Acknowledgements

My first thanks go to my grandmother, Lyudmila, who left her job and her friends to move across the world and raise me through the first years of my life. Without her, I would not have been in a position to write this dissertation.

I am grateful to my advisor, Tomoyuki Ichiba, for his guidance and support throughout my doctoral studies, and to Javier Birchenall for valuable discussions on the economic applications of this work. I thank my committee members Jean-Pierre Fouque and Michael Ludkovski for their time and feedback.

I thank my girlfriend, Viki, for her love and for taking on my physical and mental burden while I wrote. Without her, I would not have been able to finish this dissertation.

I am grateful to the rest of my family: my sister Anna, my cousin Max, my mother Lena, my father Leonard, my aunt Ira, my uncle Andrew, and my late grandfather Genrikh. Finally, I thank my friends Thiha Aung, Olivier Mulkin, Yan Lashchev, and Daniel Naylor for their encouragement and companionship throughout my time as a doctoral student.

Vita

Samuel Babichenko

Education

@p0.87r@ Ph.D. in Statistics and Applied Probability, emphasis in Financial Mathematics, UC Santa Barbara & 2026
M.A. in Statistics, UC Santa Barbara & 2026
B.S. in Mathematics, UC San Diego & 2021
A.S. in Physics, Chemistry, and Mathematics, Orange Coast College & 2019

Professional Employment

@p0.78r@ Teaching Assistant, Department of Statistics and Applied Probability, UC Santa Barbara & 2021–2026

Presentations

“Forecasting and Manipulating the Forecasts of Others.” Southern California Quantitative Finance Forum (SCQF), UC Santa Barbara, April 14, 2026.

Publications

Babichenko, Sam. “Forecasting and Manipulating the Forecasts of Others.”
arXiv preprint arXiv:2603.12140, 2026.

Fields of Study

Major Field: Statistics and Applied Probability.

Studies in dynamic games with private information, stochastic filtering, and market microstructure with Professor Tomoyuki Ichiba.

Abstract

=plus minus 1.5 Dynamic games with dispersed private information look hard, even when linearized, because each player needs forecasts of others’ forecasts, without end. The regress comes from a loop. Actions move the state, the state generates observations, observations update beliefs, and beliefs determine actions. Current approaches become tractable by cutting the loop. This dissertation builds a noise-state calculus that keeps the loop intact.

Each player keeps estimates of the primitive shocks instead of belief hierarchies about the endogenous state. Under perfect information, these estimates are the shocks themselves, and the calculus reduces to ordinary impulse responses. Beliefs, prices, and policy rules are deterministic impulse responses in continuous-time linear-quadratic-Gaussian games. Equilibrium is a fixed point in those functions and is solved numerically. A finite-deviation identity verifies a computed Nash equilibrium against arbitrary admissible deviations.

The information wedge is the shadow price of changing an opponent’s beliefs. It breaks the separation principle and vanishes when the loop is cut. In a two-player benchmark, a planner can starve an inefficient player by reallocating signal precision.

Signals and actions may arrive late without changing the calculus. Stationary systems describe kernels in terms of ages and measure distance from finite-dimensionality. In Kyle–Back markets, traders differ in signal precision within and across markets and face trader-specific price impact. On symmetric networks of local markets, shocks average out but strategic restraint does not, lowering mean orders while nearly invisible in responses to aggregate shocks. When some can identify deviations and others cannot, naive players attribute effects to ordinary shocks while privy players respond to labeled deviations. In computed markets, strategic market makers put more weight on inventory in their quotes when order flow is public and bear lower real inventory cost, with price discovery unchanged.

Notation

Times, indices, and conventions

$t, s, τ$	calendar times
$u, r$	primitive source times
$a, b$	ages; $ℓ$ the two-sided lag
$X_{t} (u)$ vs. $x (a)$	finite-horizon objects carry calendar time as a subscript and source time as an argument; stationary objects are functions of ages or lags
$X_{t}, X_{t} (u)$	a process and its kernel share a letter: the state, and its response to the shock born at $u$
$D_{t}^{i}, D_{t}^{i} (u)$	the control and its policy kernel
$B_{X}^{X} (t), G^{XX, i} (t), σ (t)$	exogenous coefficients carry time in parentheses
$X_{t}, D_{t}^{i} (u), H_{t}^{k, i} (u)$	processes, kernels, and equilibrium gains carry it as a subscript
$i, j, k \in I$	player indices, as superscripts
$x (a)$ vs. $X_{t} (u)$	lowercase letters are the stationary current-value counterparts of uppercase finite-horizon objects
$D_{W, t}^{i} (u), d_{W}^{i} (a)$	a subscript $W$ marks a kernel written in primitive-shock coordinates
$F, B, \dots$	calligraphic letters are sets, $σ$ -fields, and operators; the wedge $V_{t}^{i}$ is the one calligraphic process

Accents

$W, X$	hats are conditional expectations
$X, C$	tildes are unresolved parts
$\overset{ˉ}{D}, \overset{ˉ}{h}$	bars are deterministic means
$\overset{ˇ}{f}$	checks are transforms
$C$ ; $C_{t}^{i} (u, r)$ ; $C_{t}^{i, Y} (u)$	an observation kernel; a conditional covariance; the unresolved observation row

Letters whose meaning is local to a chapter

$z$	a control spike (Chapter 4’s raw order spike is the same object) or a transform variable, never a time
$v = B_{i}^{X} (t) z$ ; $v_{i} (s)$	the state impulse a spike induces; a deviation seed (Chapter 6)
$δ, δ_{i}$ ; $δ w_{t}^{k} (u)$	a first variation, one with respect to an origin- $i$ seed; the noise-state perturbation in density form
$Δ_{i}, Δ_{r}$ ; $Δ$	observation delays; in the appendix to Chapter 1, a finite difference
$ζ$	the noise-trader variance asymmetry, $Σ_{Z} = diag (1 + ζ, 1 - ζ)$ (Chapter 4)
$v, w$ ; $r$	vertices and a group displacement (Chapter 5)
$q$	the number of assets (Chapter 4); the delayed source date $q_{k} = s - Δ_{k}$ (Chapter 2)
$(p, q)$	Laplace variables (Chapter 3); naive and privy indices (Chapter 6)
$j ⊵ i$ ; $P_{i}, N_{i}$	player $j$ observes deviations whose origin is player $i$ ; the players privy and naive to an origin- $i$ deviation (Chapter 6)

The following table collects the objects carried across chapters and marks where each is developed. Chapter-specific objects, proof-local variables, and one-use abbreviations are defined in place.

Core finite-horizon objects

Symbol	Object	Scope
$W = (W^{0}, W^{1}, \dots, W^{n})$	primitive Brownian shock path, with common and player-specific coordinates	General
$I; i, j, k$	player set and generic player indices	General
$F_{t}, F_{t}^{i}$	primitive and player- $i$ information filtrations	General
$E^{i}$	block selector for player $i$ ’s noise channel, $W^{i} = E^{i} W$	General
$X_{t}, X_{t} (u)$	physical state and its primitive response to the shock born at $u$	Ch. 1
$B_{X}^{X} (t), B_{i}^{X} (t), σ (t)$	drift, control gains, and noise loading of the state, $d X_{t} = (B_{X}^{X} (t) X_{t} + \sum_{i} B_{i}^{X} (t) D_{t}^{i}) d t + σ (t) d W_{t}^{0}$	Ch. 1
$Y_{t}^{i}, B_{X}^{Y, i}, P^{i} (t)$	player $i$ ’s observation process, observation map, and precision $B_{X}^{Y, i ⊤} B_{X}^{Y, i}$	Ch. 1
$d I_{t}^{i}$	player $i$ ’s innovation, $d Y_{t}^{i} - B_{X}^{Y, i} X_{t}^{i} d t$	Ch. 1

Symbol	Object	Scope
$W_{t}^{i} (u)$	noise-state: player $i$ ’s conditional estimate of $W_{u}$ at time $t$	Ch. 1
$Π^{i}, F_{t}^{i} (u, s)$	direct projection and filter kernel; together, the blueprint of player $i$ ’s beliefs	Ch. 1
$X_{t}^{i} (u)$	unresolved response: the part of $X_{t} (u)$ that player $i$ ’s information does not resolve	Ch. 1
$δ w_{t}^{k} (u)$	noise-state perturbation in density form, $d_{u} δ W_{t}^{k} (u) = δ w_{t}^{k} (u) d u$	Chs. 1, 2, 4, 6
$D_{t}^{i}, D_{t}^{i} (u), D_{W, t}^{i} (u)$	control, its policy kernel, and its control kernel in primitive-shock coordinates	Ch. 1
$H_{t}^{X, i}$	state price: player $i$ ’s adjoint to the physical state	Ch. 1
$H_{t}^{k, i} (u)$	belief price: player $i$ ’s adjoint to player $k$ ’s noise-state coordinate at source time $u$	Ch. 1
$V_{t}^{i}$	information wedge: the price of moving opponents’ noise-states, entering the dynamics of the state price	Chs. 1, 2, 4, 5, 6
$B^{k}, B^{k ⊤}$	filter contraction: integrates a belief price against player $k$ ’s own filter row; and its transpose	Chs. 1, 2, 4, 6
$ρ$	exponential discount rate	Chs. 3, 4

Finite-horizon and stationary forms

The stationary chapter changes coordinates rather than economic objects. Calendar dates become ages or lags, and finite-horizon uppercase objects become lowercase current-value objects; processes keep their letters:

Finite-horizon form	Stationary form	Interpretation
$X_{t} (s)$	$x (t - s) = x (a)$	state-response kernel
$X_{t}^{i} (s)$	$\tilde{x}^{i} (t - s) = \tilde{x}^{i} (a)$	unresolved response
$D_{t}^{i} (u)$	$d^{i} (t - u) = d^{i} (a)$	noise-state policy kernel
$D_{W, t}^{i} (s)$	$d_{W}^{i} (t - s) = d_{W}^{i} (a)$	control kernel in primitive-shock coordinates
$F_{t}^{i} (u, s)$	$f^{i} (t - u, t - s) = f^{i} (b, a)$	filter kernel
$W_{t}^{i} (u)$	$W_{t}^{i} (t - a)$	noise-state (a process; stays uppercase)
$H_{t}^{X, i}$	$\overset{ˉ}{h}^{X, i}$ and $h^{X, i} (ℓ)$	mean and two-sided state price
$H_{t}^{k, i} (u)$	$\overset{ˉ}{h}^{k, i} (b)$ and $h^{k, i} (b, ℓ)$	belief price
$V_{t}^{i}$	$\overset{v}{ˉ}^{i}$ and $v^{i} (ℓ)$	information wedge

Glossary

Term	Meaning	Where
Term	Meaning	Where
noise-state	a player’s conditional estimate of the primitive shock path, $W_{t}^{i} (u) = E [W_{u} ∣ F_{t}^{i}]$ ; the coordinates in which strategies, filters, and prices are written	Introduction; Thm. 1.5
blueprint	the pair $(Π^{i}, F^{i})$ of direct projection and filter kernel, the deterministic map from primitive shocks to a player’s estimated shocks	Thm. 1.5
unresolved response	the part $X_{t}^{i} (u)$ of the state’s impulse response that a player’s information has not resolved; the filtering gain	Sec. 1.3.2
loop, chord	the state–observation–belief–action feedback cycle, and the direct edge by which an action enters what others observe without passing through the state	Introduction, Fig. 1
spike	a marginal control impulse $z δ_{t}$ ; the deviation whose first variation gives the first-order conditions (Chapter 4’s raw order spike is the same object)	Sec. 1.2.3; Sec. 4.2
belief price	$H_{t}^{k, i} (u)$ , the marginal cost to player $i$ of moving player $k$ ’s estimate of the shock born at $u$	Thm. 1.8
information wedge	$V_{t}^{i}$ , the adjoint price of moving opponents’ noise-states; the only term through which belief manipulation enters the state-price equation	Thm. 1.8
birth	the entry of a new source coordinate into a filter or adjoint at the current date (diagonal birth); a delayed public signal adds a second birth when its news arrives, at $u = t - Δ_{i}$	Thm. 1.5; Ch. 2
disclosure window	the interval between a private action and the public arrival of the news that explains it	Ch. 2, Fig. 2.1
early peak	the peak in the stationary state price of a signal-noise shock a short lag after it arrives; a noise shock harms only through the responses it triggers, and no player has responded yet when it arrives	Sec. 3.2
seed	the displacement a spike leaves in observers’ filters, an old-history density plus a birth atom; in Chapter 6, the labeled state impulse a monitored deviation creates	Sec. 4.3; Def. 6.1
execution, position	the two parts of trading profit: execution is earned at the fill, the gap between the new quote and the price paid; position is earned afterwards on shares already held; their sum, the price-profit rate, is zero-sum across the market	Sec. 4.8
book gap	$E [Q_{t}^{i} (V_{t} - P_{t})]$ , the average undervaluation of the shares held; a level, not a flow	Sec. 4.8
monitoring relation	$j ⊵ i$ : player $j$ observes player $i$ ’s deviations and knows they are player $i$ ’s; reflexive and transitive	Def. 6.1
privy, naive	a player who sees a deviation, knows who made it, and responds to it; one who does not and treats its effect as more primitive shock	Sec. 6.4
blip of irrationality	a single forced departure after which the deviating player resumes equilibrium play; pins down what the deviator does after the blip, which is what privy players respond to	Sec. 6.4.2
transparent, opaque market	the market with and without published order flow, in which the trader is privy or naive to quote deviations	Sec. 6.8