# Questions tagged [stochastic-integrals]

This tag is used for questions about stochastic integrals – especially for calculations. For questions related to more theoretical aspects of stochastic integrals such as their constructions, (stochastic-analysis) may be more appropriate.

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### Sobolev embedding for stochastic processes

Let's say I have some stochastic process $u_t \in L^2 ((0,T), H^2 (\Omega))$ with $\Omega$ as regular as you want in $\mathbb{R}^2$. If $E\left[ \int_0 ^ t ||u_t||_{H^2} ^2 \right] \leq C$, do I ...
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### Using BDG inequality to show the solution to a BSDE belongs to $S^2_{\mathscr{F}}$

For a BSDE: $$y_t = \xi + \int_{t}^{T}g_0(s)ds - \int_{t}^{T}z_sdB_s$$ Which has a fixed $\xi \in L^2(\mathscr{F}_T)$ and $g_0(\cdot)$ satisfying $E(\int_{0}^{T}|g_0(t)|dt)^2 < \infty$. There ...
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### Proving $E\bigg[\bigg(\int_a^b X_s dW_s \bigg)^2 \ \bigg\vert \ \mathcal{F}_a \bigg] =\int_a^b (X_s)^2 ds$

Suppose $(\mathcal{F}_t)_{t \geq0}$ is a filtration on a probability space and $W=(W_t)_{t \geq0}$ is a Brownian Motion with respect to this filtration. Let $(X_t)_{t \geq0}$ denote some ...
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### Show that $\mathbb{E}(X|\mathcal{F}_t)$ is a square-integrable martingale

For a BSDE : $$y_t = \xi + \int_{t}^{T}g_0(s)ds - \int_{t}^{T}z_sdB_s$$ Which has a fixed $\xi \in L^2(\mathscr{F}_T)$ and $g_0(\cdot)$ satisfying $E(\int_{0}^{T}|g_0(t)|dt)^2 < \infty$. There ...
• 141
1 vote
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### Is a Riccati BSDE explicitly solvable?

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $C\in (0;\infty)$ and $\{a_t\}_{t\in[0;T]}$ be a ...
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### $E[(d W_t)^2] =?$

I confused myself here. Let $W_t$ be Brownian motion. It is obvious that the variance of the Brownian motion increment $dW_t$ is $dt$, i.e. $E[(d W_t)^2] = dt$. However, if I write this like so, I get ...
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### Solving a linear backward stochastic differential equation

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $\{B_t\}_{t\in[0;T]}$ be an adapted process and ...
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### Define Lebesgue-Stieltjes integral for integrators of unbounded first variation

In chapter 1.5 of Shreve & Karatzas's book, after proving that for continuous, square-integrable martingales, variations greater than 0 vanish and lower variations explode, it proceeds to argue ...
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### Definitions of the Stratonovich integral and why the "average" definition is arguably correct

Notations: Herein: $\mathcal{B} := \{B(t)\}_{t \ge 0}$ denotes a standard Brownian motion, with $B(0) = 0$. $P := \{x_i\}_{i=0}^n$ denotes a partition of the interval $[0,t]$, with norm defined in ...
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### Increasing process is natural if and only if it is predictable

Let $\{A_t\}_{t \geqslant 0}$ be a progressively measurable stochastic process defined on a filtered space $(\Omega, F, \{F_t\}_{t \geqslant 0},P)$ such that every sample path is right continuous and ...
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### Possible Close-form solution for 2 dimensional $\frac{d R}{dt}R^T = \mathbf{F}R_t R_t^T + \frac{1}{2} \mathbf{G}\mathbf{G}^T$

I want to solve $\mathbf{R}_t$ for $$\frac{d \mathbf{R}_t}{dt}\mathbf{R}_t^T = \mathbf{F}\mathbf{R}_t \mathbf{R}_t^T + \frac{1}{2} \mathbf{G}\mathbf{G}^T,$$ where $\mathbf{F},\mathbf{G}$ are 2x2 ...
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### How to explicitly solve the SDE $dX_{t} = tX_{t} + \sigma \cos t X_{t}dW_{t}$?

I have the stochastic ode $dX_{t} = tX_{t} + \sigma \cos t X_{t}dW_{t}$, where $W_{t}$ is the Wiener process. I am trying to solve it and not sure if my work is correct. My attempt: Rearranging, we ...
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