# 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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### 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 ...
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### martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
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### BMO martingale and exponential martingale

Consider the BSDE, $$Y_{T}-Y_{t}=\sum_{i=1}^{n} \int_{t}^{T} Z_{s}^{i}dB_{s}^{i} - \frac{1}{2}\int_{t}^{T} \left| Z_{s}\right|^{2}ds$$ where $B$ is a standard Brownian motion on a complete ...
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### SDE with no weak solution

I'm facing the followingd d-dimensional SDE: $$dY_t=\sigma(h_t)\,dB_t$$ In addition it holds, that: $h_t\in H$ and $H$ is compact (for example the simplex of $R^n$) the proces $h_t$ is progressivley ...
i have this Equation with Condition $X\left(0\right)=a$ and $0\le t \lt T$ $$dX\left(t\right)=\frac{b-X\left(t\right)}{t-T}dt+dB\left(t\right)$$ I solved and got $$X\left(t\right)= ... 0answers 34 views ### Girsanov Measure Question. If Z_t = exp^{\int_0^t X_s dW_s - \frac{1}{2} \int_0^t (X_s)^2 ds} is a martinagle then by Girsanov's theorem, the measure P_T defined by P_T(A) = E^P(AZ_T) is mutually absolutely continuous ... 1answer 52 views ### Expectation of product of stochastic integral and brownian motion Find the covariance:$$ COV((\int_t^T(T-s)dW_s), W_t) $$I used the covariance formula: COV(X,Y) = E(XY) - E(X)E(Y) = E(XY) as E(X)=E(Y)=0 But I am stuck on figuring out the expectation of the ... 1answer 42 views ### Stochastic Integral Help Let W(t) be a Brownian Motion. Show that the integral:$$ \int_t^T W(s)ds $$can be written in terms of the stochastic integral:$$ \int_t^T (T-s)dW(S) $$Is there an error with this question? I ... 1answer 37 views ### Solutions of SDE do not explode when drift term is zero. Suppose we have dX_t = \sigma(X_t) dW_t where \sigma : \mathbb{R} \rightarrow \mathbb{R} is Borel and W_t is a standard one-dimensional Brownian motion. I am trying to show that X_t cannot ... 1answer 29 views ### Sufficient condition for time-changed quadratic covariation to vanish in probability Let (M_t^n)_{t \geq 0} be a sequence of continuous martingales of the form M^n_t = \int_0^t X^n_s \, dB_s where B_s is a Brownian motion. Let \tau^n(t) be the time change associated to M_t^n ... 1answer 78 views ### Ornstein-Uhlenbeck process and Markov property There isn't a similar question in the forum, so here it goes. Firstly, let the O-U velocity process be defined as$$ dV_t = -\beta V_t dt + \sigma dB_t $$with V_0 = v, and B = (B_t), t \geq 0 a ... 1answer 48 views ### Poisson integral and discontinuous martingale (Ito-Levy formula) Consider compounded Poisson process P given by P_t = \int_0 ^t \int _{\mathbb R}z~ N(dr,dz) where N is a Poisson random measure of intensity dt \otimes \nu and \nu  is a Levy measure. Why ... 1answer 32 views ### Elementary Malliavin Derivative question about definition. I am reading a book that defines the Malliavin derivative D_tF as follows: If F = \sum_{n=0}^{\infty} I_n(f_n) is the Wiener Chaos expansion. F is in the brownian filtration and F \in ... 1answer 77 views ### Variance of Ito Integral I want to find the variance of the Ito integral: X(t)=\displaystyle \int_0^t\sqrt{s}WdW where W is a Brownian motion and s is the variable of integration. This is what I have done so far: ... 1answer 54 views ### Determining dX_t for stochastic equations, and which of these are \mathcal{F}  - martingales? I want to write down an expression for dX_t for both: i. X_t=t^2W_t^2-2\int_0^t(sW_s^2+s^2)ds; and ii. X_t=W_t^2-tW_t What is the process I would use for differentiating these stochastic ... 1answer 36 views ### \mathbb{E}[B_t-B_s], \mathbb{E}[\exp(\sigma(B_t-B_s))] etc. This may be a duplicate but I cannot find the corresponding question. I have been asked to show: \mathbb{E}[\exp(\sigma(B_t-B_s))] = \exp\left(-\dfrac{\sigma^2}{2}(s-t)\right) As a side note I ... 1answer 40 views ### Application of Ito's Lemma to integral expression I have a problem applying Ito's lemma. I know that if: dX_t= \mu_t \, dt + \sigma_t \, dB_t then for f(t,x): df(t,X_t) =\left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial ... 0answers 19 views ### Does this Stochastic Differential Equation have a name? I came across this SDE and since I am not an expert I am wondering if this SDE is known to have an closed form solution for first passage times. The SDE is$$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$... 1answer 48 views ### Ornstein-Uhlenbeck process written explicitly I need to show that the Ornstein-Uhlenbeck process,$$ dX_t = -\theta X_tdt + dB(t) $$Where X_0=0, B(t) is Brownian motion and \theta>0 can be written explicitly as:$$ X_t=B(t) - \theta ...
I recently encountered the following object, referred to as "weak stochastic integral" in the book of SPDE's by Prevot/Rockner [PR07]: $\int_0^T { \langle \Psi dW(t), \Phi(t)\rangle }$ A few useful ...