannot see how the P[|X_t-\hat{X_t}|=0 \ \ \ \text{for all t} \in \mathbb{Q} \cap [0,T]]=1 is ... 1answer 20 views ### Finding the unique Martingale Convergence Representation of a given r.v. According to the martingale representation there exists a unique g(t,\omega) \in \mathcal{V}(0,T) such that M_t = E[M_0]+\int^{t}_{0} g(s,\omega) dB(s); \ \ \ t \in [0,T] Find g in the case ... 1answer 38 views ### More preliminaries of the Martingale Convergence Theorem Really struggling with this lemma. Not sure about the general structure of the proof. Why have we chosen g to be orthogonal to all functions of the form 4.3.1? Why should G(\lambda)=0, does it ... 1answer 59 views ### Ito formula applied to \frac{1}{t}\int_0^t W_s ds  I got this expression and I have to calculate its differential by the Ito formula, W_t denotes the Brownian motion:$$\frac{1}{t}\int_0^t W_s ds $$I calculate the derivative of ... 1answer 40 views ### Evaluating Stratonovich integral from definition \bf 3.9. Suppose f\in\mathcal V(0,T) and that t\to f(t,\omega) is continuous for a.a. \omega. Then we have shown that$$\int\limits_0^T f(t,\omega)dB_t(\omega)=\lim_{\Delta ...
This is part of the derivation of the Ito integral. In particular extending the definition to more general functions. I cannot understand why $g(.,\omega)$ is continuous for each $\omega$. $\psi$ ...