arXiv subjects

Ali Devin Sezer

Publications and source records attributed to Ali Devin Sezer.

15 recordsLinked to original sources

arXiv2023

Optimal Liquidation with Conditions on Minimum Price

The classical optimal trading problem is the closure of a position in an asset over a time interval; the trader maximizes an expected utility under the constraint that the position be fully closed by terminal time. Since the asset price is stochastic, the liquidation constraint may be too restrictive; the trader may want to relax it or slow down/stop trading depending on price behavior. We consider two additional parameters that serve these purposes within the Almgren-Chriss framework: a binary valued process $I$ that prescribes when trading takes place and a measurable set $S$ that prescribes when full liquidation is required. We give four examples for $S$ and $I$ which are defined in terms of a lower bound for the price process. The terminal cost of the control problem is $\infty$ over $S$ representing the liquidation constraint. The permanent price impact parameter enters the problem as the negative part of the terminal cost over $S^c$. $I$ modifies the running cost. A terminal cost that can take negative values implies 1) the backward stochastic differential equation (BSDE) associated with the value function of the control problem can explode to $-\infty$ backward in time and 2) existence results on minimal supersolutions of BSDE with singular terminal values and monotone drivers are not directly applicable. A key part of the solution is an assumption that balances market volume and the permanent price impact parameter and a lower bound on the BSDE based on this assumption. When liquidation costs are quadratic, the problem is convex and, under a general filtration, the minimal supersolution of the BSDE gives the value function and the optimal control. For the non-quadratic case, we assume a stochastic volatility model and focus on choices of $I$ and $S$ that are Markovian or can be broken into Markovian pieces. These give PDE/PDE-system representations for the value functions.

math.PR↗

arXiv2021

Hitting probabilities of constrained random walks representing tandem networks

Let $X$ be the constrained random walk on $\mathbb{Z}_+^d$ $d >2$, having increments $e_1$, $-e_i+e_{i+1}$ $i=1,2,3,...,d-1$ and $-e_d$ with probabilities $\lambda$, $\mu_1$, $\mu_2$,...,$\mu_d$, where $\{e_1,e_2,..,e_d\}$ are the standard basis vectors. The process $X$ is assumed stable, i.e., $\lambda < \mu_i$ for all $i=1,2,3,...,d.$ Let $\tau_n$ be the first time the sum of the components of $X$ equals $n$. We derive approximation formulas for the probability ${\mathbb P}_x(\tau_n < \tau_0)$. For $x \in \bigcup_{i=1}^d \Big\{x \in {\mathbb R}^d_+: \sum_{j=1}^{i} x(j)$ $> \left(1 - \frac{\log \lambda/\min \mu_i}{\log \lambda/\mu_i}\right) \Big\}$ and a sequence of initial points $x_n/n \rightarrow x$ we show that the relative error of the approximation decays exponentially in $n$. The approximation formula is of the form ${\mathbb P}_y(\tau < \infty)$ where $\tau$ is the first time the sum of the components of a limit process $Y$ is $0$; $Y$ is the process $X$ as observed from a point on the exit boundary except that it is unconstrained in its first component (in particular $Y$ is an unstable process); $Y$ and ${\mathbb P}_y(\tau< \infty)$ arise naturally as the limit of an affine transformation of $X$ and the probability ${\mathbb P}_x(\tau_n < \tau_0).$ The analysis of the relative error is based on a new construction of supermartingales. We derive an explicit formula for ${\mathbb P}_y(\tau < \infty)$ in terms of the ratios $\lambda/\mu_i$ which is based on the concepts of harmonic systems and their solutions and conjugate points on a characteristic surface associated with the process $Y$; the derivation of the formula assumes $\mu_i \neq \mu_j$ for $i\neq j.$

math.PR↗

arXiv2019

Backward Stochastic Differential Equations with Non-Markovian Singular Terminal Conditions with General Driver and Filtration

We consider a class of Backward Stochastic Differential Equations with superlinear driver process $f$ adapted to a filtration supporting at least a $d$ dimensional Brownian motion and a Poisson random measure on ${\mathbb R}^m- \{0\}.$ We consider the following class of terminal conditions $ξ_1 = \infty \cdot 1_{\{τ_1 \le T\}}$ where $τ_1$ is any stopping time with a bounded density in a neighborhood of $T$ and $ξ_2 = \infty \cdot 1_{A_T}$ where $A_t$, $t \in [0,T]$ is a decreasing sequence of events adapted to the filtration ${\mathcal F}_t$ that is continuous in probability at $T$. A special case for $ξ_2$ is $A_T = \{τ_2 > T\}$ where $τ_2$ is any stopping time such that $P(τ_2 =T) =0.$ In this setting we prove that the minimal supersolutions of the BSDE are in fact solutions, i.e., they attain almost surely their terminal values. We further show that the first exit time from a time varying domain of a $d$-dimensional diffusion process driven by the Brownian motion with strongly elliptic covariance matrix does have a continuous density; therefore such exit times can be used as $τ_1$ and $τ_2$ to define the terminal conditions $ξ_1$ and $ξ_2.$ The proof of existence of the density is based on the classical Green's functions for the associated PDE.

math.PR↗

arXiv2019

Approximation of the Exit Probability of a Stable Markov Modulated Constrained Random Walk

Let $X$ be the constrained random walk on ${\mathbb Z}_+^2$ having increments $(1,0)$, $(-1,1)$, $(0,-1)$ with jump probabilities $λ(M_k)$, $μ_1(M_k)$, and $μ_2(M_k)$ where $M$ is an irreducible aperiodic finite state Markov chain. The process $X$ represents the lengths of two tandem queues with arrival rate $λ(M_k)$, and service rates $μ_1(M_k)$, and $μ_2(M_k)$. We assume that the average arrival rate with respect to the stationary measure of $M$ is less than the average service rates, i.e., $X$ is assumed stable. Let $τ_n$ be the first time when the sum of the components of $X$ equals $n$ for the first time. Let $Y$ be the random walk on ${\mathbb Z} \times {\mathbb Z}_+$ having increments $(-1,0)$, $(1,1)$, $(0,-1)$ with probabilities $λ(M_k)$, $μ_1(M_k)$, and $μ_2(M_k)$. Let $τ$ be the first time the components of $Y$ are equal. For $x \in {\mathbb R}_+^2$, $x(1) + x(2) < 1$, $x(1) > 0$, and $x_n = \lfloor nx \rfloor$, we show that $P_{(n-x_n(1),x_n(2)),m)}( τ< \infty)$ approximates $P_{(x_n,m)}(τ_n < τ_0)$ with exponentially vanishing relative error as $n\rightarrow \infty$. For the analysis we define a characteristic matrix in terms of the jump probabilities of $(X,M).$ The $0$-level set of the characteristic polynomial of this matrix defines the characteristic surface; conjugate points on this surface and the associated eigenvectors of the characteristic matrix are used to define (sub/super) harmonic functions which play a fundamental role both in our analysis and the computation / approximation of $P_{(y,m)}(τ< \infty).$

math.PR↗

arXiv2018

Excessive Backlog Probabilities of Two Parallel Queues

Let $X$ be the constrained random walk on ${\mathbb Z}_+^2$ with increments $(1,0)$, $(-1,0)$, $(0,1)$ and $(0,-1)$; $X$ represents, at arrivals and service completions, the lengths of two queues working in parallel whose service and interarrival times are exponentially distributed with arrival rates $λ_i$ and service rates $μ_i$, $i=1,2$; we assume $λ_i < μ_i$, $i=1,2$, i.e., $X$ is assumed stable. Without loss of generality we assume $ρ_1 =λ_1/μ_1 \ge ρ_2 = λ_2/μ_2$. Let $τ_n$ be the first time $X$ hits the line $\partial A_n = \{x \in {\mathbb Z}^2:x(1)+x(2) = n \}$. Let $Y$ be the same random walk as $X$ but only constrained on $\{y \in {\mathbb Z}^2: y(2)=0\}$ and its jump probabilities for the first component reversed. Let $\partial B =\{y \in {\mathbb Z}^2: y(1) = y(2) \}$ and let $τ$ be the first time $Y$ hits $\partial B$. The probability $p_n = P_x(τ_n < τ_0)$ is a key performance measure of the queueing system represented by $X$ (probability of overflow of a shared buffer during system's first busy cycle). Stability of $X$ implies $p_n$ decays exponentially in $n$ when the process starts off $\partial A_n.$ We show that, for $x_n= \lfloor nx \rfloor$, $x \in {\mathbb R}_+^2$, $x(1)+x(2) \le 1$, $x(1) > 0$, $P_{(n-x_n(1),x_n(2))}( τ< \infty)$ approximates $P_{x_n}(τ_n < τ_0)$ with exponentially vanishing relative error. Let $r = (λ_1 + λ_2)/(μ_1 + μ_2)$; for $r^2 < ρ_2$ and $ρ_1 \neq ρ_2$, we construct a class of harmonic functions from single and conjugate points on a characteristic surface of $Y$ with which $P_y(τ< \infty)$ can be approximated with bounded relative error. For $r^2 = ρ_1 ρ_2$, we obtain $P_y(τ< \infty) = r^{y(1)-y(2)} +\frac{r(1-r)}{r-ρ_2}\left( ρ_1^{y(1)} - r^{y(1)-y(2)} ρ_1^{y(2)}\right).$

math.PR↗

arXiv2018

Approximation of Excessive Backlog Probabilities of Two Tandem Queues

Let $X$ be the constrained random walk on ${\mathbb Z}_+^2$ taking the steps $(1,0)$, $(-1,1)$ and $(0,-1)$ with probabilities $λ< (μ_1\neq μ_2)$; in particular, $X$ is assumed stable. Let $τ_n$ be the first time $X$ hits $\partial A_n = \{x:x(1)+x(2) = n \}$ For $x \in {\mathbb Z}_+^2, x(1) + x(2) < n$, the probability $p_n(x)= P_x( τ_n < τ_0)$ is a key performance measure for the queueing system represented by $X$. Let $Y$ be the constrained random walk on ${\mathbb Z} \times {\mathbb Z}_+$ with increments $(-1,0)$, $(1,1)$ and $(0,-1)$. Let $τ$ be the first time that the components of $Y$ equal each other. We derive the following explicit formula for $P_y(τ< \infty)$: \[ P_y(τ< \infty) = W(y)= ρ_2^{y(1)-y(2)} + \frac{μ_2 - λ}{μ_2 - μ_1} ρ_1^{ y(1)-y(2)} ρ_1^{y(2)} + \frac{μ_2-λ}{μ_1 -μ_2} ρ_2^{y(1)-y(2)} ρ_1^{y(2)}, \] where, $ρ_i = λ/μ_i$, $i=1,2$, $y \in {\mathbb Z}\times{ \mathbb Z}_+$, $y(1) > y(2)$, and show that $W(n-x_n(1),x_n(2))$ approximates $p_n(x_n)$ with relative error {\em exponentially decaying} in $n$ for $x_n = \lfloor nx \rfloor$, $x \in {\mathbb R}_+^2$, $0 < x(1) + x(2) < 1$. The steps of our analysis: 1) with an affine transformation, move the origin $(0,0)$ to $(n,0)$ on $\partial A_n$; let $n\nearrow \infty$ to remove the constraint on the $x(2)$ axis; this step gives the limit {\em unstable} /{\em transient} constrained random walk $Y$ and reduces $P_{x}(τ_n < τ_0)$ to $P_y(τ< \infty)$; 2) construct a basis of harmonic functions of $Y$ and use it to apply the superposition principle to compute $P_y(τ< \infty).$ The construction involves the use of conjugate points on a characteristic surface associated with the walk $X$. The proof that the relative error decays exponentially uses a sequence of subsolutions of a related HJB equation on a manifold.

math.PR↗

arXiv2017

Stationary analysis of a single queue with remaining service time dependent arrivals

We study a generalization of the $M/G/1$ system (denoted by $rM/G/1$) with independent and identically distributed (iid) service times and with an arrival process whose arrival rate $λ_0f(r)$ depends on the remaining service time $r$ of the current customer being served. We derive a natural stability condition and provide a stationary analysis under it both at service completion times (of the queue length process) and in continuous time (of the queue length and the residual service time). In particular, we show that the stationary measure of queue length at service completion times is equal to that of a corresponding $M/G/1$ system. For $f > 0$ we show that the continuous time stationary measure of the $rM/G/1$ system is linked to the $M/G/1$ system via a time change. As opposed to the $M/G/1$ queue, the stationary measure of queue length of the $rM/G/1$ system at service completions differs from its marginal distribution under the continuous time stationary measure. Thus, in general, arrivals of the $rM/G/1$ system do not see time averages. We derive formulas for the average queue length, probability of an empty system and average waiting time under the continuous time stationary measure. We provide examples showing the effect of changing the reshaping function on the average waiting time.

math.PR↗

arXiv2016

Backward Stochastic Differential Equations with Nonmarkovian Singular Terminal Values

We solve a class of BSDE with a power function $f(y) = y^q$, $q > 1$, driving its drift and with the terminal boundary condition $ ξ= \infty \cdot \mathbf{1}_{B(m,r)^c}$ (for which $q > 2$ is assumed) or $ ξ= \infty \cdot \mathbf{1}_{B(m,r)}$, where $B(m,r)$ is the ball in the path space $C([0,T])$ of the underlying Brownian motion centered at the constant function $m$ and radius $r$. The solution involves the derivation and solution of a related heat equation in which $f$ serves as a reaction term and which is accompanied by singular and discontinuous Dirichlet boundary conditions. Although the solution of the heat equation is discontinuous at the corners of the domain the BSDE has continuous sample paths with the prescribed terminal value.

math.PR↗

arXiv2015

Exit Probabilities and Balayage of Constrained Random Walks

Let $X$ be the constrained random walk on ${\mathbb Z}_+^d$ representing the queue lengths of a stable Jackson network and $x$ its initial position. Let $τ_n$ be the first time the sum of the components of $X$ equals $n$. $p_n \doteq P_x(τ_n < τ_0)$ is a key performance measure for the queueing system represented by $X$, stability implies $p_n\rightarrow 0$ exponentially. Currently the only analytic method available to approximate $p_n$ is large deviations analysis, which gives the exponential decay rate of $p_n$. Finer results are available via rare event simulation. The present article develops a new method to approximate $p_n$ and related expectations. The method has two steps: 1) with an affine transformation, move the origin onto the exit boundary of $τ_n$, take limits to remove some of the constraints on the dynamics, this yields a limit unstable constrained walk $Y$ 2) Construct a basis of harmonic functions of $Y$ and use them to apply the classical superposition principle of linear analysis. The basis functions are linear combinations of $\log$-linear functions and come from solutions of "harmonic systems," which are graphs whose vertices represent points on the "characteristic surface" of $Y$, the edges between the vertices represent conjugacy relations between the points, the loops represent membership in "the boundary characteristic surfaces." Using our method we derive explicit, simple and almost exact formulas for $P_x(τ_n < τ_0)$ for $d$-tandem queues, similar to the product form formulas for the stationary distribution of $X$. The same method allows us to approximate the Balayage operator mapping $f$ to $x \rightarrow {\mathbb E}_x \left[ f(X_{τ_n}) 1_{\{τ_n < τ_0\}} \right]$ for a range of stable constrained random walks in $2$ dimensions. We indicate how the ideas of the paper relate to more general processes and exit boundaries.

math.PR↗

arXiv2014

Analysis of Push-type Epidemic Data Dissemination in Fully Connected Networks

Consider a fully connected network of nodes, some of which have a piece of data to be disseminated to the whole network. We analyze the following push-type epidemic algorithm: in each push round, every node that has the data, i.e., every infected node, randomly chooses $c \in {\mathbb Z}_+$ other nodes in the network and transmits, i.e., pushes, the data to them. We write this round as a random walk whose each step corresponds to a random selection of one of the infected nodes; this gives recursive formulas for the distribution and the moments of the number of newly infected nodes in a push round. We use the formula for the distribution to compute the expected number of rounds so that a given percentage of the network is infected and continue a numerical comparison of the push algorithm and the pull algorithm (where the susceptible nodes randomly choose peers) initiated in an earlier work. We then derive the fluid and diffusion limits of the random walk as the network size goes to $\infty$ and deduce a number of properties of the push algorithm: 1) the number of newly infected nodes in a push round, and the number of random selections needed so that a given percent of the network is infected, are both asymptotically normal 2) for large networks, starting with a nonzero proportion of infected nodes, a pull round infects slightly more nodes on average 3) the number of rounds until a given proportion $λ$ of the network is infected converges to a constant for almost all $λ\in (0,1)$. Numerical examples for theoretical results are provided.

math.PR↗

arXiv2014

Joint Hitting-Time Densities for Finite State Markov Processes

For a finite state Markov process and a finite collection $\{ Γ_k, k \in K \}$ of subsets of its state space, let $τ_k$ be the first time the process visits the set $Γ_k$. We derive explicit/recursive formulas for the joint density and tail probabilities of the stopping times $\{ τ_k, k \in K\}$. The formulas are natural generalizations of those associated with the jump times of a simple Poisson process. We give a numerical example and indicate the relevance of our results to credit risk modeling.

math.PR↗

arXiv2009

Approximation of Bounds on Mixed Level Orthogonal Arrays

Mixed level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao and Gilbert-Varshamov type bounds for mixed level orthogonal arrays. The computational complexity of the terms involved in these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provide the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm uses a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton Jacobi Bellman equation associated with the limit problem.

math.ST↗

arXiv2008

Asymptotically Optimal Importance Sampling for Jackson Networks with a Tree Topology

Importance sampling (IS) is a variance reduction method for simulating rare events. A recent paper by Dupuis, Wang and Sezer (Ann. App. Probab. 17(4):1306- 1346, 2007) exploits connections between IS and stochastic games and optimal control problems to show how to design and analyze simple and efficient IS algorithms for various overflow events for tandem Jackson networks. The present paper uses the same approach to build asymptotically optimal IS schemes for stable open Jackson networks with a tree topology. Customers arrive at the single root of the tree. The rare overflow event we consider is the following: given that initially the network is empty, the system experiences a buffer overflow before returning to the empty state. Two types of buffer structures are considered: 1) A single system-wide buffer of size $n$ shared by all nodes, 2) each node $i$ has its own buffer of size $β_i n$, $β_i \in (0,1)$.

math.PR↗

arXiv2007

Dynamic importance sampling for queueing networks

Importance sampling is a technique that is commonly used to speed up Monte Carlo simulation of rare events. However, little is known regarding the design of efficient importance sampling algorithms in the context of queueing networks. The standard approach, which simulates the system using an a priori fixed change of measure suggested by large deviation analysis, has been shown to fail in even the simplest network setting (e.g., a two-node tandem network). Exploiting connections between importance sampling, differential games, and classical subsolutions of the corresponding Isaacs equation, we show how to design and analyze simple and efficient dynamic importance sampling schemes for general classes of networks. The models used to illustrate the approach include $d$-node tandem Jackson networks and a two-node network with feedback, and the rare events studied are those of large queueing backlogs, including total population overflow and the overflow of individual buffers.

math.PR↗

arXiv2002

A remark on unified error exponents: Hypothesis testing, data compression and measure concentration

Let A be finite set equipped with a probability distribution P, and let M be a "mass" function on A. A characterization is given for the most efficient way in which A^n can be covered using spheres of a fixed radius. A covering is a subset C_n of A^n with the property that most of the elements of A^n are within some fixed distance from at least one element of C_n, and "most of the elements" means a set whose probability is exponentially close to one (with respect to the product distribution P^n). An efficient covering is one with small mass M^n(C_n). With different choices for the geometry on A, this characterization gives various corollaries as special cases, including Marton's error-exponents theorem in lossy data compression, Hoeffding's optimal hypothesis testing exponents, and a new sharp converse to some measure concentration inequalities on discrete spaces.

math.PR↗