Questions tagged [stochastic-control]

Stochastic control is widely used in finance since it is about decision taking in a random environment. Option replication, portfolio construction, intraday trading are typical applications of stochastic control.

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14
votes
1answer
451 views

How to, from various hypotheses on the P&L, get known models (BS, Heston etc …)

Usually models in quantitative finance are taught by giving, let's say, stochastic differential equations, initial conditions, and then pricing, under the model, various derivatives written on the ...
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3answers
6k views

Application of Control Theory in Quantitative Finance

I have recently completed an MSc in Control Systems from a top university. It seems to me that control theory must have an application within quantitative finance. I would like to apply my degree ...
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0answers
395 views

How do you actually solve a stochastic HJB equation in practice?

I've read a number of recent papers on market making. Nearly all of the more recent papers focus on defining the problem in terms of a state and action space, deriving the relevant HJB equations and ...
5
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1answer
329 views

Lipschitz condition in mathematical finance

I am interested in a rigorous explanation on why the Lipschitz condition plays a major part in stochastic calculus, most significantly in mathematical finance. To be specific, suppose we want to ...
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0answers
125 views

Reference Request: Control Theory Prerequisites for Quantitative Finance

Right now, even though I have a mathematical background, I did not take up control theory in college. I'm looking for an introductory text on (stochastic?) control theory as applicable to quantitative ...
3
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0answers
86 views

Boundary condition in perpetual american option problem

I am trying to solve the perpetual American option problem. Currently I'm following this (slide 9). The stock price is modelled as Ito's process. $dS_t = (\mu-D_0)S_tdt\ +\ \sigma S_tdW_t $ where $...
3
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0answers
42 views

Bounded solution for a SDE

I have this SDE $$ dX(t) = [X(t)(u(t)(\delta-r)+r-\beta(t))+\theta(t)(1-\alpha(t))]dt+X(t)u(t)\sigma dW(t), t \in [0,T] \\ X(0) = X_0(1-\alpha(0)) $$ I've checked some books and I find the solution ...
3
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0answers
115 views

Pre-requisites for Finance Mathematics

I would like to pursue research in the areas of Financial Mathematics. Hoping to look into Operations Research, Risk Management and Stochastic Modeling. Anyone got some suggestions on useful resources ...
3
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0answers
59 views

Price of a stochastic game between an agent and the market

In the article Pricing via utility maximization and entropy from Richard Rouge and Nicole El Karoui, they define the value function of the optimization problem as \begin{align} V(x,C) = \dfrac{1}{\...
3
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0answers
91 views

Portfolio insurance strategy with path dependence

I have the following problem. Let us assume that $S_t$, the stock price follows, geometric Brownian moation with parameters $(\mu,\sigma^2)$. We are given an amount of money $M$ and at each point in ...
2
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1answer
616 views

Code examples of solving Stochastic Optimal Control problems

I'm currently reading a book demonstrating how Stochastic Optimal Control can solve common optimization problems encountered within quantitative finance. I haven't covered much continuous mathematics ...
2
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1answer
188 views

example Hamilton-Jacobi-Bellman Equation - clarification of $dX_t$ derivation using $\pi_t$, $\Pi_t$

I have a market with safe rate r and risky asset S $$ \frac{dS_t}{S_t}=(r+Y_t)dt+\sigma dW_t \quad \quad (1)$$ $$ dY_t = - \lambda Y_t +dB_t \quad \quad (2)$$ where W, B are Brownian Motions with ...
2
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1answer
128 views

Merton portfolio allocation problem proportions/weights >1 or <0?

In the classical Merton portfolio problem, lets assume: $$ dX_t \, = \, \frac{\pi_t X_t}{S_t} S_t(\mu dt +\sigma dW_t) = \pi_t X_t (\mu dt +\sigma dW_t) $$ ie: zero interest rates for simplicity. ...
2
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1answer
116 views

The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market

I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They talk about the relative entropy of a probability measure $Q$ with respect to the ...
2
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3answers
335 views

Understanding the HJM drift condition's dimensions

In an HJM model the forward rate dynamics follow $$ df_t(T) =a_t(f_t(T))dt+b_t(f_t(T))dW_t $$ where $W_t$ is a $d$-dimensional brownian motion, $b_t$ takes values in $\mathbb{R}^{d\times d}$ and $a_t$ ...
2
votes
1answer
93 views

investor terminal value of portfolio with two risky assets 1) correlated 2)not correlated $\phi_t^1=S^{2}_{t}, \ \phi_t^2=S^{1}_{t}$

I am analyzing a problem where I have two stocks described by the equations $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t}$$ $$ \frac{dS^{2}_{t}}{S^{2}_{t}}=\mu_{2} dt + \sigma_{2}...
2
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0answers
19 views

Differentiation of value function in perpetual american option

I am trying to solve the perpetual American option problem. Currently I'm following this (slide 9). The stock price is modelled as Ito's process. $dS_t = (\mu-D_0)S_tdt\ +\ \sigma S_tdW_t $ where $...
2
votes
0answers
67 views

Interchange Expectation and Supremum in Snell Envelope/American Options

I had a question about the properties of a snell envelope, $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, which came to me while studying American options. I know that in general,...
1
vote
1answer
74 views

optimal strategy problem (using Jensen's inequality)

I have a strategy in Samuelson model with zero safe rate defined as $$Z_t^{\Pi}=\frac{X_t^{\Pi}}{X_t^{\rho}} \quad \quad (1)$$ where $$\frac{dX_t^{\Pi}}{X_t^{\Pi}} = \mu \pi dt + \sigma \pi \ dW_t \...
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0answers
93 views

Proving Flow Property of Stochastic Differential Equation

I am trying to show that $X_t^{s,x} = X_t^{r, X_r^{s,x}}$ for $0 \leq s \leq r \leq t$, $x \in \mathbb{R}^n$ is a given initial condition for time $s$, for some SDE: \begin{equation*} d X(u)=b(X(u))d ...
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0answers
54 views

Model of asset substitution/risk shifting in continuous time

Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion: $$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$ where $r>0$ is the risk-free ...
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0answers
134 views

How to Implement an optimal Stochastic Control Optimization? [closed]

I'm currently working on an stochastic optimal control problem applied to a portfolio asset allocation. In principle, the problem is to maximize the return of a fixed income portfolio under certain ...
1
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0answers
55 views

Flow Variable and Stock Variable

I am new to stochastic control and I need your help! Suppose that we are a trader and we are trading based two sources of signal. One comes from the stock's flow of dividends as well as another trader'...
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0answers
22 views

Is there any theoretical work to find an optimum size for the size of horizon in finite-horizon optimization or control?

we learn a lot about finite and infinite horizon control in dynamic programming. but I was wondering if we want to minimize the cost per time(discrete time) is there any work to find the optimum size ...
0
votes
1answer
149 views

Closed-form solution to optimal single assset position sizing with predicted returns

Say that I observe a predictor $w_t \sim N(0,\sigma_1)$ for the returns in a single asset over the next time interval: $$ r_t = \alpha w_{t-1} + z_t $$ where $z_t \sim N(0,\sigma_2)$ is unobserved ...
0
votes
1answer
56 views

trading strategy problem - initial capital x buys S over time [0,T] at the constant rate of x/T euros per unit of time

I am looking for clarification to the trading strategy problem where the number of stocks is depending on time. In the Market with zero safe rate and stock dynamics defined as $$\frac{dS_t}{S_t}=\...
0
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0answers
37 views

Portfolio of Assets with non-constant correlation - Maximum Drawdown

How could I calculate the maximum drawdown (given a specified confidence interval, ie. 99%) of a portfolio whose assets have non-constant (deterministic or stochastic) correlation? Is there a ...