RESEARCH STATEMENT ANDREY SARANTSEV

RESEARCH STATEMENT ANDREY SARANTSEV My research in Stochastic Finance and Stochastic Analysis is focused on the following topics: 1. Interacting Brownian particles with rank-dependent dynamics. This model was originally introduced for stock market modeling in [4] by Banner, Fernholz, Karatzas. It can serve as scaling limits of exclusion processes on the integer lattice Z. In addition, it is related to the McKean-Vlasov equation, which has widespread applications ranging from physics (plasma modeling) to social interactions studies; and to Sherrington-Kirkpatrick model of a spin glass (a disordered magnet). Similar models of ranked particles are related to random matrix theory and queueing theory. We study various aspects: whether these particles collide with each other (this question is related to potential theory and set capacity); stationary gap distributions and questions of convergence (this is related to ergodic theory); and scaling limits of these systems, which turn out to be stochastic heat equation or a Stefan free-boundary problem for the heat equation. 2. Stochastic Portfolio Theory. This theory was developed by Fernholz and presented in his book [23]. This theory does not require no arbitrage assumptions, as opposed to Arbitrage Pricing Theory. Unlike Modern Portfolio Theory developed by Markowitz, or Capital Asset Pricing Model, this theory deals with portfolios outperforming a benchmark almost surely, not on average. In joint work [38] with Karatzas, we use Stochastic Portfolio Theory to model systems with splits and mergers, with non-constant number of stocks. 3. Systemic risk in financial systems. We continue research by Carmona, Fouque, Sun in [12]. Banks exchange cash flows, borrow money from the outside economy at a certain interest rate r, and invest in (correlated) risky assets. We are interested in systemic risk: the probability that many banks will default: that is, will have net worth below a certain threshold. We study dependence of systemic risk on correlation between these risky assets, and on the interest rate r. 1. Interacting Brownian particles with rank-dependent dynamics Consider a finite system of N Brownian particles X i = (X i (t), t 0), i = 1,..., N, on the real line. Rank them from bottom to top: X (1) (t)... X (N) (t). As particles move, they can collide and exchange ranks. Assume the particle with current rank k moves as a Brownian motion with drift and diffusion coefficients g k, σk 2. If two particles exchange ranks, they also exchange drift and diffusion coefficients. Let us present the stochastic differential equation governing these particles, with 1(A) being the indicator function of an event A, and W 1,..., W N independent Brownian motions: N (1) dx i (t) = 1 (X i has rank k at time t) (g k dt + σ k dw i (t)), i = 1,..., N. This is called a system of competing Brownian particles, introduced in [4] for stock market modeling by Banner, Fernholz, Karatzas, see also [23, 24]. These systems serve as scaling limits of certain types of exclusion processes (asymmetrically colliding random walks), see [37] by Karatzas, Pal, Shkolnikov; are related to spin-glass models in physics, see [2] by Arguin, Aizenman, and [46] by Ruzmaikina, Aizenman; to the McKean-Vlasov equation (used for plasma modeling and population dynamics studies), see [35] by Jourdain, Reygner, and [16] by Dembo, Shkolnikov, Varadhan, Zeitouni; to eigenvalues of random matrices, see [6] by Baryshnikov, [50] by Sasamoto, Spohn, and [53] by Warren; and to queueing theory, [51] by Seppalainen. In particular, we would like to mention the following financial application. Let the capitalization of the ith stock at time t be S i (t) := exp(x i (t)). Define market weights: µ i (t) = S i (t), i = 1,..., N. S 1 (t) +... + S N (t) We can rank them from top to bottom: µ (1) (t)... µ (N) (t). An observed feature of real-world markets is that the log-log plot of ranked market weights (log n, log µ (n) (t)), n = 1,..., N, is linear and independent of time t. This property was captured by the model of competing Brownian particles in [13] by Chatterjee, Pal.

2 Andrey Sarantsev Research Statement Infinite systems X = (X i ) i 1 have bottom-to-top ranking X (1) (t) X (2) (t)... and an infinite sum in the right-hand side of (1). Motivation to study them comes from similar discrete systems on Z from [3] by Arratia, and the Harris model from [30], with infinitely many i.i.d. Brownian motions, starting from points at distance i.i.d. Exp(λ) from each other, for a fixed constant λ > 0. Here, Exp(λ) is the exponential distribution with rate λ and density λe λx dx on the positive half-line. A particlular case of the infinite system was first considered in [44] by Pal, Pitman, for the parameters: (2) g 1 = 1, g 2 = g 3 =... = 0, σ 1 = σ 2 =... = 1. We can rewrite (1) for parameters from (2) as follows: (3) dx i (t) = 1 ( X i (t) = X (1) (t) ) dt + dw i (t), i = 1, 2,... This system is called the infinite Atlas model. Here, the bottom particle moves as a Brownian motion with drift 1, all other particles move as driftless Brownian motions. The name comes from the ancient Atlas hero, which supported the sky on his shoulders, as the bottom particle X (1) pushes all other particles via its drift 1. Similarly, one can define the finite Atlas model with N particles. In a related Aldous problem: a unit drift is distributed among a finite collection of Brownian particles on R +, which are annihilated once they reach the origin. As proved in [52], the push-the-laggard strategy of distributing the drift maximizes the total number of surviving particles. We have studied various aspects of such systems. Instead of discussing all our papers on this topic, let us only mention the following results. A triple collision happened at time t if A simultaneous collision happened at time t if X (k 1) (t) = X (k) (t) = X (k+1) (t) for some k = 2,..., N 1. X (k) (t) = X (k+1) (t) and X (l) (t) = X (l+1) (t) for some k < l. This question is related to potential theory of Markov processes, which answers the question on whether a given process ever hits a given subset of its state space. A related concept in real analysis is capacity, see [7] by Bass. Another motivation to study this question is: For a given probability space, strong existence and pathwise uniqueness for (1) hold until the first triple collision. Whether they hold after this remains an open question. Partial results in this direction were obtained in papers [32, 33] by Ichiba, Karatzas, Shkolnikov. following necessary and sufficient condition was proved in my paper [47]: Theorem 1.1. There are a.s. no triple and simultaneous collisions at any time t > 0 if and only if (4) σk 2 1 ( σ 2 2 k 1 + σk+1 2 ), k = 2,..., N 1. If the condition (4) does not hold for a certain k = 2,..., N 1, then with positive probability there is a triple collision between particles with ranks k 1, k, and k + 1. The following corollary seems unexpected: Corollary 1.2. If there are a.s. no triple collisions, then there are a.s. no simultaneous collisions. As an example, for N = 4, σ 2 1 = σ 2 4 = 1, σ 2 2 = σ 2 3 = 2, there are a.s. no triple and simultaneous collisions. However, for σ 2 1 = σ 2 4 = 2, σ 2 2 = σ 2 3 = 1, there can be collisions of particles with ranks 1, 2, 3, and of particles with ranks 2, 3, 4. Condition (4) establishes the absence of all triple collisions. One can also seek the absence only of a triple collision of a particular kind, say between the three lowest-ranked particles; or collisions of four, five, or more particles. On this topic, see our papers [9, 34]. However, we could not find necessary and sufficient conditions, and this is still an open question. Let us present one of our results from [34] with Ichiba. Theorem 1.3. Fix an n 4. There are a.s. no collisions of n particles at any time t > 0 if max,...,n σ2 k < n 1 min 2,...,N σ2 k. The gap process for the system (1) is defined as an R N 1 + -valued process Z = (Z(t), t 0), with (5) Z(t) = (Z 1 (t),..., Z N 1 (t)), Z k (t) = X (k+1) (t) X (k) (t), k = 1,..., N 1, t 0. For the infinite system, this process takes values in R +. An important problem is whether there exists a stationary gap distribution: a probability measure π such that if Z(0) π, then Z(t) π for every t 0. This problem was The

Research Statement Andrey Sarantsev 3 solved in [44, 5] for finite systems. In particular, if the stationary distribution exists, then it is unique. For the finite Atlas model with N particles, this stationary distribution has the following form: N 1 ( (6) π (N) = Exp 2 N k ), N The Lebesgue density of (6) is a solution to a certain PDE in the positive orthant with Neumann boundary conditions. For infinite systems, we studied stationary gap distributions in [49] in our joint work with Tsai. As an example, consider the infinite Atlas model. Then for every a 0, the following is a stationary gap distribution: (7) π a := Exp(2 + ka). One particular case of a distribution from (7) is (8) π 0 := In fact, let N formally in (6). Since Exp(2). 2 N k 2 as N for every k, N it is natural to conjecture that π 0 from (8) serves as a stationary gap distribution for the infinite Atlas model. This was formally proved earlier in the paper [44] by Pal, Pitman, with a conjecture that π 0 is the unqiue stationary gap distribution for the infinite Atlas model. Thus, in our work [49] this conjecture was resolved in the negative. However, it is still an open question whether the distributions π a, a 0, from (7), and their mixtures, represent all stationary gap distributions for the infinite Atlas model. An interesting feature of the distribution π 0 from (8) is that it is invariant under shifts: all exponential components have the same rate 2. This is despite the intrinsic asymmetry of the infinite Atlas model: only the bottom-ranked particle has drift 1. For the distributions π a from (7) for a > 0, there are exponentially many particles between X (1) (t) and X (1) (t)+l, as L. This creates a pressure from above: although all particles, except the bottom one, have zero drift, the dense crowd of fluctuating Brownian particles creates an induced drift. Moreover, this induced negative drift overwhelmes the positive drift of the bottom particle, and E [ X (k) (t) X (k) (0) ] = a t, k = 1, 2,..., t 0. 2 For the finite Atlas model started from any initial distribution, the gap process Z(t) converges in law to π (N) as t. For the infinite Atlas model, this is an open question to find a limit in law (if it exists) of the gap process Z(t) as t. Since there are multiple stationary gap distributions, this limit must depend on the initial distribution. This problem was partially resolved in my paper [48]: For the initial value of the gap process stochastically larger than π 0, the gap process converges weakly to π 0 as time goes to infinity. See also the work [15] by Dembo, Jara, Olla on this question. Consider the particle X (1) (t) in the infinite Atlas model. We are interested in its long-term behavior as t. In my joint work [10] with Cabezas, Dembo, Sidoravicius, we solve the following problem. Assume we start the infinite Atlas model out of equilibrium: from the gap distribution Exp(λ) for λ 2. Then X (1) (t) t 1/2 k(λ) in law as t. where k(λ) can be found explicitly from λ. The scaled empirical distribution of particles converges to a solution to a certain Stefan free-boundary problem for the heat equation. One can compare this result with the scaling limit from [17] by Dembo, Tsai for initial gap distribution π 0. For H (0, 1), define the fractional Brownian motion B H = (B H (t), t 0) with Hurst parameter H as follows: the vector (B H (t 1 ),..., B H (t k )) is Gaussian for every t 1,..., t k 0, and EB H (t) = 0, EB H (t)b H (s) = 1 ( t 2H + s 2H t s 2H), t, s 0. 2 For H = 1/2, this is a Brownian motion. For H 1/2, this process has dependent increments. If the infinite Atlas model starts from the stationary gap distribution π 0, the bottom particle X (1) can be scaled as (9) (ε 1/4 X (1) (t/ε), t 0) cb 1/4, weakly, as ε 0.

4 Andrey Sarantsev Research Statement Here, c > 0 is some explicit constant. In particular, X (1) (t) is of order t 1/4 for large t. Moreover, the scaled empirical distribution of particles converges to a solution of the stochastic heat equation. The result (9) is reminiscent of the Harris model in [30]: start independent Brownian motions from the Poisson point process of constant intensity on the real line. Then the particle with a constant rank has fluctuations of order t 1/4. In other words, the unit drift of the bottom particle in the infinite Atlas model serves as a substitute for the lower half of the Harris model. 2. Stochastic portfolio theory A market model in Stochastic Portfolio Theory is a collection of N stochastic processes S 1 (t),..., S N (t), where S i (t) is the capitalization of the ith stock at time t 0. A market weight for the ith stock is µ i (t) = S i (t), i = 1,..., N, t 0. S 1 (t) +... + S N (t) We would like to outperform the market, that is, to construct a portfolio that almost surely (as opposed to on average, as in Modern Portfolio Theory by Markowitz or Capital Asset Pricing Model) yields greater return than a share of the whole market. We say that such portfolio provides arbitrage relative to the market. A market model is called diverse if for some constant δ (0, 1), we have: µ i (t) 1 δ for all t 0 and i = 1,..., N. It was shown in [24, 25] by Fernholz, Karatzas, Kardaras that if the market model is diverse, then (under an additional technical condition which is called sufficient intrinsic volatility) there, in fact, exists arbitrage relative to the market. Note that we did not specify the model, for example with a stochastic differential equation for S 1,..., S N. Neither we imposed any assumptions on the existence of an equivalent martingale measure. In fact, such measure cannot exist, otherwise we would not have any arbitrage. Take a system (X 1,..., X N ) of competing Brownian particles, and consider the market model (10) S i (t) = e Xi(t), i = 1,..., N, t 0. As mentioned in Section 1, one can rank market weights from top to bottom: µ (1) (t)... µ (N) (t); then an observed feature of real-world markets is: The log-log plot of ranked market weights (log n, log µ (n) (t)), n = 1,..., N, is linear and independent of time t. This property was captured by the model (10) in [13] by Chatterjee, Pal. The model (10), however, is not diverse: it admits an equivalent martingale measure and is therefore arbitrage-free. Now, suppose two companies can merge into one, and, conversely, a company can split into two companies. We would like to ensure that the resulting model is diverse. We need to impose the following restrictions: If the market weight of a company reaches 1 δ, then it splits into two parts of random size. If a merger results in a company with market weight exceeding 1 δ, such merger is suppressed. Such market model is diverse: by construction, no market weight can exceed 1 δ. One can view this as a consequence of antitrust legislation. In the paper [28], Fouque and Strong considered a model with splits and mergers: When a company splits into two parts, then at the same moment two companies with the smallest capitalizations merge. This keeps the overall number of stocks constant. A surprising result is that in this model, diversity does not lead to arbitrage. This is in stark constrast with models with a constant number of stocks. However, the rule governing mergers seemed quite restrictive to us. We wanted to relax assumptions on mergers to allow the number of stocks to change over time. In our paper [38] with Karatzas, we considered a market model of the form (10) based on competing Brownian particles, but with splits and mergers. In this model, the next merger occurs at an exponentially distributed time after the last split or merger. The quantity of stocks is not constant, but the model is diverse. Under some technical conditions, we proved existence and uniqueness of this model on the infinite time horizon t [0, ). By constructing an equivalent martingale measure, we showed that in this model, diversity does not lead to arbitrage. This is an arbitrage-free model. Possible generalizations include a company being split into more than two parts, or a split becoming progressively more likely when the market weight approaches 1 (rather than at the threshold 1 δ). 3. Systemic risk in financial systems Modeling financial and banking systems is a focus of much recent research. Of particular interest is the concept of systemic risk: the probability of a collapse (default) of a significant part of the system. Without attempting to give an exhaustive survey, let us mention the handbook [27] by Fouque, Langsam; [20] by Eisenberg, Noe on modeling contagion in a non-random system using a clearing mechanism; using risk measures to quantify systemic risk in [8] by Biagini, Fouque, Frittelli, Meyer-Brandis and [21, 22] by Feinstein; using random graphs in [1] by Amini, Cont, Minca, and in [18] for modeling default contagion; constructing interacting particle system models for default contagion in [42] by Nadtochiy, Shkolnikov; Bayesian approach in [14] of Chong, Kluppelberg; a system of

Research Statement Andrey Sarantsev 5 stochastic differential equations with Bessel-type diffusion coefficients in [26] by Fouque, Ichiba; mean-field games with optimal stopping in [11] by Carmona, Delarue, Lacker, continuing the research of the classic paper [19] by Diamong, Dybvig. We model the financial system as a collection of N continuous-time stochastic processes X i := (X i (t), t 0), i = 1,..., N, with X i (t) the net worth of the ith financial player (we call them simply banks) at time t 0. We let Y i (t) := log X i (t), and if Y i (t) D for some constant D > 0, we say that the ith bank is in default. The total number of defaulted banks on the time interval [0, T ] is then the following random variable: N ( ) D := 1 inf Y i(t) D. 0 t T i=1 A systemic event is defined as {D M}, with M a fixed large number (for example, M = N/3). Systemic risk is the probability of this systemic event. In joint paper [41] with Maheshwari, we build our work upon the model studied in the paper [12] by Carmona, Fouque, Sun, which studies the following model: (11) dy i (t) = dw i (t) + a ( Y (t) Y i (t) ) dt, i = 1,..., N, with Y (t) = 1 N Y i (t), N where W 1,..., W N, are correlated Brownian motions with certain drift and diffusion coefficients. In [12], these Ornstein-Uhlenbeck-type drifts are generated by the decisions of banks to borrow money from one another. Their decisions are made by minimizing a certain cost functional, which measures, roughly speaking, the preference of a bank to borrow from other banks, as opposed to borrowing from the central bank. This model was further studied in the book [27, Chapter 16, Chapter 17], and in [29] by Garnier, Papanicolaou, Yang. The novelty of our approach is as follows: In our model, in addition to interbank flows as in (11), each private bank (continuously) borrows money from the outside economy, pays back interest, and invests in risky assets. The (common) interest rate r is controlled by the central bank. Each private bank wants to maximize its terminal logarithmic wealth: (12) E log X i (T ) = EY i (T ) sup, i = 1,..., N. Given the choice of the interest rate r, the problem becomes a stochastic N-player game for private banks. We can solve the stochastic optimal control problem explicitly for each private bank. This is due to the special choice of logarithmic utility function for private banks in (12). For other choices of utility function, it is probably impossible to solve the stochastic game explicitly. Then one could try to use mean-field limits, as in [40] by Lacker. The results are as follows. As long as most banks stay above the threshold, their interaction through Ornstein- Uhlenbeck drift terms keeps each of them from default. But as long as many banks are in default, the other banks are pushed towards default too. An increase in correlation between driving Brownian motions W 1,..., W N from (11) increases the probability of large defaults, but reduces the small default probabilities, similar to flocking behaviour. An increase in the interest rate r leads to a decrease in the number of defaults (of any size). References [1] Hamed Amini, Rama Cont, Andreaa Minca (2016). Resilience to Contagion in Financial Networks. Math. Fin. 26 (2), 329-365. [2] Louis-Pierre Arguin, Michael Aizenman (2009). On the Structure of Quasi-Stationary Competing Particle Systems. 37 (3), 1080-1113. [3] Richard Arratia (1983). The Motion of a Tagged Particle in the Simple Symmetric Exclusion system on Z. Ann. Probab. 11 (2), 362-373. [4] Adrian D. Banner, E. Robert Fernholz, Ioannis Karatzas (2005). Atlas Models of Equity Markets. Ann. Appl. Probab. 15 (4), 2996-2330. [5] Adrian D. Banner, E. Robert Fernholz, Tomoyuki Ichiba, Ioannis Karatzas, Vassilios Papathanakos (2011). Hybrid Atlas Models. Ann. Appl. Probab. 21 (2), 609-644. [6] Yuliy Baryshnikov (2001). GUE and Queues. Probab. Th. Rel. Fields 119 (2), 256-274. [7] Richard F. Bass (1995). Probabilistic Techniques in Analysis. Springer. [8] Francesca Biagini, Jean-Pierre Fouque, Marco Frittelli, Thilo Meyer-Brandis (2015). A Unified Approach to Systemic Risk Measures via Acceptance Sets. Available at arxiv:1503.06354. [9] Cameron Bruggeman, Andrey Sarantsev (2018). Multiple Collisions in Systems of Competing Brownian Particles. Bernoulli 24 (1), 156-201. [10] Manuel Cabezas, Amir Dembo, Andrey Sarantsev, Vladas Sidoravicius (2017). Brownian Particles with Rank-Dependent Drifts: Out-of-Equilibrium Behavior. Available at arxiv:1708.01918. [11] Rene Carmona, Francois Delarue, Daniel Lacker (2017). Mean-Field Games of Timing and Models for Bank Runs. Available at arxiv:1606.03709. [12] Rene Carmona, Jean-Pierre Fouque, Li-Hsien Sun (2013). Mean-Field Games and Systemic Risk. Available at arxiv:1308.2172. [13] Sourav Chatterjee, Soumik Pal (2010). A Phase Transition Behavior for Brownian Motions Interacting Through Their Ranks. Probab. Th. Rel. Fields 147 (1), 123-159. i=1

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