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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Differentiability of Wiener integral with respect to a parameter

Let $f:[0,T]\times\Theta\to\mathbb R$ and let $\{B_t\}_{t\in [0,T]}$be a Brownian motion. Consider the Wiener integral $$\int_0^T f(t,\theta)dB_t.$$ I am looking for conditions that ensure that $$\...
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When is an SDE solution differentiable in its starting value?

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, $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ ...
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Fundamental theorem of calculus equivalent for stochastic integrals

Let $B$ be a standard Brownian motion in one dimension and let $H$ be a continuous, adapted, bounded process. Prove that $$\frac{\int_t^{t+h}H_sdB_s}{B_{t+h}-B_t}\to H_t$$ in probability as $h\...
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1 answer
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Question on use of Ito's formula with integral in the function

This question is being asked in the context of the Feynman-Kac formula. Suppose the real-valued process $Z$ satisfies the SDE $$dZ_t=b(Z_t)dt+\sigma(Z_t)dW_t.$$ Suppose we have functions $f:\mathbb{R}\...
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Solution to a linear Backward SDE

In my last question, Jose and I discussed about the solution to a linear backward SDE. I followed his steps and made it clear. Besides, I read a paper from Professor Peng talk about the linear ...
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How to calculate this stochastic integral? [closed]

I have this integral: (from the Leland model) Integral F(V) is the following function: F(V) Where the answer should be: Answer Not sure how to get to the answer - I think I Should be using Itô's ...
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3 votes
1 answer
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Constructing the solution to a linear backward SDE

For a linear backward stochastic differential equation (BSDE): for any given $\xi \in L^2(\mathscr{F}_T)$, $$-dY_s = (a_s Y_s + b_s Z_s +c_s)ds-Z_sdB_s$$ Where $a_t,b_t,c_t \in L^2_\mathscr{F}(0,T;R)$,...
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  • 141
2 votes
1 answer
40 views

Expected value of maximum of two numbers, where one is normal distributed

I am searching something similar to the first order loss function ( $\mathbb{E}[max({y_{i}-y_{i},0})])$ (where $y_{i}$ is normally distributed) but for $\mathbb{E}[min(y_{i},d_{i})]$ , where $y_{i}$ ...
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4 votes
1 answer
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Using SDE uniqueness in law to show that two processes have the same distribution

Let $B$ and $\tilde{B}$ be independent standard Brownian motions defined on the same probability space with $B_0=\tilde{B}_0=0$. Let $$X_t=e^{B_t}\int_0^te^{-B_s}d\tilde{B}_s,\hspace{1cm}Y_t=\sinh(B_t)...
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Finding a change of measure so that rescaled local martingale is a local martingale

Let $\mu,\sigma:[0,\infty)\to\mathbb{R}$ be deterministic continuous functions, assume that $\sigma$ is bounded below by a strictly positive constant and that $\mu$ has compact support. Suppose that $...
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2 answers
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Expectation of product of brownian motion and stochastic integral

Let $f:[0,\infty)\to\mathbb{R}$ be a deterministic continuous function and $B$ a Brownian motion with $B_0=0$. I need to prove that $$\mathbb{E}\left(B_t\int_0^tf(s)dB_s\right)=\int_0^tf(s)ds.$$ I ...
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2 votes
1 answer
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Finding a weak solution to an SDE

Consider the SDE $$dX_t=\text{sign}(X_t)dB_t$$ with $X_0=0$ and where $$\text{sign}(x)=\begin{cases}-1&\text{if }x\leq0\\1&\text{if }x>0\end{cases}.$$ I am asked to find a weak solution to ...
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1 answer
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Using the BDG inequality to show a process is uniformly integrable

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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Distribution of stochastic integral to stopping time

Let $B$ be a standard Brownian motion, and let $H$ be a continuous adapted process with $\int_0^\infty H_s^2ds=\infty$. For $\sigma>0$, let $T_\sigma=\inf\{t\geq0:\int_0^tH_s^2ds>\sigma^2\}$. ...
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1 answer
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For stochastic integral $I$ of simple process $X$, $0 \le s < t< \infty$, show $E[I_t(X) | \mathscr{F}_s] = I_s(X)$ a.s.

For stochastic integral $I$ of simple process $X$, $0 \le s < t< \infty$, show $E[I_t(X) | \mathscr{F}_s] = I_s(X)$ a.s. This is Karatzas + Shreve 2nd Edition, Chapter 3.2.B, equation (2.13) ...
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Integrating factor and homogeneous equation for SDEs.

Can someone explain how an integrating factor is obtained when solving SDEs. An example would be when finding the solution for a general linear SDE: $dX_t = (a(t)X_t +b(t))dt + (c(t)X_t +d(t))dB_t, ...
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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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1 vote
1 answer
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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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  • 141
2 votes
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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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1 answer
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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 ...
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  • 141
1 vote
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An application of the multidimensional version of Itô's formula

I am starting to study the multidimensional version of Ito's lemma . The book shows an exercise that I don't understand. The exercise is: $(X_t^1,X_t^2)$ is a 2-dimensional Brownian motion and $F(t,...
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Explicit form of expectation of a function of sum of geometric Brownian motion

Let $X^a$ and $X^b$ be geometric Brownian motions, i.e. for any $i\in \{a,b\}$, $$ dX_t^i/X_t^i=\mu_idt+\sigma_idB^i_t $$ with $X_0^i=x_i>0$, where $B^a$ and $B^b$ are independent Brownian motions. ...
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2 votes
2 answers
101 views

The convexity of a stochastic integral with respect to the initial condition

Let $B(t)$ be the standard Brownian motion, $$dx(t) = \mu(t,x(t))\,dt+\sigma(t,x(t))\,dB(t),$$ where $\mu(x,t)$ and $\sigma(x,t)$ satisfy the usual conditions laid out e.g. on the Wikipedia page on ...
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$E\left[e^{\mu B_t} \max_{0\le s\le t} B_t\right] = \mu^{-1}e^{-\mu^2 ~t/2}(1+\epsilon_t)$ [closed]

Prove that for $\mu > 0,$ $$E\left[e^{\mu B_t} \max_{0\le s\le t} B_t\right] = \mu^{-1}e^{-\mu^2 ~t/2}(1+\epsilon_t), $$ where $\epsilon_t\to \infty$ as $t\to \infty.$ It seems that we need to ...
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3 votes
1 answer
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Inequality on Brownian motions and Ito integrals

I know that Ito's integral isn't monotonic, i.e. if $X \le Y$ almost surely, then $\int_0^t X_s \, dX_s \le \int_0^t Y_s \, dY_s$ almost surely for $X,Y$ semi-martingales. However, is it true that $\...
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2 votes
0 answers
62 views

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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  • 495
1 vote
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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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3 votes
2 answers
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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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  • 495
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0 answers
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Stochastic Integral and Standard Brownian motion

Let $B_t (t\geq 0)$ be a standard Brownian Motion, where $B_0=0$. I have $d(B_s^2)= 2B_sdB_s$. I want to find the integral $ \int_0^tB_sdB_s$. Therefore, I get $ \int_0^tB_sdB_s= \frac{1}{2}\int_0^td(...
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1 answer
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Pricing a call option in the Black Scholes Market - calculation steps

I am working on computing the price of a standard European call option under a Black-Scholes market. Using knowledge of the payoff, I can split the calculation into: $ e^{-rT}(E[S_t] \mathbb{1}_{S_T &...
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3 votes
1 answer
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Properties of the time integral of Ito process

Consider the Ito process $X_t$ defined by $$ dX_t = a(t,X_t) dt + b(t,X_t) dW_t $$ where $W_t$ is the standard continuous-time Wiener process. Let's define the process $Y_t$ to be some integral of $...
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3 votes
1 answer
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A definition of Stochastic integral that is both a martingale and preserves chain rule?

my question is straightforward: is there any definition of Stochastic integral (so I assume at least some kind of Riemann sum compatibility and linearity) that is both a local martingale and preserves ...
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4 votes
1 answer
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Proof verification: conditional distribution of integral of brownian motion

I am looking to compute the conditional distribution of $$S_T=\int_0^T W_t dt$$ given $W_T=x$. Thanks to this question and using the fact that $d(tW_t)=W_tdt + tdW_t$ by Itô's formula, we get \begin{...
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the mean of a stochastic differential equation

Let $(X_{t} , t \geq 0)$ be a processus and solution of the sifferential equation : \ $ X_{t}= X_{0} + \int_{0}^{t} \mu (1-2X_s)ds+ \int_{0}^{t} \sqrt{X_s(1- X_s)} dB_s$ with $ \mu > 0$ and B a ...
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63 views

optimal spread modeling

Here is an interesting case. A bacterium either doubles or transforms into an infectious form with a time-dependent probability $p_n$ and $1-p_n$. Let $X_n$ be the number of duplicate bacteria and $...
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0 votes
1 answer
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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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6 votes
1 answer
111 views

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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0 votes
2 answers
37 views

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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1 vote
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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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1 vote
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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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3 votes
1 answer
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Find the variance $Var[\int_0^t(aB_s+bs)^2dB_s]$

How to find the variance $Var[\int_0^t(aB_s+bs)^2dB_s],$ where $B_s$ are standard Brownian motion. My thoughts: $Var[\int_0^t(aB_s+bs)^2dB_s] = E[(\int_0^t(aB_s+bs)^2dB_s)^2] - E[\int_0^t(aB_s+bs)^...
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  • 601
1 vote
0 answers
56 views

Transforming a martingale into Brownian motion by integration

Let $(\mathcal F_t)$ be an arbitrary filtration on $[0,T]$ and $(X_t)$ a sample continuous stochastic process on $[0,T]$ that is a martingale with respect to $(\mathcal F_t)$. The quadratic variation ...
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  • 2,040
2 votes
1 answer
76 views

Integral of a deterministic function with respect to a martingale is a martingale?

Let $(X_t)$ be a sample continuous stochastic process on $[0,T]$ and a martingale with respect to a given filtration $(\mathcal F_t)$. Let $f:[0,T] \rightarrow \mathbb R$ be a smooth function. Is $ ...
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  • 2,040
0 votes
1 answer
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Why is twice ddifferential of running maximum zero?

I have saw the following statement in my notes. But I cannot see it clearly, if someone could explain? Consider an Itô process given by $$ \mathrm{d} Z_{t}=a_{t} \mathrm{~d} t+b_{t} \mathrm{~d} W_{t}, ...
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0 answers
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Compute the quadratic variation of $N_t = W_t^2 - t$

The question is to compute the quadratic variation of $N_t = W_t^2 - t$, where $W$ is a standard Brownian motion and quadratic variation is defined as the unique process such that $N^2_t - \langle N \...
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  • 379
3 votes
1 answer
74 views

Prove that $\mathbb{E}\left[ \exp\left(\int_0^t X_s dW_s - \int_0^t X_s^2 ds \right)\right] \leq 1$ $\forall t > 0$

Let $(X_t, t \geq 0)$ be a predictable square integrable process. Let \begin{align*} Y_t := \exp\left(\int_0^t X_s dW_s - \int_0^t X_s^2 ds \right) \end{align*} Prove that $\mathbb{E}[Y_t] \leq 1$, $\...
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  • 368
1 vote
1 answer
46 views

Differential of a stochastic process $I(t) = e^{\sigma W(t)}$

I need to compute the differential $dI$ of $I(t) = e^{\sigma W(t)}$ where $W(t)$ is a brownian motion. With $I(W_t) = e^{\sigma W_t}$ I can apply Itô's lemma and get $dI =\frac{\partial I}{\partial ...
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  • 65
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0 answers
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Question regarding to sotchastic integral and Lebesgue-Stieltjes integral

I am recently self-studying stochastic integral based on the textbook written by $Øksendal$. I have a question that why can't we also use Leb-Stj integral to define the stochastic integral $\int_0^T ...
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2 votes
0 answers
41 views

applying Ito's lemma to complex logarithm

This is a question I got while reading the proof of Lemma 4 from the following post: https://almostsuremath.com/2010/06/16/continuous-processes-with-independent-increments/ Here, let $X$ be a ...
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4 votes
1 answer
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A question about Ito formula

As well known, Ito formula is an equality that $$f(W_t)=f(W_0)+\int_o^t f'(W_s)dW_s +\frac{1}{2}\int_o^t f''(W_s)ds,$$ where $f''$ is countunous and $(W_t)_{t>0}$ is a standard brownian motion. We ...
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