arXiv subjects

Umut Çetin

Publications and source records attributed to Umut Çetin.

At least 19 recordsLinked to original sources

arXiv2026

When large trades are not (automatically) news: liquidity tail risk and price discovery

We examine how heavy-tailed liquidity demand changes price discovery in a sequential limit order book with asymmetric information. In our setting, liquidity suppliers observe aggregate order flow, not its decomposition into informed demand and uninformed liquidity shocks. With heavy-tailed uninformed aggregated order flow, large trades remain plausibly uninformed over a wider range of depths, flattening price impact and slowing learning; sufficiently extreme trades can nevertheless become informative. We characterize equilibrium through a non-linear fixed point equation for the marginal-cost schedule; heavy-tailed uninformed aggregated order flow invalidates the monotonicity and compactness arguments available under Gaussianity. Therefore, we establish fixed-point existence within a tail-controlled class, prove posterior consistency for liquidity suppliers in the presence of endogenous dependent order flow, and derive tail asymptotics for marginal costs, informed demand, and aggregate order flow. Additionally, we obtain eventual informed-demand dominance and eventual monotonicity of the book in the far tails. Empirically, using 10-level AAPL data, we document farther-out crossover diagnostics and persistent bid-ask spreads following large heavy-tailed trades.

q-fin.TR↗

arXiv2026

A class of Markov processes with resetting and applications to cybersecurity

We introduce a class of piecewise deterministic Markov processes with resetting, motivated by self-exciting models of cyber attacks. Under assumptions reminiscent of ruin theory, we prove the existence and uniqueness of an invariant distribution and derive its density explicitly, thereby establishing ergodicity of the process. We then formulate an associated long-run average control problem in which resetting acts as the intervention mechanism. For exponentially distributed jump sizes, the model becomes analytically tractable, allowing an explicit characterization of the invariant distribution and of the optimal intervention policy.

math.PR↗

arXiv2026

A Limit Order Market with Uncertain Informed Trading Participation

We study a one period limit order market with informed traders, noise traders, and competitive liquidity suppliers, in which the number of informed traders is random. Liquidity suppliers know the distribution of the informed trader count, but not its realization, and therefore face uncertainty about both the presence and the intensity of informed trading. We characterize equilibrium by a fixed point integral equation for the marginal cost function and establish existence of equilibrium for bounded asset values. We then analyse large order asymptotics. For bounded asset values with power law endpoint behaviour, equilibrium price impact follows a power law whose exponent is determined jointly by the asset value tail and the full distribution of the informed trader count. In particular, this exponent is not determined by the expected number of informed traders alone. In the light endpoint regime, price impact is instead logarithmic. Finally, we solve the fixed point numerically across several asset value and informed trader count distributions. The numerical results are consistent with the theoretical asymptotics in the cases covered by the theory and provide comparative statics beyond them.

q-fin.TR↗

arXiv2026

Schrödinger's problem with constraints

Motivated by the connection between the Kyle equilibrium with static private signal and the Brownian bridge, we study a much broader class of bridges that allow one to consider more general equilibrium models, for example ones including trading costs and default risk. We show that such bridges are solutions to problems of the Schrödinger-type. Leveraging this connection, we obtain that the equilibria in models with trading costs converge to equilibria in the classical Kyle model.

math.PR↗

arXiv2023

Minimal subharmonic functions and related integral representations

A Choquet-type integral representation result for non-negative subharmonic functions of a one-dimensional regular diffusion is established. The representation allows in particular an integral equation for strictly positive subharmonic functions that is driven by the Revuz measure of the associated continuous additive functional. Moreover, via the aforementioned integral equation, one can construct an {\em \Ito-Watanabe pair} $(g,A)$ that consist of a subharmonic function $g$ and a continuous additive functional $A$ is with Revuz measure $μ_A$ such that $g(X)\exp(-A)$ is a local martingale. Changes of measures associated with \Ito-Watanabe pairs are studied and shown to modify the long term behaviour of the original diffusion process to exhibit transience.

math.PR↗

arXiv2023

Insider trading with penalties, entropy and quadratic BSDEs

Kyle model in continuous time where the insider may be subject to legal penalties is considered. In equilibrium the insider internalises this legal risk by trading less aggressively. The equilibrium is characterised via the solution of a backward stochastic differential equation (BSDE) whose terminal condition is determined as the fixed point of a non-linear operator in equilibrium. The insider's expected penalties in equilibrium is non-monotone in the fee structure and is given by the relative entropy of the law of a particular h-transformation of Brownian motion.

math.PR↗

arXiv2022

Order routing and market quality: Who benefits from internalisation?

We analyse two models of liquidity provision to determine the retail traders' preference for marketable order routing. Order internalization is captured by a model of market makers competing for the retail order flow in a Bertrand fashion. On the other hand, the price-taking competitive liquidity providers characterize the open exchange model. We show that, when liquidity providers are risk averse, routing of the marketable orders to the wholesalers is preferred by all retail traders: informed, uninformed and noisy. The unwillingness of liquidity providers to bear risk causes the strategic trader (informed or not) to absorb large shocks in their inventories. This results in mean reverting inventories, price reversal, and lower market depth. The equilibria in both models coincide with Kyle (1985) when liquidity providers are risk neutral. We also identify a universal parameter that allows comparison of market liquidity, profit and value of information across different markets.

q-fin.TR↗

arXiv2021

Speeding up the Euler scheme for killed diffusions

Let $X$ be a linear diffusion taking values in $(\ell,r)$ and consider the standard Euler scheme to compute an approximation to $\mathbb{E}[g(X_T)\mathbf{1}_{[T<ζ]}]$ for a given function $g$ and a deterministic $T$, where $ζ=\inf\{t\geq 0: X_t \notin (\ell,r)\}$. It is well-known since \cite{GobetKilled} that the presence of killing introduces a loss of accuracy and reduces the weak convergence rate to $1/\sqrt{N}$ with $N$ being the number of discretisatons. We introduce a drift-implicit Euler method to bring the convergence rate back to $1/N$, i.e. the optimal rate in the absence of killing, using the theory of recurrent transformations developed in \cite{rectr}. Although the current setup assumes a one-dimensional setting, multidimensional extension is within reach as soon as a systematic treatment of recurrent transformations is available in higher dimensions.

math.NA↗

arXiv2021

On pricing rules and optimal strategies in general Kyle-Back models

The folk result in Kyle-Back models states that the value function of the insider remains unchanged when her admissible strategies are restricted to absolutely continuous ones. In this paper we show that, for a large class of pricing rules used in current literature, the value function of the insider can be finite when her strategies are restricted to be absolutely continuous and infinite when this restriction is not imposed. This implies that the folk result doesn't hold for those pricing rules and that they are not consistent with equilibrium. We derive the necessary conditions for a pricing rule to be consistent with equilibrium and prove that, when a pricing rule satisfies these necessary conditions, the insider's optimal strategy is absolutely continuous, thus obtaining the classical result in a more general setting. This, furthermore, allows us to justify the standard assumption of absolute continuity of insider's strategies since one can construct a pricing rule satisfying the necessary conditions derived in the paper that yield the same price process as the pricing rules employed in the modern literature when insider's strategies are absolutely continuous.

q-fin.TR↗

arXiv2020

Informed trading, limit order book and implementation shortfall: equilibrium and asymptotics

We propose a static equilibrium model for limit order book where profit-maximizing investors receive an information signal regarding the liquidation value of the asset and execute via a competitive dealer with random initial inventory, who trades against a competitive limit order book populated by liquidity suppliers. We show that an equilibrium exists for bounded signal distributions, obtain closed form solutions for Bernoulli-type signals and propose a straightforward iterative algorithm to compute the equilibrium order book for the general case. We obtain the exact analytic asymptotics for the market impact of large trades and show that the functional form depends on the tail distribution of the private signal of the insiders. In particular, the impact follows a power law if the signal has fat tails while the law is logarithmic in case of lighter tails. Moreover, the tail distribution of the trade volume in equilibrium obeys a power law in our model. We find that the liquidity suppliers charge a minimum bid-ask spread that is independent of the amount of `noise' trading but increasing in the degree of informational advantage of insiders in equilibrium. The model also predicts that the order book flattens as the amount of noise trading increases converging to a model with proportional transactions costs.. Competition among the insiders leads to aggressive trading causing the aggregate profit to vanish in the limiting case $N\to\infty$. The numerical results also show that the spread increases with the number of insiders keeping the other parameters fixed. Finally, an equilibrium may not exist if the liquidation value is unbounded. We conjecture that existence of equilibrium requires a sufficient amount of competition among insiders if the signal distribution exhibit fat tails.

q-fin.TR↗

arXiv2018

Integral representation of subharmonic functions and optimal stopping with random discounting

An integral representation result for strictly positive subharmonic functions of a one-dimensional regular diffusion is established. More precisely, any such function can be written as a linear combination of an increasing and a decreasing subharmonic function that solve an integral equation \[ g(x)=a + \int v(x,y)μ_A(dy) + κs(x), \] where $a>0$, $κ\in \mathbb{R}$, $s$ is a scale function of the diffusion, $μ_A$ is a Radon measure, and $v$ is a kernel that is explicitly determined by the scale function. This integral equation in turn allows one construct a pair $(g,A)$ such that $g$ is a subharmonic function, $A$ is a continuous additive functional with Revuz measure $μ_A$ and $g(X)\exp(-A)$ is a local martingale. The changes of measures associated with such pairs are studied and shown to modify the long term behaviour of the original diffusion process to exhibit transience. Theory is illustrated via examples that in particular contain a sequence of measure transformations that render the diffusion irregular in the limit by breaking the state space into distinct regions with soft and hard borders. Finally, the theory is applied to find an "explicit" solution to an optimal stopping problem with random discounting.

math.PR↗

arXiv2018

Diffusion transformations, Black-Scholes equation and optimal stopping

We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the first exit times of diffusions, which is recently characterised by Karatzas and Ruf \cite{KR} as the minimal solution of an appropriate Cauchy problem under more stringent conditions. A particular limit of these transformations also turn out to be instrumental in characterising the stochastic solutions of Cauchy problems defined by the generators of strict local martingales, which are well-known for not having unique solutions even when one restricts solutions to have linear growth. Using an appropriate diffusion transformation we show that the aforementioned stochastic solution can be written in terms of the unique classical solution of an {\em alternative} Cauchy problem with suitable boundary conditions. This in particular resolves the long-standing issue of non-uniqueness with the Black-Scholes equations in derivative pricing in the presence of {\em bubbles}. Finally, we use these path transformations to propose a unified framework for solving explicitly the optimal stopping problem for one-dimensional diffusions with discounting, which in particular is relevant for the pricing and the computation of optimal exercise boundaries of perpetual American options.

math.PR↗

arXiv2017

Path transformations for local times of one-dimensional diffusions

Let $X$ be a regular one-dimensional transient diffusion and $L^y$ be its local time at $y$. The stochastic differential equation (SDE) whose solution corresponds to the process $X$ conditioned on $[L^y_{\infty}=a]$ for a given $a\geq 0$ is constructed and a new path decomposition result for transient diffusions is given. In the course of the construction of the SDE the concept of {\em recurrent transformation} is introduced and {\em Bessel-type motions} as well as their SDE representations are studied. A remarkable link between an $h$-transform with a minimal excessive function and recurrent transformations is found, which, as a by-product, gives a useful representation of last passage times as a mixture of first hitting times. Moreover, the Engelbert-Schmidt theory for the weak solutions of one dimensional SDEs is extended to the case when the initial condition is an entrance boundary for the diffusion. This extension was necessary for the construction of the Bessel-type motion which played an essential part in the SDE representation of $X$ conditioned on $[L^y_{\infty}=a]$.

math.PR↗

arXiv2017

Financial equilibrium with asymmetric information and random horizon

We study in detail and explicitly solve the version of Kyle's model introduced in a specific case in \cite{BB}, where the trading horizon is given by an exponentially distributed random time. The first part of the paper is devoted to the analysis of time-homogeneous equilibria using tools from the theory of one-dimensional diffusions. It turns out that such an equilibrium is only possible if the final payoff is Bernoulli distributed as in \cite{BB}. We show in the second part that the signal of the market makers use in the general case is a time-changed version of the one that they would have used had the final payoff had a Bernoulli distribution. In both cases we characterise explicitly the equilibrium price process and the optimal strategy of the informed trader. Contrary to the original Kyle model it is found that the reciprocal of market's depth, i.e. Kyle's lambda, is a uniformly integrable supermartingale. While Kyle's lambda is a potential, i.e. converges to $0$, for the Bernoulli distributed final payoff, its limit in general is different than $0$.

q-fin.MF↗

arXiv2016

Linear inverse problems for Markov processes and their regularisation

We study the solutions of the inverse problem \[ g(z)=\int f(y) P_T(z,dy) \] for a given $g$, where $(P_t(\cdot,\cdot))_{t \geq 0}$ is the transition function of a given Markov process, $X$, and $T$ is a fixed deterministic time, which is linked to the solutions of the ill-posed Cauchy problem \[ u_t + A u=0, \qquad u(0,\cdot)=g, \] where $A$ is the generator of $X$. A necessary and sufficient condition ensuring square integrable solutions is given. Moreover, a family of regularisations for the above problems is suggested. We show in particular that these inverse problems have a solution when $X$ is replaced by $ξX + (1-ξ)J$, where $ξ$ is a Bernoulli random variable, whose probability of success can be chosen arbitrarily close to $1$, and $J$ is a suitably constructed jump process.

math.PR↗

arXiv2016

Markovian Nash equilibrium in financial markets with asymmetric information and related forward-backward systems

This paper develops a new methodology for studying continuous-time Nash equilibrium in a financial market with asymmetrically informed agents. This approach allows us to lift the restriction of risk neutrality imposed on market makers by the current literature. It turns out that, when the market makers are risk averse, the optimal strategies of the agents are solutions of a forward-backward system of partial and stochastic differential equations. In particular, the price set by the market makers solves a nonstandard "quadratic" backward stochastic differential equation. The main result of the paper is the existence of a Markovian solution to this forward-backward system on an arbitrary time interval, which is obtained via a fixed-point argument on the space of absolutely continuous distribution functions. Moreover, the equilibrium obtained in this paper is able to explain several stylized facts which are not captured by the current asymmetric information models.

math.PR↗

arXiv2015

Markov bridges: SDE representation

Let $X$ be a Markov process taking values in $\mathbf{E}$ with continuous paths and transition function $(P_{s,t})$. Given a measure $μ$ on $(\mathbf{E}, \mathscr{E})$, a Markov bridge starting at $(s,\varepsilon_x)$ and ending at $(T^*,μ)$ for $T^* <\infty$ has the law of the original process starting at $x$ at time $s$ and conditioned to have law $μ$ at time $T^*$. We will consider two types of conditioning: a) {\em weak conditioning} when $μ$ is absolutely continuous with respect to $P_{s,t}(x,\cdot)$ and b) {\em strong conditioning} when $μ=\varepsilon_z$ for some $z \in \mathbf{E}$. The main result of this paper is the representation of a Markov bridge as a solution to a stochastic differential equation (SDE) driven by a Brownian motion in a diffusion setting. Under mild conditions on the transition density of the underlying diffusion process we establish the existence and uniqueness of weak and strong solutions of this SDE.

math.PR↗

arXiv2015

On certain integral functionals of squared Bessel processes

Let $X$ be a squared Bessel process. Following a Feynman-Kac approach, the Laplace transforms of joint laws of $(U, \int_0^{R_y}X_s^p\,ds)$ are studied where $R_y$ is the first hitting time of $y$ by $X$ and $U$ is a random variable measurable with respect to the history of $X$ until $R_y$. A subset of these results are then used to solve the associated small ball problems for $\int_0^{R_y}X_s^p\,ds$ and determine a Chung's law of iterated logarithm. $(\int_0^{R_y}X_s^p\,ds)$ is also considered as a purely discontinuous increasing Markov process and its infinitesimal generator is found. The findings are then used to price a class of exotic derivatives on interest rates and determine the asymptotics for the prices of some put options that are only slightly in-the-money.

math.PR↗