Questions about stochastic analysis or stochastic calculus, for example the Ito integral. See https://en.wikipedia.org/wiki/Stochastic_calculus

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Thinning a Renewal Process - Poisson Generalization

If we have a Poisson point process with rate $\lambda$ and we keep each of its point with probability $p$, we obtain another Poisson point process with rate $\lambda p$. Does this result holds for a ...
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289 views

“Continuity” of stochastic integral wrt Brownian motion

I'd like to prove a nice property of a stochastic integral with respect to Brownian motion. Let $(H_t)_{t\geq0}$ be a progressive and bounded process that is continuous at $0$ and $B$ a standard ...
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1k views

Is continuous L2 bounded local martingale a true martingale?

I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...) I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
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110 views

Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?

Suppose we have an sde of the form: \begin{eqnarray} dX_t=b(X_t)dX_t + \sigma (X_t)dB_t \end{eqnarray} where $b$ and $\sigma$ are Lipschitz. Then we have existence and uniqueness of the solution $X_t$...
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752 views

Hermite Polynomials and Brownian motion

I am asked to prove the following : Let $B_t$ be a standard brownian motion. The $n$th Hermite polynomial is $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
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670 views

What is the difference between stochastic calculus and stochastic analysis?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
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214 views

Random diffusion coefficient in the Fourier equation

I'm stuck on the following simple problem: It's given the Fourier equation: $$\partial_t{u(x,t)}=\partial_x[k(t)\partial_xu(x,t)]$$where the diffusion coefficient $k(t)$ is a random variable with a ...
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27 views

Characterizing superposition of two renewal processes

This is a follow-up question of "When superposition of two renewal processes is another renewal process?". How can we characterize the superposition of two renewal processes? The superposition ...
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39 views

Given a $C_c^∞(G)$-valued random variable, is $C_c^∞(G)∋φ↦\text E[\langle\xi,φ\rangle]$ an element of the dual space of $C_c^∞(G)$?

Let $G\subseteq\mathbb R^d$ and $$\mathcal D:=C_c^\infty(G)$$ be equipped with some topology $\tau$ $\mathcal D'$ be the dual space of $\mathcal D$ and $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the ...
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207 views

solution of SDE: $dS_t=(\alpha S_t+f(t))dW_t$

does someone know how to solve the following SDE $$dS_t=(\alpha S_t+f(t))dW_t, S_0=s$$ where $f(t)$ is a deterministic function and $W_t$ is a standard brownian motion. Is there a explicit solution ...
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37 views

Integral representation $B_T^3$

I have to find a $F_t$ such that $B_T^3=E[B_T^3]+\int_0^T F_t dB_t$. I have shown by ito formula that $B_T^3=\int_0^T 3 B_s^2 dB_s+\int_0^T 3 B_s ds$. Could you please help me?
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84 views

A Stochastic Integral Inequality

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ Is there a pair $(\mu,\sigma)$ such that $$\infty&...
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69 views

Convergence in $L^2$ of the stochastic integral $\int\limits^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s$

Let $e\in \mathbb{R}^+$ and $B_t$ 1-dimensional Brownian motion. Consider $$X_t=\int^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s.$$ How to show that $X_t \to 0$ in $L^2$ as $e\to0$? Obviously the ...
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31 views

Quadratic Variation of Brownian motion martingale

Let $B_t$ be a standard Brownian motion and let $M_t = B_t^2 -t$. I want to show that $[M]=[B^2]$. Therefore I want to use the linearity of quadratic variations \begin{align} [\alpha X + \beta Y, Z] =...
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71 views

Monte-Carlo simulation with sampling from uniform distribution

I used to work with Monte-Carlo simulations for a while. In my case, I generated random data for a variety of input parameters according to uniform distributions (with non-negative support), say for ...
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149 views

Representation theorem for local martingales

I want to prove the following local martingale representation theorem. For the statement of the theorems to come we fix a filtered probability space $(\Omega,\mathcal{A},\mathcal{F},\mathbb{P})$ where ...
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132 views

Strong solutions SDE inequality with an application of Gronwall's inequality

Suppose that we have a general SDE on a probability space $(\Omega,\mathcal{F},P)$ defined by: $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) d W_t, $$ where $W$ is a Brownian motion and $b$ and $\sigma$ are ...
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1answer
260 views

Brownian Motion with Optional Stopping Theorem (OST)

Let $(B_t)_{t \geq 0}$ be a standard Brownian Motion and let $T:=\inf\{t \geq 0: B_t=at-b\}$ for some positive constant $a,b>0$. Calculate $\mathbb{E}[T]$. How do i begin it?
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167 views

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(W_t)_{t\...
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95 views

A question related to reflection principle

Question: $$P(X_1\gt 0, ..., X_n\gt 0, X_n=a-b)=?$$ Its Answer: $= (1,1) \rightarrow (n,a-b) $ that meet neither touch nor cross paths. $=[(1,1) \rightarrow (n,a-b) \ \ \text{all paths}]-[(...
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513 views

Further Reading on Stochastic Calculus/Analysis

I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal. So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on ...
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103 views

Quadratic variation - Semimartingale

We know that any Semimartingale has Quadratic variation. I am interested to know if the converse is also true i.e. if a process has quadratic variation then it is semimartingale. Can some one prove/...
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133 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
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112 views

Weak convergence of stochastic integral

Consider a sequence of processes $Z_t^n$ and a procoss $Z_t$, $t\in[0,1]$ such that all $\int_0^1 Z^n dW$ and $\int_0^1 Z dW$ are martingales. Assume $$\int_0^1 Z_t^n \mathrm dW_t \xrightarrow{d} \...
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1answer
475 views

Rigorous Book on Stochastic Calculus

I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been ...
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55 views

Slight generalisation of the distribution of Brownian integral

I think I have seen once that if the processes $\sigma$ and $W$, a Brownian motion, are independent then one has that $$ E \left[\exp \left(iu\int_t^T \sigma_s \, dW_s\right) \mid \mathcal{F} \right] ...
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301 views

A book/text in Stochastic Differential Equations

Somebody know a book/text about Stochastic Differential Equations? I'm in the last period of the undergraduate course and I have interest in this field, but my university don't have a specialist in ...
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35 views

Uniqueness of the trajectories of the solution of an SDE

Consider an SDE \begin{equation} dX_t=f(X_t,t)dt+b(X_t,t)dW_t \end{equation} Suppose firstly that the coefficient are Lipschitz continuous. So by the theorem of existence and unicity I have that exist ...
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357 views

Relation between Hermite polynomials and Brownian motion (on martingale property) [duplicate]

Let us define Hermite polynomials as $H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n} e^{-x^2/2}.$ One can prove that $e^{\theta x-\frac{1}{2} \theta^2}=\sum_{n=0}^{\infty} \frac{1}{n!}H_n(x)\theta^n \quad (*)...
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43 views

Some Kind of Generalized Brownian Bridge

Let $\displaystyle X(t) = \int_0^t f(s)dB(s)$ where $B(t)$ is a Brownian motion and $f(t)\in L^2[0,1]$. What is a simple representation for $Y(t):=(X(t)|X(1))$ in terms of $B(t)$? Note, I am not ...
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285 views

Demonstrate that every martingale is a local martingale.

The Original Question: Demonstrate that every martingale is a local martingale. Attempt at a Solution: Consider the standard setup of this problem: $\mathscr{F}_t$ is the filtration that satisfies ...
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160 views

When do we have $E[X_{n+1}\mid X_n] = E[X_{n+1}\mid\mathscr{F}_n]$?

Under what conditions does it hold that $$E[X_{n+1}\mid X_n] = E[X_{n+1}\mid\mathscr{F}_n]$$ if we are given a stochastic process $X = (X_n)_{n \geq 0}$ on a filtered probability space $(\Omega, \...
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65 views

Positivity of a stochastic process

I want to simulate the paths of a stochastic process $$ dS_t = r S_t dt + \sigma S_t dW_t$$ Using the Forward Euler method, we can write: $$ S_{n+1} = (1 + r \Delta t_n + \sigma \Delta W_{n}) S_n $$ ...
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277 views

Find the Stationary Distribution of an infinite state Markov chain

A Markov Chain on states 0,1,..... has transition probabilities $P_{ij}=1/(i+2)$ for j=0,1,....,i,i+1. I'm supposed to find the stationary distribution. So do I take the limit as n goes to ...
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170 views

What is the right invariant $\sigma$-algebra for the Birkhoff ergodic theorem?

I have been reading stuff about ergodic theory, and I have encountered two versions of the involved "invariant sigma field". let the underlying probability space be $(\Omega,\mathcal{F},P)$, and let's ...
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65 views

Example of a Continuous-Time Markov Process which does NOT have Independent Increments

1. Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a ...
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867 views

Expectation of exponential martingale and indicator function.

Let $W$ be a Wiener process, $r,\sigma \in \mathbb{R}_+$ and $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t) + \sigma(W(T)-W(t))}$. I want to evaluate $$A:=E[e^{- \frac12 \sigma^2 (T-t) - \sigma(W(T)-W(t))...
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145 views

Ito Integral representation for bounded claims

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be Probability space and let $\{(X_t) : 0 \leq t \leq T \}$ be a continuous semimartingale on it. Let $\xi$ be $\mathcal{F}_T^X$ measurable and bounded. Does it ...
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121 views

Second order linear partial differential equation: $\partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0$

Is there a way to solve $$ \partial_t u(t,x)+\frac12 \partial_{x,x}u(t,x)+u(t,x)v(x)=0? $$ This appeared as a condition for $$ X_t=u(t,B_t)e^{\int_0^tv(B_s)ds} $$ to be a martingale. With $B$ a ...
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Semigroup associated to a Markov process

I'm studying the transition semigroup associated to a Markov Process, in particular the Hille-Yosida theorem and the Martingale Problem. In my notes I found : "If $\{T_t\}_t$ is a strongly continuous ...
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62 views

What is the “distributional derivative” of a Brownian motion?

Let $\emptyset\ne I\subseteq\mathbb R$ be an open interval and $A:C_0^\infty(I)\to\mathbb R$ be a distribution. Then, $$\langle{\rm D}A,\varphi\rangle:=-\langle A,\varphi'\rangle\;\;\;\text{for }\...
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How can I prove the equivalence of these two Ito's lemma notations?

Let $X_t=(X_1, \dots , X_T), t \in [0,T] $ be a continuous semimartingale and $f$ a function of class $C^{1,2}$ (continuous and differentiable). Then, $f(t,X)$ is a semimartingale and we have, $\...
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43 views

Scaling in utility maximisation

If I have the wealth process $$dw_t=rw_tdt+n_tS_t(\sigma dB_t+(\mu-r)dt)-c_tdt,$$ where $n$ is number of $S_t$ and $B_t$ is Brownian motion. If we define the admissible set $A$ as follows: $(n_t,c_t)\...
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102 views

$\sin(W_T)$ and Ito / Martingale Representation Theorem

I've been solving some exercises which require a function to be represented as an adapted stochastic process such that $$ X = \mathbb{E}[X] + \int_0^T \Theta(s)\,dW(s) $$ For example, $X = W(T)$ ...
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49 views

I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
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2answers
50 views

Approximation of $\int_0^tF_x(s,X_s)Φ_0dW_s$ where $dX_s=φ_sds+Φ_sdW_s$ and $F_x$ is the Fréchet derivative of some $F:[0,t]×H→\mathbb R$

Let $U$ and $H$ be Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ be equipped with the usual inner product $(\Omega,\mathcal A,\operatorname P)$ ...
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90 views

Distribution of sum of $n$ i.i.d. symmetric Pareto distributed random variables

Let $X$ be a random variable which follows the symmetric Pareto distribution. For a fix, real parameter set $\alpha > 0$ and $L>0$, its PDF is defined as $$ p_X(x) = \left\{ \begin{array}{...
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1answer
60 views

Expectation of a Wiener process at a Stopping Time

I am working through an answer to the following question and do not understand an expectation which takes place at the end. $\textbf{Question:}$ Define the following stochastic process \begin{align} ...
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1answer
29 views

What is a generalized stochastic process? I've found two different definitions. Are they equivalent?

Let $\mathcal D:=C_c^\infty(\mathbb R^d)$ and $\mathcal D'$ be the dual space of $\mathcal D$. What is a generalized stochastic process? I've found two different definitions in some textbooks: ...
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1answer
50 views

Quadratic Variation Brownian motion martingale (2)

Let $B_t$ be a standard Brownian motion and $M_t = B_t^2 -t$. From here we are aware of the identity \begin{align} [M]=[B^2]. \end{align} Now, I want to apply Itô's formula to $B_t^2$ and from that ...