# Tagged Questions

Questions on the calculus of stochastic processes, or processes that have a random component.

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### Integral with respect to Brownian motion, Variance

Good day. Imagene we have a martingale $M(t)=\int_0^t f(s)dB(s)$ which satisfies Dambis-Dubins-Schwarz Theorem. At the same time $M(t)^2 - <M>(t)$ is a Martingale starting in $0$ as well. If i ...
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### What is a good book for learning Stochastic Calculus?

I am in search of a good book for learning Stochastic Calculus from a purely mathematical/statistical point of view. Almost all the books I see are based on Finance. Also, please specify the ...
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### How can I solve this stochastic system of equation?

$(B_1(t),B_2(t))$ is a 2-dimensional standard Brownian motion. $\alpha , \beta$ are constant. The system of equations is: $$dX_1(t)=X_2(t)dt+\alpha dB_1(t)\\dX_2(t)=-X_1(t)dt+\alpha dB_2(t)$$ I tried ...
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I have read a paper about numerical SDE. After deriving the method, it uses the method to calculate the following nonlinear cases: $$\begin{cases} dX_t=ud\tau+\sigma ... 0answers 11 views ### ODE where the derivative is a function of a stochastic process Suppose we have a linear ODE, \frac{dv}{dt} = 1 + xv , where the coefficient x is an Ornstein-Uhlenbeck process, dx = (x_0 - x)dt + \sigma d\omega. Is there a way to express this ODE as an ... 0answers 16 views ### Finding a function to use for Ito's Lemma The original problem was to show the following stochastic process has a global solution:$$dx_i = x_i\left(b_i-\sum_{j=1}^4 a_{ij}x_j \right)dt+\sigma_ix_idW_t$$To do so, they considered the ... 0answers 18 views ### Compute the Gibbs energy I have a question about Gibbs distribution in Stochastic theory. In which, it defined a clique as a a subset C in the whole image \Omega if two different element of C are neighbors. Figure 2 ... 1answer 49 views ### How to find exact solution of this volterra equation? I was working on numerical solution of this equation (by block pulse).$$x(t)=1+\int_{0}^{t}s^2x(s)ds+\int_{0}^{t}sx(s)dB(s)\\t \in[0,\frac{1}{2}]$$B(t) is standard brownian motion. Author of the ... 0answers 31 views ### What does it means of Normalization term of Gibbs distribution? I am studying about Gibbs distribution concept and I am confusing about the term" normalization ". According to the Hammersley–Clifford theorem, an random x can equivalently be characterized by a ... 0answers 36 views ### Stochastic exponentials Let X be a good integrator with X_0=0, then the process \begin{equation*} Z_t=\exp(X_t-\frac{1}{2}[X,X]_t)\prod_{0\leq s \leq t}(1+\Delta X_s) \exp(-\Delta X_s + \frac{1}{2}(\Delta X_s)^2) ... 2answers 49 views ### Estimate mean and variance for a truncated sample set Assume there is a normally distributed random variable X \tilde{} N(\mu, \sigma) I want to estimate \mu and \sigma. So far the standard setting. Assume I am given a sample (X_i)_{i=1}^N of ... 0answers 26 views ### Show Y(t)=X^{(1)}(t)-X^{(2)}(t) and \lim_{t\to\infty} \mathbb{E}Y^2(t)=0 , for dX^{(i)}=\mu X^{(i)}dt+\sigma X^{(i)}dW I am trying to solve this exercise which my professor has "solved" (he says what the result but not how he gets it). This is in a problem sheet which is about the Euler-Maruyama scheme. What I get ... 1answer 28 views ### interchanging spatial integral and time integral in the Brownian context The problem is the following My attempt is inspired in the following: Consider$$F_n(x) = \int_{-\infty}^\infty h(a) u_n(x - a)\,da By Itô's formula: \begin{align} &F_n(W_t) = F_n(W_0) + ... 1answer 28 views ### Doob Meyer decomposition for Super-martingales Let Z be a super-martingale with usual Doob-Meyer decomposition: Z=M-A. Is it true that : A\leq M and therefore: \mathbb{E}[A^2]\leq \mathbb{E}[M^2] ? 0answers 62 views ### Ito's Lemma / Expected Value / Variance - Mathematical Finance Assume an asset price S_t follows the geometric Brownian motion\Bbb dS_t = \mu S_t\Bbb dt + \sigma S_t\Bbb dWt,$$where \mu and \sigma are constants and r is the risk-free rate. ... 1answer 73 views ### Calculation with Ito processes, what is ds \, dt, dW_t \, ds and dW_s \, dW_t? I am working on an exercise and I am not sure how to deal with these 3 cases... For example, is ds \, dt=0? I know (dt)^2=0, but I am not sure when it is 2 different variables. And what about ... 2answers 71 views ### Gradient descent method with random perturbation Suppose there is a function f:\mathbb R^n \to \mathbb R. One way to find a stationary value is to solve the ODE \dot x = - \nabla f(x), and look at \lim_{t\to\infty} x(t). However I want to ... 2answers 54 views ### Show that \mathbb{E}\left(\int_0^1X_n(t)dW(t)-\int_0^1X(t)dW(t)\right)^2\to 0 I tried to shove this question in another one of my posts as a follow-up question, but I deleted my comment and will post it instead. I would really appreciate if someone could help me spot out and ... 1answer 35 views ### Burkholder's inequality for elementary stochastic integral An elementary Burkholder's inequality for simple stochastic integral says that given nonnegative martingale M and simple bounded predictable process H, it holds that for all c>0, the tail ... 1answer 18 views ### Show that X_n\in\mathcal{H}, where \mathcal{H}:=\{h(t):h(t)\text{ is an adapted process, }\mathbb{E}[\int_0^{\infty}h^2(t)dt]<\infty\} I am not sure if I got this exercise right... I have 2 questions: Have I obtained the final result correctly? If so, I used Wolfram Alpha to obtain the value of the series, but how else can I obtain ... 0answers 43 views ### Differentiating Stochastic Integral I was wondering how to write the following integral in differential form:$$\int^t_0 f(s,t)dW_s$$where W_s is a standard Brownian Motion. In my understanding, if f(s,t) can be written as ... 0answers 36 views ### Generator of a stochastic process I have a question about the generator of a stochastic process. T>0: fix Let b: \mathbb{R} \to \mathbb{R} be a bounded measurable function. Let \left( (X_{t})_{t \in [0,T]}, \left(P_{x} ... 0answers 45 views ### Novikov condition, martingale I have a question about Novikov condition and martingale. T>0: fix. Let (\Omega, \mathcal{F}, \left(\mathcal{F}_{t}\right)_{t \in [0,T]}, P) be a filtered probability space and (B_{t})_{t ... 1answer 42 views ### Application of Ito's isometry in deduction of Wiener Ito Chaos expansion I am trying to learn about the Wiener Ito Chaos expansion and starting reading Oksendal's notes on Malliavin calculus where it is treated in Chapter 1. For a link to the notes, please see ... 1answer 33 views ### Calculate \mathbb{E}[M_{\alpha}^{p}(t)] for all p>0 and t>0, where M_{\alpha}(t):=e^{\alpha W_t-\frac{\alpha^2}{2}t}, t\ge 0 I am going through this solved problem but I don't understand some steps. My professor is notorious for making errors very often so don't hold back if you think he's wrong... Or if I am wrong. I am ... 1answer 28 views ### Evaluate \mathbb{E}\left(\left[W\left(\frac{k}{n}\right)-W(t)\right]^2\right) for all t\in\left(\frac{k}{n},\frac{k+1}{n}\right] I am trying to do a past exam paper to practice, but I don't know if I have answered this question properly... I would really appreciate it if someone could double check it. Thanks a lot! QUESTION: ... 0answers 37 views ### Probabilistic interpretation for Fokker-Planck equation It is well known that if X_t is a stochastic process that solves the SDE$$dX_t = \mu(X_t,t)\,\mathrm{d}t + \sigma(X_t,t)\,\mathrm{d}W_t,$$with W_t a Wiener process, then the associated ... 1answer 38 views ### Absolute continuity counterexample of a stochastic process This example is from Stochastic Modelling and Applied Probability by Sören Asmussen (2010) p.358. The setup is the following: Let \{Z_{t}\} be stochastic process on a Skorokhod space D and a ... 0answers 28 views ### Is the space of all adapted processes with Càdlàg paths a Banach space? Consider first the definition of a stochastic integral for simple predictable processes.$$I:\mathbb{S}\rightarrow\mathbb{D},\ H\mapsto I_X(H):=H_0X_0+\sum_{i=1}^nH_i(X_{T_{i+1}}-X_{T_i})$$The ... 1answer 49 views ### Itō Integral multiplied by Riemann Integral I was wondering whats the result of an Itō integral multiplied by a Riemann Integral. For example, what is$$\left(\int_0^T f(u)\ \mathsf dW_u\right)\left(\int_0^T g(v)\ \mathsf dv\right)$$where W ... 0answers 32 views ### Impose initial condition on partial differential equation After solving a Fokker-Planck equation (using expansion in eigenfunctions) I have obtained the following, general solution for the probability density: p(x,t) = \int_0^\infty dk~ ... 0answers 52 views ### Fundamental theorem for Malliavin derivative and Lebesgue integral I am interested in some kind of fundamental theorem of calculus for the Malliavin derivative: My notations are mainly taken from the Book Nualart: The Malliavin Calculus and Related Topics. Let ... 4answers 199 views ### Uncountable increasing family of \sigma-algebras Could someone give an example of what an uncountable increasing family of \sigma-algebras \{\mathcal{F}_t\}_{t\geq 0}, (\mathcal{F}_s \subset \mathcal{F}_t for s<t) might look like? For ... 0answers 39 views ### Simple Stratonovich product for physical system I was reading a physical textbook and they used the "Stratonovich product" referred to v_1 \circ dW_1 = \frac{1}{2}[v_1 + (v_1+dv_1)]dW_1. I think this product is from the Stochastic process, thus ... 1answer 60 views ### Using Feynman-Kac, compute the following: [closed] Let B(t) be Brownian Motion and let \alpha be a constant and T>0. Compute \mathbb{E}_{B_{0} = x}\left[\exp\left(-\alpha \int_0^T B(s)^2 ds\right)\right]. I'm just having a hard time with ... 1answer 91 views ### Analytic solution to stochastic differential equations I need help to to find the analytic solution (if it exists) of the following system of SDE. Usually, I use Matlab as software but in this case I'm unable to use it in order to figure out the problem. ... 1answer 27 views ### Change from stochastic exponential to exponential of Lévy process - Applebaum In the book "Lévy Processes and Stochastic Calculus (2 edition)" of prof. Applebaum, Theorem 5.1.6 introduce how to change stochastic exponential to exponential of a Lévy process. I am not sure about ... 1answer 38 views ### Stationary distribution for Kolmogorov Forward Equation Given X_t which satisfies the following SDE,$$ dX_t = f'(X_t)dt + \sigma dW_t $$where f is an infinitely differentiable function, and f' above is the first derivative of f. We know that ... 1answer 89 views ### How to prove that this process is always positive? I would like to ask is there any way to prove that following process$$ \mathrm dY_t=\left(a+\frac{b}{Y_t}\right)\mathrm dt +\mathrm dW_t, \ \ Y_0=y_0>0, $$where a\neq 0 and b\geq 1/2, is ... 0answers 38 views ### Ito's formula and Infinitesmal generator Consider an Ito process$$ dX_t = \sigma_t dB_t  where $\sigma_t$ is a two-state continuous-time Markov chain with state space $\{ \sigma_1, \sigma_2 \}$ that switches state with Poisson ...
In the book "Malliavin Calculus and related topics", the author states that $\|F\|_{k,p}=((E(|F|^p)+\sum_{n=1}^k E(\|D^n F\|^p_{H^k}))^{\frac{1}{p}}$ has monotonicity property, i.e. \$\|F\|_{k,p}\leq ...