This tag is used for questions about stochastic integrals - especially for calculations . For questions related to more theoretic aspects of stochastic integrals such as its construction. Stochastic-analysis may be a more appropriate tag.

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2
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1answer
235 views

Deriving SDE(s) and Expectation from Given PDE

We want to solve the PDE $u_t + \left( \frac{x^2 + y^2}{2}\right)u_{xx} + (x-y^2)u_y + ryu = 0 $ where $r$ is some constant and $u(x,y,T) = V(x,y)$ is given. Write an SDE and express $u(x,y,0)$ as the ...
4
votes
1answer
315 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
1
vote
2answers
296 views

Expectation of Stochastic Process Given First Hitting Time Information

Let $V_t$ satisfy the SDE $dV_t = -\gamma V_t dt + \alpha dW_t$. Let $\tau$ be the first hitting time for 0, i.e., $\tau $ = min$(t | V_t = 0)$. Let $s =$ min$(\tau, 5)$. Let $\mathcal{F}_s$ be the ...
4
votes
1answer
1k views

How to derive the Ornstein-Uhlenbeck Stochastic Integral Equation?

I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-a\int^t_0 X_s\,ds +B_t$. B is the Brownian motion. And its analytic ...
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0answers
228 views

Show that this is the unique solution of that Stochastic Differential Equation

Reading through a paper, I stumbled across the stochastic differential equation $ dS_t = \sigma S_{t-} dX_t $. The claim there was that $ S_t = S_0 \exp(\alpha N_t - \beta t) $ should be its unique ...
3
votes
1answer
253 views

How shall I prove this Stochastic integral equation?

I want to prove $$ \int_0^T B_t^2 dB_t = \frac{B_T^3}{3} - \int_0^T B_t dt $$ by the definition of Ito integral. I have tried this so far. Given a partition $0=t_0 < t_1 < ... < t_n=T$, I ...
6
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1answer
330 views

Does Itō isometry have different versions?

Itō isometry from Wikipedia: Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical real-valued Wiener process defined up to time $T > 0$, and let $X : [0, T] \times \Omega \to ...
2
votes
1answer
322 views

First exit time

How to calculate the first exit time of the process $$dx=-M\;dt+dw,$$ where $M$ is a positive constant, and $w$ is a Wiener process? Start from $d>0$, to the boundary $0$. I solved the ...
0
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0answers
378 views

Solution to nonlinear Stochastic Differential Equation

$dX_t=(\sqrt{1+X_t^2}+\frac{1}{2}X_t)dt+\sqrt{1+X_t^2}dW_t, X_0=0$, where $W_t$ is brownian. I tried using $X_t=\sinh(W_t)$ but then when I apply Ito's lemma to it, I can't get the first sqrt term. ...
4
votes
1answer
270 views

Solution to the stochastic differential equation

Let $X_o=x$, $dX_t=\frac{1}{X_t}dt+X_tdW_t$, $W_t$ is a brownian motion i am thinking of trying $Y_t=\frac{X_t^2}{2}$ and apply ito's lemma on $Y_t$
2
votes
1answer
320 views

expected value of product of stochastic processes

Let $X_t=\sigma \int_0^t e^{-a(t-s)} dW_s$, where $\sigma , a $ are constants. How can I find the expected value of the product of $X_t, X_s$ For t>s, $\mathbb{E}[X_t, X_s]$, and $\mathbb{E}[X_t, ...
6
votes
1answer
2k views

Expected value of the stochastic integral $\int_0^t e^{as} dW_s$

I am trying to calculate a stochastic integral $\mathbb{E}[\int_0^t e^{as} dW_s]$. I tried breaking it up into a Riemann sum $\mathbb{E}[\sum e^{as_{t_i}}(W_{t_i}-W_{t_{i-1}})]$, but I get expected ...
3
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1answer
347 views

what's the difference between RDE and SDE?

what's the difference between random differential equation and stochastic differential equation? does stochastic differential equations include random differential equation?
4
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0answers
165 views

Observable and unobservable parameters of stochastic processes

Consider the following diffusion process $$ dX_t = \mu\,dt+\sigma(t,X_t)\,dW_t $$ where $X,W$ are 1-dimensional and. Is it true that given a history $(X_s,s\leq t)$ for each $s< t$ one can find ...
3
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3answers
166 views

Which courses before Stochastics?

I would like to know which maths course I need to take before studying stochastics. Thx for helping, Stephane
4
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1answer
568 views

Brownian hitting time of a _very_ simple linear boundary

I realize that general results on the hitting times of a curve are practically nonexistant, but I am hoping that someone can string together a sequence of tricks to tell me what $$ \Pr\left( ...
5
votes
3answers
3k views

Expectation of geometric brownian motion

I was deriving the solution to the stochastic differential equation $$dX_t = \mu X_tdt + \sigma X_tdB_t$$ where $B_t$ is a brownian motion. After finding $$X_t = x_0\exp((\mu - \frac{\sigma^2}{2})t + ...
3
votes
3answers
351 views

How to evaluate the following stochastic integral?

How to evaluate the following stochastic integral? $$\int_0^t M_{s^-}^2 dM_s$$ where $M_t = N_t - \lambda t$ is a compensated Poisson process. I tried to apply Ito's formula to $M_t^2$ but still ...
2
votes
0answers
138 views

Stochastic differential equation

Using stochastic(!) methods find explicit solution to each of the two ($i = 1, 2$) initial value problems $$\partial_t u(t, x) = \frac{1}{2} \beta^2 \partial_x^ 2 u(t, x) + (−\alpha x + \gamma ...
4
votes
0answers
194 views

stochastic differential equation

Xt is a weak solution to the SDE with dXt = ( −αXt + γ )dt + β dBt , ∀t ≥ 0 X0 = x0. α, β , and γ constants, and Bt is a brownina motion. need to find the PDE for the transition density of X at ...
1
vote
1answer
202 views

Integration Order Reversal

I have a question regarding integration order reversal in a stochastic integral. This is a homework problem of the form "Show this is true". My problem is 1) my results are not exactly the same as the ...
1
vote
1answer
123 views

Definite integral

Please I need help with the evaluation of this integral. I've tried with both mathematica and maple, but to no avail. Here is the integral: $$ ...
4
votes
1answer
519 views

Martingale problem

If $X_t$ is an $\mathbb{R}$- valued stochastic process with continuous paths, show that the following two conditions are equivalent: (i) for all $f\in C^2(\mathbb{R})$ the process $$f(X_t) − f(X_0) ...
6
votes
1answer
729 views

Is this a martingale?

Let $W_t$ be a standard Brownian motion with $W_0 = 0$ and let $Z_t$ solve the stochastic differential equation $dZ_t = 2 Z_t W_t \mathrm{d}W_t$. This has solution $$ ...
5
votes
3answers
288 views

Stochastic integral and Stieltjes integral

My question is on the convergence of the Riemann sum, when the value spaces are square-integrable random variables. The convergence does depend on the evaluation point we choose, why is the case. Here ...
4
votes
0answers
297 views

Ito's Lemma application

$Z(t) = \int_0^t g(s)\,dW(s)$, where $g$ is an adapted stochastic process. Find $dZ$ ?
3
votes
0answers
696 views

Doleans-Dade exponential formula

How do I apply the Doleans-Dade exponential formula for the following levy stochastic differential equation: $$dZ_t=Z_t\left(\theta_1(t)dW_t^{(1)} +\theta_2(t)dW_t^{(2)}+\int_\mathbb R ...
1
vote
2answers
547 views

Compute a stochastic integral

I am trying to do the following stochastic integral $$ \int_0^T \mu(B_s) dB_s - \frac{\int_0^T (\mu(B_s))^2 ds}{2} $$ where $ \{ B_t \}$ is a standard Brownian motion, and $\mu(x) = ...