Questions tagged [stochastic-calculus]

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

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Find the distribution of sum

Find the distribution of $W_1 + W_2 + W_3$, where $W_t$ - standard Wiener process
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If $E(X_n^2) = \infty$ then $\limsup \frac{|X_n|}{\sqrt{n}} \geq a$ almost surely.

We have given $X_1,X_2,\ldots$ an i.i.d. sequence of random variables such that $$\Bbb{E}(X_1^2)=\infty$$ I claim that for all $a>0$ $$\Bbb{P}\left(\limsup_{n\rightarrow \infty} \frac{|X_n|}{\sqrt{...
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Find the expected Lebesgue measure

Define a random set $$ Z(\omega) = \{ t > 0 : W_t > t \} $$ $W_t$ - standard Wiener process. Find $E|Z|$ - the expected Lebesgue measure of $Z$.
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2 votes
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Proving uniqueness of weak solution to SDE

This is the SDE: $$dX_t=\operatorname{sign}(X_t+1)dt + dB_t$$ This is a $\mathbb{Q}$-Brownian motion: $$W_t=B_t -\int_{0}^{t}\operatorname{sign}(B_s+1)ds$$ I've already shown that $B_t$ under measure $...
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1 answer
32 views

Bounding the Probability of the L^2-Norm

Let $Y$ be an random Variable, and $f(x,Y)$ be a measurable function. I have for every $x\in[0,1]$ $$ \mathbb{P}\left(|f(x,Y)|\geq \varepsilon\right)\leq \delta $$ Is there any chance to bound $\|f(.,...
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Calculating the transition density function by finding the partial derivative of a conditional probability.

I've been given a process $Y_t=(1+t)B_t^2,t\ge0$ where $B_t$ is a standard Brownian motion and asked to find the transition density function $f(y,t|x,s)$. I've been instructed that $f(y,t|x,s)$ can be ...
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3 votes
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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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2 votes
1 answer
36 views

Exponential submartingale inequality

In a paper I am reading I found the following: "Applying the exponential martingale inequality we derive that $$P\Big(\omega: \sup_{0 \leq t \leq k}[M(t)-1/2 \epsilon \langle M(t) \rangle] \leq 2 ...
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$exp$ of Half-Normal Distribution

I know that the Half-Normal Distribution has moments of all orders - that is, if $X\sim\mathcal{N}(\mu,\sigma)$, then, $$ E[|X|^p]<\infty $$ However, do we also have $$ E[e^{|X|}]<\infty $$ ? ...
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Find the mean and the variance of $X(1)$ for stochastic differential equation: $dX(t)=-1.5X(t)dt+0.85dW(t)$ with $X(0)=0.7$

Suppose that $X(t)$ satisfies $\hspace{5cm}$ $dX(t)=-1.5X(t)dt+0.85dW(t)$ with $X(0)=0.7.$ Find the mean and the variance of $X(1).$ I know that $E[X(1)]$ will result in mean and $E[(X(1))^{2}]$ in ...
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How to prove or calculate $E[\int_{t_{i-1}}^{t_i} e^{-μ({t_i}-s)}\sigma B_s |x_{t_{i-1}}]=0$?

$B_s$ is brownian motion. Because $\int_{t_{i-1}}^{t_i} e^{-μ({t_i} -s)}\sigma dB_s $ has a Brownian component, it is normally distributed with the mean zero according to Taylor & Karin 1998 's ...
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3 votes
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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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Radon–Nikodym derivative of two multivariate Gaussians

Let two Gaussian distributions $P_1$, $P_2$ with mean $0$ and covariance matrix $\boldsymbol{\Sigma}_1,\boldsymbol{\Sigma}_2\in\mathbb{R}^d$ be given. I want to calculate the Radon–Nikodym derivative ...
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Bound Hypergeometric distribution by Gauss distribution

I want to lower bound the pdf of a Hypergeometric distribution $H(N,K,K)$, which has equal number of success states and number of draws with the pdf of a normal distribution. With the central limit ...
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1 answer
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$E[|T\cap S|^2]$ for random sets $S$, $T$ with fixed number of member $|S|=|T|=d$

Let two sets $T,S\subset\{1,\dots, p\}$ be given. Both sets have $d<p$ elements, i.e. $|S|=|T|=d$ Find $E[|T\cap S|^2]$ for random sets $S$, $T$, i.e. if $\mathcal{S}$ is the set over all those ...
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european call with binomial model in python

I have this exercise to resolve. Calculate the prices of a European and an American call option in an N = 3 period binomial model with $ S_0 = $ 1, $ u = \frac{1}{2} = $ 2 and $ r = - \frac{1}{4} $. ...
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3 votes
1 answer
30 views

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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Randomness in iterative algorithm

Let $(H, \langle, \rangle)$ be a Hilbert space. take a closed and convex set $K \subset H$ and $f: K\times K \longrightarrow \mathbb{R}$. The equilibrium problem $EP(f,K)$ consists of finding $\...
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42 views
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Can I get the following estimate for a family of solutions of SDE's?

Fix a time horizon $T>0$, consider a discretization parameter $\Delta>0$ and divide $[0,T]$ into intervals of the form $[K \Delta, (K+1)\Delta]$ for $K=0...\lfloor T/ \Delta \rfloor - 1$. Now ...
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6 votes
1 answer
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Exercise 3.13 Paolo Baldi stochastic calculus: bounds on Brownian motion

I have been brushing up on stochastic analysis before the start of my PhD, and I encountered this exercise on the book "Stochastic Calculus with applications" by Paolo Baldi. The text is as ...
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How to calculate the variance of stochastic integral of stock price?

question picture How to calculate the variance of stochastic integral of stock price?
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0 answers
21 views

Optional Sampling THM for random walker

Say $S_n = X_1 + ... + X_n$ where $X_i$ can take values -1 and 1 each with probability 0.5. Let stopping time $T = min( n : |S_n| = K)$ where K is a positive integer. Since $S_T$ is bounded ($-K \leq ...
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1 vote
1 answer
63 views

How do I prove that the weak law of large numbers holds?

We have given $X_1,X_2,...$ an i.i.d. sequence of random variables with $$\Bbb{P}(X_1=1)=\Bbb{P}(X_1=-1)=\frac{1}{2}$$ From class we know that then the characteristic function is $\Phi_{X_i}(t)=\cos(t)...
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3 votes
1 answer
233 views

Are characteristic functions always differentiable?

I am thinking about the statement if the characteristic function of a random variable $X$, $\Phi_X$, is always differentiable. By definition, $$\Phi_X(t)=\int_{\Bbb{R}^d}e^{i\langle t,y \rangle}P_X(dy)...
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31 views

Black-Scholes Model for options pricing

The Black-Scholes pricing formula for a European call option is given by: $$C(S,t) = SN(d_+) - E\exp(-r(T-t))N(d_-)$$ where $S\ge0$ is the spot price, $t\le T$ is the time,$T\ge0$ is the expiry date, $...
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4 votes
1 answer
42 views

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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0 answers
28 views

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
39 views

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
45 views

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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3 votes
0 answers
67 views

How to use Ito's formula to show that $ K_N(s,t)-K(t,t)=\int_s^t[-U' K_N(u,t)+<\mathbf{J}x_u, x_t>]du+\frac{1}{N}\sum x_t^i(B_s^i-B_t^i) $?

I am reading one lecture note Dynamics for Spherical Models of Spin-Glass and Aging. On page 124, it shows that for $s\ge t$, $$ K_N(s,t)-K(t,t)=\int_s^t[-U' K_N(u,t)+<\mathbf{J}x_u, x_t>]du+\...
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0 votes
1 answer
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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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1 vote
0 answers
43 views

A family of probability measures satisfying an integral equation

Consider the problem of finding a family of probability measures $(\mu_t)_{t \geq 0}$ on $(\mathbb{R}, \mathscr{B}( \mathbb{R}))$ such that $$ \int_{\mathbb{R}} \varphi(x) \mu_t(dx) = \int_{\mathbb{R}}...
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How do I prove that the distribution of $x_t$ in $x_t=x_{t-1}e^{-μt}+ θ(1-e^{-μΔt}) + \int_{t-1}^t e^{-μ(t-s)}\sigma dB_s $ is a normal distribution?

$B_s$ is brownian motion. Because $\int_{t-1}^t e^{-μ(t-s)}\sigma dB_s $ has a Brownian component, it is normally distributed according to Taylor & Karin 1998 's Introduction to Stochastic ...
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2 votes
1 answer
84 views

Stochastic Differential Equation after a Change in the Time Parameter

How does the stochastic differential equation for a stochastic process change under a change in the time parameter? For example, consider the Bessel Squared Process $$ dR_t = m \, dt + 2 \sqrt{R_t} \, ...
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  • 1,353
2 votes
1 answer
39 views

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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3 votes
1 answer
42 views

Question about Ito formula and BSDE

When I was reading the paper from Peng, I saw an equation which I had no idea about how to get it. The details are shown below: For a BSDE : $$y_t = \xi + \int_{t}^{T}g_0(s)ds - \int_{t}^{T}z_sdB_s$$ ...
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  • 141
2 votes
1 answer
80 views

The stochastic differential of $\cos (B_t^{(1)}B_t^{(2)})$

Let ($B_1$, $B_2$) be a bi-dimensional correlated Brownian motions Calculate the stochastic differential equation of the process $\cos(B_{1,t}B_{2,t})$. Attempt: Let $X_t$ be the stochastic process ...
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3 votes
1 answer
42 views

Hitting time for 2-dimensional Brownian motion

Fix $a>0$ and let $(A_t^-)_{t\geq0}$ and $(A_t^+)_{t\geq 0}$ be two independent one-dimensional Brownian motions, starting from $-a$ and $a$, respectively. Set $$T=\inf\{t\geq0:A_t^-=A_t^+\}.$$ I ...
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1 vote
1 answer
59 views

How do I prove that the expectation value of the following two random variables converge to zero?

I have given $X_n,Y_n$ two sequences of real valued random variables in the same probability space. I assume that $X_n\Rightarrow X$ in distribution and $|X_n-Y_n|\rightarrow 0$ in probability. In ...
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1 vote
0 answers
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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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3 votes
1 answer
92 views

Stopping time and super-martingale

Consider a right-continuous super-martingale $(X_u,\mathcal{F}_u)_{u \in \mathbb{R}_+}.$ Let $\theta_1$ and $\theta_2$ be two stopping times such that $\theta_1 \leq \theta_2.$ Prove that $(X_{u\wedge\...
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0 votes
1 answer
26 views

Convergence in distribution of a vector random variable implies convergence in distribution of the sum.

If $X_n,X,Y_n, Y$ ar real valued random variables and I assume that $(X_n,Y_n)\Rightarrow (X,Y)$ in distribution I need to show that $X_n+Y_n\Rightarrow X+Y$ in distribution Since $(X_n,Y_n)\...
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  • 1,029
0 votes
0 answers
17 views

Showing that a composition of a non-explicit function is Lipschitz

Let $b$ be a bounded and continuous function and let $W$ be a scalar Brownian motion. Consider the SDE $$dX_t=b(X_t)dt+dW_t.$$ I was tasked with showing that there exists a strictly increasing ...
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22 views

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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  • 148
4 votes
0 answers
56 views

Radon-Nikodym derivative of pushforward measures and Girsanov theorem

Let $\mu$ and $\nu$ be two measures on a measure space $(\Omega, \Sigma)$, and $\mu$ is absolute continuous w.r.t. $\nu$. Also let $X\colon \Omega \to H$ be a measurable functions mapping to another ...
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0 votes
1 answer
33 views

Why can we use this density for random variables with normal distribution?

We have given $N\sim \mathfrak{N}(0;1)$, $x\in \Bbb{R}$ and $\epsilon>0$. Furthermore $f$ is continuous and bounded. Then we want to compute $E(f(x+\epsilon N))$ We have just proven that for a ...
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  • 1,284
1 vote
1 answer
53 views

Dynkin's martingale formula

Dynkin's formula (from Wikipedia): for an Ito diffusion $X_t$ having infinitesimal generator $A$, $$ \mathbb{E}^x[f(X_\tau)] = f(x) + \mathbb{E}^x\left[\int_0^\tau Af(X_s)ds\right].$$ Here, $f$ is a ...
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  • 1,394
2 votes
1 answer
31 views

Properities of sine(and cosine) of Brownian Motion

Suppose $(B_t, \mathcal{F}_t)_{t \geq 0}$ is a classical Brownian Motion (with the canonical filtration) and consider the process $X_t=\sin(B_t)$. What properties does our new process share with a ...
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  • 586
2 votes
1 answer
48 views

Lebesgue-Stieltjes integral and Dynkin $\pi-\lambda$ theorem

I am studying the Lebesgue-Stieltjes integral from this PDF: https://www.math.utah.edu/~li/L-S%20integral.pdf. In Theorem 8 the authors claim to use Dynkin's theorem in a way that I do not understand. ...
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  • 356
-1 votes
0 answers
23 views

$E\left[e^{\mu B_t} \max_{0\le s\le t} B_t\right] = \mu^{-1}e^{-\mu^2 ~t/2}(1+\epsilon_t)$ [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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