# Tagged Questions

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### Write down the HJB equation

Suppose that we have to solve the following optimal control problem \begin{align} V(t,x) = \min_{\alpha}\mathbb{E} \left[\int_{0}^{T}L(t,x,\alpha)dt + F(e^{-\beta t}X^{\alpha}_{T})\right] ...
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### Angle bracket and sharp bracket for discontinuous processes

The question is quite simple actually. I am trying to understand the differences between the angle bracket $\left<X,Y\right>$ of two processes with jumps $X,Y$, and the sharp bracket of $[X,Y]$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$E\int_0^tX_sdB_s\neq E\int_0^tX_sdW_s.$$
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### 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 ???
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
$\bf 3.9.$ 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 $$\int\limits_0^T f(t,\omega)dB_t(\omega)=\lim_{\Delta ... 1answer 30 views ### Continuity problem in derivation of general ito integral This is part of the derivation of the Ito integral. In particular extending the definition to more general functions. I cannot understand why g(.,\omega) is continuous for each \omega. \psi ... 1answer 20 views ### Extensions of the Ito integral This is an extract from Oksendal's Stochastic Differential Equations (end of chapter 3). I cannot understand why we have taken the intersection, surely the union would have been more appropriate? 1answer 19 views ### A Property of the Ito Integral Let f,g \in \mathcal{V}(0,T) and let 0 \leq S < T. Then E[\int^{T}_{S}f dB_t]=0 Apparently this holds clearly for elementary functions, (Im not so sure), and can be obtained by taking ... 1answer 20 views ### Continuity theorem in Itô integral explanation What is the continuity theorem used here in the explanation of the Itô integral? I cannot seem to find anything that would be exactly useful in my measure and integration text. 1answer 17 views ### Expectations of certain Brownian motion equations B_t is Brownian motion. It is assumed that motion starts at 0. I do not understand how the highlighted equalities hold true. Is the first one equivalent to ... 0answers 52 views ### First hitting time Geometric Brownian motion I have the following problem: My Process underlies the SDE  d W_t = \mu W_t dt + \sigma W_t d B_t  with B_t being a standard Brownian motion, \mu,\sigma >0, i.e. W_t = S_0 \exp\Big( ... 0answers 25 views ### generator of a function (stochastic) [closed] How do I find a generator of$$g(Y_t)=Y_t^2-10Y_t+25 \, ,$$where Y_t is a geometric BM:$$dY_t=-1Y_tdt+2Y_tdW_t \, ,$$and W_t is white noise 0answers 27 views ### Matlab code for Simulation of SDE [duplicate] I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated.. Q: "Simulate ... 1answer 47 views ### Solve Itô integral with power$$\int_0^t e^{Ws} W_s^r dW_s$$where W_s is Wiener process and r> in \mathbb{Z} My first approach would be to use Ito's lemma, however, coming up with the function g(t,x) is difficult The ... 0answers 44 views ### Stochastic Differential equations with \sin(x^2) as drift. Can somebody help me how to solve the following SDE analytically or suggest me to go through some literature to understand this or can give me a little bit hint to work by myself. Thanks in advance. ... 0answers 35 views ### Brownian motion starts fresh variant It is a standard result that if W_t is a Brownian Motion and S is a stopping time of the standard filtration F_t then we have that B_t = W_{S+t} - W_S is a Brownian Motion. I quote the ... 0answers 49 views ### What is a.e. a.s I am reading a paper which uses almost everywhere almost surely (a.e.,a.s.) simultaneously, I am not quite sure what it means then. To be specific, they consider a stochastic process \{X_t\} such ... 0answers 69 views ### Multipe Ito Integrals Im working on a Lemma 10.8 in the Book "Numerical Solution of Stochastic Differential Equations by Kloeden And Platen" I have been stuck on one point. Can somebody help me to understand how he moved ... 1answer 35 views ### Strong solution of stochastic differential equation Consider the stochastic differenctial equation: dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t. How to find a strong solution? 0answers 36 views ### Second (centered) moment for martingales Take the process {x}_t following geometric Brownian motion (GBM)$$x_t=\mu x_t \,dt+\sigma x_t \,dW_t$$with x_0>0 known. It has first moment equal to$$\text{E}[x_t]=x_0 e^{\mu t}$$and second ... 0answers 48 views ### Poisson/ jump process distribution for process z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k For the process: z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k, where X(t) is a poisson process with paramater \lambda, and: J_k are i.i.d . random variables (jumps). B(t)=brownian motion. I want to ... 0answers 34 views ### Girsanov theorem conditions If we have an adapted function f(t) such that \int_0^t f(s)ds\,<\infty, then the Girsanov exponent can be defined:$$ Z(t):=\exp\left( \int_0^t f(s)dW(s) - \frac{1}{2} \int_0^t ...
Let ${W_t}$ be 1 dim Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. My try is below. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why ...